What is the minimum number of binary bits needed to represent each of the following unsigned decimal integers? a. 65

Answer:

See the proof below.

Step-by-step explanation:

Assuming this complete question: "For each given p, let Z have a binomial distribution with parameters p and N. Suppose that N is itself binomially distributed with parameters q and M. Formulate Z as a random sum and show that Z has a binomial distribution with parameters pq and M."

Solution to the problem

For this case we can assume that we have N independent variables [tex]X_i[/tex] with the following distribution:

[tex] X_i Bin (1,p) = Be(p)[/tex] bernoulli on this case with probability of success p, and all the N variables are independent distributed. We can define the random variable Z like this:

[tex] Z = \sum_{i=1}^N X_i[/tex]

From the info given we know that [tex] N \sim Bin (M,q) [/tex]

We need to proof that [tex] Z \sim Bin (M, pq)[/tex] by the definition of binomial random variable then we need to show that:

[tex] E(Z) = Mpq[/tex]

[tex] Var (Z) = Mpq(1-pq)[/tex]

The deduction is based on the definition of independent random variables, we can do this:

[tex] E(Z) = E(N) E(X) = Mq (p)= Mpq[/tex]

And for the variance of Z we can do this:

[tex] Var(Z)_ = E(N) Var(X) + Var (N) [E(X)]^2 [/tex]

[tex] Var(Z) =Mpq [p(1-p)] + Mq(1-q) p^2[/tex]

And if we take common factor [tex]Mpq [/tex] we got:

[tex] Var(Z) =Mpq [(1-p) + (1-q)p]= Mpq[1-p +p-pq]= Mpq[1-pq][/tex]

And as we can see then we can conclude that   [tex] Z \sim Bin (M, pq)[/tex]

To show that X has a binomial distribution with parameters p*n and q*m, we can formulate X as a random sum of indicator random variables Zi = 1 if the i-th trial is a success and 0 otherwise.

Let X be a binomial random variable with parameters p and n, and let Y be a binomial random variable with parameters q and m. To show that X has a binomial distribution with parameters p*n and q*m, we can formulate X as a random sum of indicator random variables Z1, Z2, ..., Zn, where Zi = 1 if the i-th trial is a success and 0 otherwise. Then X can be expressed as X = Z1 + Z2 + ... + Zn. Since each Zi is independent and follows a Bernoulli distribution with parameter p, the sum of n independent Bernoulli random variables is a binomial random variable with parameters p and n. Therefore, X has a binomial distribution with parameters p*n and q*m.

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Answer:there are 16 goats and 22 chickens.

Step-by-step explanation:

Let x represent the number of goats in the field.

Let y represent the number if chicken in the field.

After a quick count, you determine there are 38 heads and 108 feet in the field. A goat has one head. A chicken also has one head. It means that

x + y = 38

A goat has four legs. A chicken has 2 legs. It means that

4x + 2y = 108 - - - - - - - - - - - 1

Substituting x = 38 - y into equation 1, it becomes

4(38 - y) + 2y = 108

152 - 4y + 2y = 108

- 4y + 2y = 108 - 152

- 2y = - 44

y = - 44/ - 2

y = 22

x = 38 - y = 38 - 22

x = 16

a) The surface area, S, as a function of radius r is given by S = 2πrh + 2πr².

b) The value of S (surface area) approaches infinity as r approaches infinity.

a) The surface area of a closed cylindrical can consists of two parts: the lateral surface area and the two circular base areas.

Let's denote the height of the cylinder as 'h'.

The volume V of the cylinder is given by:

V = πr²h.

We want to express the surface area S in terms of the radius r.

The lateral surface area is given by:

Lateral Surface Area = 2πrh.

Each circular base has an area of πr².

So, the total surface area S is:

S = 2πrh + 2πr².

Hence,  the total surface area S is S = 2πrh + 2πr².

b) As r approaches infinity, let's analyze the behavior of the surface area S:

S = 2πrh + 2πr².

As r gets larger and larger, the term 2πr² dominates the expression.

This is because the term 2πrh (the lateral surface area) is proportional to both 'r' and 'h', whereas the term 2πr² (the circular base areas) is solely proportional to 'r²'.

So, as 'r' approaches infinity, the value of 2πr² will also approach infinity, and this will greatly outweigh the influence of the term 2πrh.

Therefore, the value of S will tend toward infinity as 'r' approaches infinity.

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a) S(r) = 2πr^2 + (2V/r)

b) As r→∞, S(r)→∞, the surface area increases without bound, so it approaches infinity.

a) To find the surface area, S, of a closed cylindrical can as a function of its radius r, you can use the formula for the surface area of a closed cylinder:

S = 2πr^2 + 2πrh

Where:

S is the surface area,

π (pi) is a mathematical constant approximately equal to 3.14159,

r is the radius of the cylinder,

and h is the height (or length) of the cylinder.

However, you mentioned that the volume V is fixed, so we can express the height (h) of the cylinder in terms of its radius (r) and fixed volume (V). The volume of a cylinder is given by:

V = πr^2h

Solving for h:

h = V / (πr^2)

Now, substitute this expression for h into the formula for the surface area:

S = 2πr^2 + 2πr(V / (πr^2))

Simplify:

S = 2πr^2 + 2V/r

So, the surface area, S, as a function of the radius r for a closed cylindrical can with a fixed volume V is:

S(r) = 2πr^2 + 2V/r

b) As r approaches infinity, let's analyze what happens to the value of S(r):

S(r) = 2πr^2 + 2V/r

The first term, 2πr^2, is a quadratic term in r. As r becomes very large (approaching infinity), this term dominates the expression, and S(r) grows without bound. In other words, it approaches infinity.

The second term, 2V/r, is inversely proportional to r. As r becomes very large, this term approaches zero. However, the first term dominates the behavior of S(r), so the overall behavior is dominated by the growth of the quadratic term.

So, as r approaches infinity, the value of S(r) approaches infinity (i.e., S(r) tends toward infinity).

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Answer:

a ) 0.1403604645 and 0.1368

b) 0.3464961 and 0.3485

c) 0.802671982 and 0.8018

Step-by-step explanation:

Y~ B (15,0.45)

Y~ N (15*0.45, 15*0.45*0.55) = Y~ N (6.75, 3.7125)

a) P(Y=5) = 15C5 (0.45)^5 * (0.55)^10 = 0.1403604645

   For normal approximation

P(Y = 5 ) = P ( 4.5 < Y < 5.5 )  ......... continuity correction

Hence,

[tex]P ( 4.5 < Y < 5.5 ) = P ( \frac{4.5 - 6.75}{\sqrt{3.7125} } < Z < \frac{5.5 - 6.75}{\sqrt{3.7125} } ) = P ( -1.16775 < Z < -0.64875 )[/tex]

The probability P ( 4.5 < Y < 5.5 ) = 0.1368

b) P(Y>7) = 15C8 (0.45)^ 8 (0.55)^7 + 15C9 (0.45)^9 * (0.55)^6 + 15C10 (0.45)^10 * (0.55)^5 + 15C11 (0.45)^11 * (0.55)^4 + 15C12 (0.45)^12 * (0.55)^3 + 15C13 (0.45)^13 * (0.55)^2 + 15C14 (0.45)^14 * (0.55) + (0.45)^15

= 0.3464961

  For normal approximation

P(Y > 7 ) = P (Y > 7.5 )  ......... continuity correction

Hence,

[tex]P (Y > 7.5) = P (Z > \frac{7.5-6.75}{\sqrt{3.7125} } ) = P (Z > 0.389249)\\[/tex]

The probability P ( Y>7.5 ) = 0.3485

c) P (4 < Y < 10) = 15C5 (0.45)^5 (0.55)^10 + 15C6 (0.45)^ 6 (0.55)^9 + 15C7 (0.45)^7 (0.55)^8 + 15C8 (0.45)^ 8 (0.55)^7 + 15C9 (0.45)^9 * (0.55)^6

= 0.802671982

For normal approximation

P( 4 < Y < 10 ) = P (4.5< Y < 9.5 )  ......... continuity correction

Hence,

[tex]P ( 4.5 < Y < 9.5 ) = P ( \frac{4.5 - 6.75}{\sqrt{3.7125} } < Z < \frac{9.5 - 6.75}{\sqrt{3.7125} } ) = P ( -1.167748416 < Z < 1.427248064 )[/tex]

The probability P (4.5< Y < 9.5 ) = 0.8018

Answer:

a) 68% of the investments had a return of between 10% and 30%.

b) 32% of investments had a return that was either less than 10% or morethan 30%.

Step-by-step explanation:

We can use the Empirical Rule to solve this question:

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean. This also means that 32% of the measures are more than 1 standard deviation from the mean.

95% of the measures are within 2 standard deviation of the mean. This also means that 5% of the measures are more than 2 standard deviations from the mean.

99.7% of the measures are within 3 standard deviations of the mean. This also means that 0.3% of the measures are more than 3 standard deviations from the mean.

In this problem, we have that:

Mean = 20%.

Standard deviation = 10%.

a. What proportion of the investments had a return of between 10% and 30%?

10 is the mean subtracted by 1 one standard deviation

30 is one standard deviation added to the mean.

So 10 and 30 are within 1 standard deviation of the mean. So 68% of the investments had a return of between 10% and 30%.

b. What proportion of investments had a return that was either less than 10% or morethan 30%?

This is the proportion of investments that were farther than one standard deviation of the mean.

By the Empirical Rule, 32% of investments had a return that was either less than 10% or morethan 30%.

The formula used is: [tex]d = \frac{C}{ \pi }[/tex]

The diameter is 17.83 feet

Solution:

Given that,

Circumference of pool = C = 56 feet

To find: diameter (d)

The circumference of circle is given as:

[tex]C = 2 \pi r[/tex]

Where "r" is the radius of circle

We know, that diameter is twice the radius

[tex]d = 2r[/tex]

Thus the formula becomes,

[tex]C = \pi d[/tex]

Rearrange for "d"

[tex]d = \frac{C}{ \pi }[/tex]

Substituting the values we get,

[tex]d = \frac{56}{3.14} = 17.83[/tex]

Thus the diameter is 17.83 feet

Answer:

The perimeter is 48 units

Step-by-step explanation:

The picture of the question in the attached figure

we know that

The perimeter of the octagon is the sum of its length sides

so

[tex]P=AB+BC+CD+DE+EF+FG+GH+HA[/tex]

we have

[tex]BC=10\ units\\CD=6\ units\\EF=4\ units\\GH=8\ units[/tex]

substitute

[tex]P=AB+10+6+DE+4+FG+8+HA[/tex]

Combine like terms

[tex]P=AB+DE+FG+HA+28[/tex]

we know that

[tex]BC=DE+FG+HA[/tex] ---> by segment addition postulate

[tex]BC=10\ units[/tex]

so

[tex]DE+FG+HA=10\ units[/tex]

substitute in the expression of perimeter

[tex]P=AB+(DE+FG+HA)+28[/tex]

[tex]P=AB+10+28\\P=AB+38[/tex]

Since

[tex]DC= 6\ units[/tex]

and

[tex]EF = 4\ units[/tex]

The distance between F and line BC must be

[tex]6-4=2\ units[/tex]

so

[tex]AB = HG + 2 = 10\ units[/tex]

substitute

[tex]P=AB+38\\P=10+38=48\ units[/tex]

Answer:

The volume of remaining sphere is 489.84 cubic inches.

Step-by-step explanation:

We are given the following in the question:

A hole 2 inches in radius is drilled out of a solid sphere of radius 5 inches.

Radius of sphere = 5 inches

Radius of hole = 2 inches

Volume of sphere =

[tex]\dfrac{4}{3}\pi r^3[/tex]

where r is the radius of sphere.

Volume of sphere =

[tex]\displaystyle\frac{4}{3}\pi (5)^3\\\\=\frac{4}{3}\times 3.14\times (5)^3\\\\=523.33\text{ cubic inches}[/tex]

Volume of hole =

[tex]\displaystyle\frac{4}{3}\pi (2)^3\\\\=\frac{4}{3}\times 3.14\times (2)^3\\\\=33.49\text{ cubic inches}[/tex]

Volume of remaining solid =

Volume of sphere - Volume of hole

[tex]=523.33 - 33.49\\=489.84\text{ cubic inches}[/tex]

The volume of remaining sphere is 489.84 cubic inches

Answer:Mean = $248

Median = $252

Quartile 3 = $265

Mode = $265

range = $69

Standard deviation= 20.126

Coefficient of variation = 8.115

Negatively

Coefficient of skewness = -0.596

Quartile 2 = $252

Step-by-step explanation:

The detailed explanation can be found in the attached pictures

Answer:

84.9

Step-by-step explanation:

The weighted mean is given by the sum of the products of each grade by its respective weight. If the first four grades correspond to 15% of the final grade each, and the final exam is equivalent to 40% of the final grade, Michael's final grade is:

[tex]G= (73+77+82+86)*0.15+(93*0.4)\\G= 84.9[/tex]

Michael's final weighted mean is 84.9.

Michael's final grade, calculated using the weighted mean formula, taking into consideration the individual weights of each test (15%) and the final exam (40%), is approximately 85.4.

The subject of this problem is weighted mean, which is used quite often in the field of statistics. The formula to calculate the weighted mean is *(w1*x1 + w2*x2 + ... + wn*xn) / (w1 + w2 + ... + wn), where w represents the weights and x represents the values.

In Michael's case, his weights for the test grades are 15% for each test, and 40% for the final exam. Thus, the calculation will be as follows:

(0.15 * 73 + 0.15 * 77 + 0.15 * 82 + 0.15 * 86 + 0.4 * 93) / (0.15 + 0.15 + 0.15 + 0.15 + 0.4)

After calculation, the weighted mean, or the final grade, is approximately 85.4, rounded to one decimal place.

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Answer:

51

Step-by-step explanation:

Substitute the value of a

3(2)³+10(2)²-3(2)-7

3(8)+10(4)-3(2)-7

24+40-6-7

64-13

51

Answer:

C. Total flow

Step-by-step explanation:

The total amount of fluid that has passed a designated point is designated as the total flow.

The flow rate is the rate at which fluid passes through a certain point, not the total amount of fluid.

Laminar and turbulent flow are different types of fluid flow patterns and don't characterize a volume of fluid.

Therefore, the correct answer is C. Total flow.

Answer:

You didn't give the differential equations, but I'll explain how to identify the independent variable, dependent variable, how to know the order, linearity, and nonlinearity of a differential equation.

Step-by-step explanation:

DIFFERENTIAL EQUATION

This is any equation that involves differential coefficients. It is a relationship between an independent variable, x, a dependent variable, y, and one or more derivatives of y with respect to x.

Examples

(1) xd²y/dx² + 7dy/dx = 0

(2) y²dy/dx + 2x = 0

(3) xd³y/dx³ = y½ + 1

(4) 2xy'' - 3y' + 5y = 0

(5) (y''')² + 30xy = 0

Note that the dependent variable is always the numeratior, and the independent, denominator, in a different coefficient. In the case of our examples, y is the dependent variable, and x is the independent.

Example (4) is another way of writing a differential coefficient, y' (read as y-prime) is the same as dy/dx (read as dee-y dee-x). In some cases when the independent variable is time t, it is written as ÿ, which is the same as d²y/dt² (read as dee-two-y dee-t-squared)

ORDER

This is the order of the highest derivative in a differential equation. You need not consider other derivatives, just the highest.

In the examples, the orders are

(1) two

(2) one

(3) three

(4) two

(5) three

LINEAR DIFFERENTIAL EQUATION

This is the kind of differential equation in which the functions of the dependent variable are linear. There are no powers of the dependent variable and/or its derivatives, there are no products of the dependent variable and its derivative, there are no functions of the dependent variable like cos, sin, exp, etc.

NONLINEAR DIFFERENTIAL EQUATION

If any condition for linearity is not met, then it is nonlinear.

(1) Linear

(2) Nonlinear because y is the dependent variable, and y² is nonlinear, and even still, it multiplies a derivative.

(3) Nonlinear because y½ is nonlinear

(4) Linear

(5) Nonlinear because (y''')² in nonlinear.

Understanding this, you can determine the order, linearity or nonlinearity of any differential equation. Cheers!

Equations 1 and 3 are nonlinear due to their respective terms, while Equation 2 is linear. The orders are first for Equations 1 and 2, and second for Equation 3.

Let's analyse each of the given differential equations step by step.

Equation 1: y' = y - x²

Equation 2: xy' = 2y

Equation 3: x'' + 5x = e-x

Thus,

1. [tex]\( y' = y - x^2 \)[/tex]

  (a) Independent variable: x, Dependent variable: y

  (b) First order

  (c) Nonlinear because of the term -x².

2. xy' = 2y

  (a) Independent variable: x, Dependent variable: y

  (b) First order

  (c) Nonlinear because of the product xy.

3. [tex]\( x'' + 5x = e^{-x} \)[/tex]

  (a) Independent variable: t (not explicitly given), Dependent variable: x

  (b) Second order

  (c) Linear because it has no products or powers of x other than x and x''.

Complete question: In problems below

(a) identify the independent variable and the dependent variable of each equation (use 't' for the independent variable if an independent variable is not given explicitly): (b) give the order of each differential equation (enter '1' for first order. '2' for second order and so on: do not include the quotes); and (C) state whether the equation is linear or nonlinear If your answer to (C) is nonlinear, make sure that you can explain why this is true

Equation:

1. y' = y-x²

2. xy' = 2y

3. [tex]\( x'' + 5x = e^{-x} \)[/tex]

Answer:

A. x = -4 + 3t, y = 3 - 4t: - 2 lessthanorequalto t lessthanorequalto 3

Step-by-step explanation:

Given that a line in two dimension passes through (-4,3) and (2,-5) oriented in the direction of increasing x

We can write the line equation in parametric form as

[tex]\frac{x-x_1}{x_2-x_1} =\frac{y-y_1}{y_2-y_1} \\\frac{x+4}{6} =\frac{y-3}{-8} =t\\x=-4+6t\\y = 3-8t[/tex]

The values of t when x=-4 is 0 and when x =2 is 1

So t varies from 0 to 1

If instead of t we give t' say which is t +2

then we have

t he parametric equations as

x =-4+3t and y = 3-4t

For  x=-4, t =2 and for x = 2 , t =3

So option A is right.

Answer:

a) [tex] \frac{ds}{dt}= v(t) = 3t^2 -18t +15[/tex]

b) [tex] v(t=3) = 3(3)^2 -18(3) +15=-12[/tex]

c) [tex] t =1s, t=5s[/tex]

d)  [tex] [0,1) \cup (5,\infty)[/tex]

e) [tex] D = [1 -9 +15] +[(5^3 -9* (5^2)+ 15*5)-(1-9+15)]+ [(6^3 -9(6)^2 +15*6)-(5^3 -9(5)^2 +15*5)] =7+ |32|+7 =46[/tex]

And we take the absolute value on the middle integral because the distance can't be negative.

f) [tex] a(t) = \frac{dv}{dt}= 6t -18[/tex]

g) The particle is speeding up [tex](1,3) \cup (5,\infty)[/tex]

And would be slowing down from [tex][0,1) \cup (3,5)[/tex]

Step-by-step explanation:

For this case we have the following function given:

[tex] f(t) = s = t^3 -9t^2 +15 t[/tex]

Part a: Find the velocity at time t.

For this case we just need to take the derivate of the position function respect to t like this:

[tex] \frac{ds}{dt}= v(t) = 3t^2 -18t +15[/tex]

Part b: What is the velocity after 3 s?

For this case we just need to replace t=3 s into the velocity equation and we got:

[tex] v(t=3) = 3(3)^2 -18(3) +15=-12[/tex]

Part c: When is the particle at rest?

The particle would be at rest when the velocity would be 0 so we need to solve the following equation:

[tex] 3t^2 -18 t +15 =0[/tex]

We can divide both sides of the equation by 3 and we got:

[tex] t^2 -6t +5=0[/tex]

And if we factorize we need to find two numbers that added gives -6 and multiplied 5, so we got:

[tex] (t-5)*(t-1) =0[/tex]

And for this case we got [tex] t =1s, t=5s[/tex]

Part d: When is the particle moving in the positive direction? (Enter your answer in interval notation.)

For this case the particle is moving in the positive direction when the velocity is higher than 0:

[tex] t^2 -6t +5 >0[/tex]

[tex] (t-5) *(t-1)>0[/tex]

So then the intervals positive are [tex] [0,1) \cup (5,\infty)[/tex]

Part e: Find the total distance traveled during the first 6 s.

We can calculate the total distance with the following integral:

[tex] D= \int_{0}^1 3t^2 -18t +15 dt + |\int_{1}^5 3t^2 -18t +15 dt| +\int_{5}^6 3t^2 -18t +15 dt= t^3 -9t^2 +15 t \Big|_0^1 + t^3 -9t^2 +15 t \Big|_1^5 + t^3 -9t^2 +15 t \Big|_5^6[/tex]

And if we replace we got:

[tex] D = [1 -9 +15] +[(5^3 -9* (5^2)+ 15*5)-(1-9+15)]+ [(6^3 -9(6)^2 +15*6)-(5^3 -9(5)^2 +15*5)] =7+ |32|+7 =46[/tex]

And we take the absolute value on the middle integral because the distance can't be negative.

Part f: Find the acceleration at time t.

For this case we ust need to take the derivate of the velocity respect to the time like this:

[tex] a(t) = \frac{dv}{dt}= 6t -18[/tex]

Part g and h

The particle is speeding up [tex](1,3) \cup (5,\infty)[/tex]

And would be slowing down from [tex][0,1) \cup (3,5)[/tex]

b) The velocity after 3 seconds is -12 feet per second. c) The particle is at rest when t = 1 and t = 5. d) The particle is moving in the positive direction between the critical points of the velocity function. e) The total distance traveled during the first 6 seconds can be found by integrating the absolute value of the velocity function. f) The acceleration at time t is given by the derivative of the velocity function. h) The particle is speeding up when its acceleration is positive and slowing down when its acceleration is negative.

b) To find the velocity after 3 seconds, we need to find the derivative of the function f(t). The derivative of f(t) is v(t), the velocity function. So, v(t) = f'(t), which is equal to 3t^2 - 18t + 15. Now, to find v(3), we substitute t = 3 into the velocity function:

v(3) = 3(3)^2 - 18(3) + 15

= 27 - 54 + 15

= -12 feet per second

c) The particle is at rest when its velocity is zero. So, to find when the particle is at rest, we need to find the time when v(t) = 0:

0 = 3t^2 - 18t + 15

Solving this quadratic equation, we find that the particle is at rest when t = 1 and t = 5

d) The particle is moving in the positive direction when its velocity is positive. So, we need to find the time intervals when v(t) > 0. We can do this by finding the critical points of the velocity function and determining the sign of v(t) in between those critical points. By analyzing the sign of v(t), we can determine the intervals when the particle is moving in the positive direction.

(e) To find the total distance traveled during the first 6 seconds, we need to find the definite integral of the absolute value of the velocity function from 0 to 6:

Distance = ∫06 |v(t)| dt

(f) The derivative of the velocity function v(t) gives us the acceleration function a(t), which is the rate of change of velocity. So, a(t) = v'(t), which is equal to 6t - 18.

(h) The particle is speeding up when its acceleration is positive, and slowing down when its acceleration is negative. So, we need to find the intervals when a(t) > 0 and a(t) < 0 to determine when the particle is speeding up and when it is slowing down.

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Answer:

Step-by-step explanation:

Given

height of tank [tex]h=5\ ft[/tex]

Width of tank [tex]w=3\ ft[/tex]

length of tank [tex]L=2\ ft[/tex]

suppose a layer of water at height h of thickness dh from bottom needed to be pump out

So distance moved by this layer to come out of tank is [tex]\Delta h=5-h[/tex]

weight density of water [tex]\rho =50\ pounds/ft[/tex]

Force required to hold this layer up [tex]F_s=2\times 3\times \Delta h\times 50=300\Delta h[/tex]

Work done to remove the water

[tex]W=\int_{0}^{5}300\Delta hdh[/tex]

[tex]W=\int_{0}^{5}300\left ( 5-h\right )dh[/tex]

[tex]W=3750\ Pound-ft[/tex]

The work does it take to pump all of the liquid out of the top of the tank is 3750-pound feet.

Given that

A tank in the shape of a right rectangular prism has a height of 5 feet, width of 3 feet, and length of 2 feet.

It is full of a liquid weighing 50 pounds per cubic foot.

How much work does it take to pump all of the liquid out of the top of the tank?

A tank in the shape of a right rectangular prism has a height of 5 feet, width of 3 feet, and length of 2 feet.

The distance moved by this layer to come out of the tank is;

[tex]\rm \triangle h = 5-h[/tex]

The force required to hold this layer up is;

[tex]\rm Force = \triangle Height \times Width \times length \times liquid \ weighing\\ \\ Force = 2 \times 3 \times (5-h) \times 50\\ \\ Force = 300 (5-h)[/tex]

The work does it take to pump all of the liquid out of the top of the tank is calculated by;

[tex]\rm Work = \int\limits^5_0 {300(5-h)} \, dh\\ \\ Work = 300(5\int\limits^5_0 {} \, dh - \int\limits^5_0 {h} \, dh) \\ \\ Work = 300(5[h]^5_0- [\dfrac{h^2}{2}]^5_0)\\ \\ Work = 300(5(5-0)-\dfrac{5^2}{2}-\dfrac{0^2}{2})\\ \\ Work = 300(25-\dfrac{25}{2})\\ \\ Work = 300\times \dfrac{50-25}{2}\\ \\ Work = 300 \times \dfrac{25}{2}\\ \\ Work = 300 \times (12.5)\\ \\ Work = 3750 \ pounds \ feet[/tex]

Hence, The work does it take to pump all of the liquid out of the top of the tank is 3750-pound feet.

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Answer:

Step-by-step explanation:

-2x + 7x + 5 = 0

-2x +7x = 5

5x = 5

divide both side by 5

5x/5 = 5/5

x = 1

Answer:

1. Linear relation

2. Linear relation

3. Non Linear relation

4. Non Linear relation

Step-by-step explanation:

i) the number of pages a copy machine can print over time, expressed by n = 30m is a linear relation where the value of 'n' is directly proportional to 'm'.

ii) The amount of money an hourly worker makes over time, expressed by a = 15h, is a linear relation where the value of 'a' is directly proportional to 'h'.

iii) The temperature outside at noon each day for a year is a non-linear relation because the temperature will be different every day and not by the same amount.

iv) The height increase of a child over 3 years is also a non-linear relation as the height increase will vary from year to year and not by the same amount.

Answer:

E (Y) = 3

Step-by-step explanation:

If a 4-sided die is being rolled repeatedly; and the odd-numbered rolls (1st 3rd,5th, etc.)

The probability of odd number roll will be, p(T) = [tex]\frac{1}{2}[/tex]

However, on your even-numbered rolls, you are victorious if you get a 3 or 4. Also, the  probability of even number roll, p(U) = [tex]\frac{1}{2}[/tex]

In order to  calculate: E (Y); We can say Y to be the number of times you roll.

We know that;

E (Y) = E ( Y|T ) p(T) + E ( Y|U ) p(U)

Let us calculate E ( Y|T ) and E ( Y|U )

Y|T ≅ geometric = [tex]\frac{1}{4}[/tex]

Y|U ≅ geometric = [tex]\frac{1}{2}[/tex]

also; x ≅ geometric (p)

∴ E (x) = [tex]\frac{1}{p}[/tex]

⇒ [tex]\frac{Y}{T}[/tex] = 4 ; also [tex]\frac{Y}{U}[/tex] = 2

E (Y) = 4 × [tex]\frac{1}{2}[/tex] + 2 ×

        = 2+1

E (Y) = 3

let P(-2,0) & Q(0,2)

slope = 2-0/0+2

= 1

the slope is 1 which will be 45°

Answer: slope = 1

Step-by-step explanation:

The formula for finding slope is given as :

slope = [tex]\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]

[tex]x_{1}[/tex] = -5

[tex]x_{2}[/tex] = 3

[tex]y_{1}[/tex] = -3

[tex]y_{2}[/tex] = 5

substituting the values into the formula , we have :

slope = [tex]\frac{5-(-3)}{3-(-5)}[/tex]

slope = [tex]\frac{5+3}{3+5}[/tex]

slope = [tex]\frac{8}{8}[/tex] = 1

Therefore : the slope of the line is 1

To calculate the resultant force and its direction given two forces at angles, you decompose each force into x and y components, sum these components separately to find the resultant vector, and then use Pythagorean theorem and inverse tangent to find magnitude and direction.

To find the magnitude of the resultant force and the angle it makes with the positive x-axis, given forces at 45 degrees and 30 degrees below the x-axis with magnitudes 28 lb and 12 lb respectively, we break each force into its x and y components. For force a at 45 degrees, the components are 28cos(45) in the x-direction and 28sin(45) in the y-direction. For force b at -30 degrees, the components are 12cos(-30) in the x-direction and 12sin(-30) in the y-direction, since it is below the x-axis.

To find the resultant force (Fres), we add the x-components and y-components separately: Fres,x = 28cos(45) + 12cos(-30) and Fres,y = 28sin(45) + 12sin(-30). The total magnitude is calculated using the Pythagorean theorem: |Fres| = sqrt(Fres,x² + Fres,y²). The angle θ with the positive x-axis is found using the inverse tangent of the y-component over the x-component (θ = atan(Fres,y/Fres,x)).

Answer:

[tex](x-4)^2+(y-7)^2=4[/tex]

z=6

Step-by-step explanation:

The equation of a circle is satisfied by every point on it. The equation of a circle of radius r and point (h,k,l) parallel to the xy-plane is:

[tex](x-h)^2+(y-k)^2=r^2[/tex]

we can just substitute values into the equation:

[tex](x-4)^2+(y-7)^2=r^2[/tex] where z=6

The radius is 2. Therefore:

[tex](x-4)^2+(y-7)^2=4[/tex]

Final answer:

The equations describing the circle with a radius of 2 that is centered at (4, 7, 6) and parallel to the xy-plane are (x - 4)² + (y - 7)² = 4 and z = 6.

Explanation:

The equation that describes a circle of radius 2 centered at (4, 7, 6) that is parallel to the xy-plane can be found using the general equation of a circle in three dimensions, considering that the circle lies in a plane parallel to the xy-plane, hence the z-coordinate will remain constant. The standard equation for a circle in two dimensions is (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is its radius.

For a circle of radius 2, the equation becomes (x - 4)² + (y - 7)² = 2². Since the circle is parallel to the xy-plane and the center has a z-coordinate of 6, the equation does not change with respect to z, remaining constant at z = 6. Therefore, the complete set of equations describing the circle are (x - 4)² + (y - 7)² = 4 and z = 6.

Answer:

c). Two tailed test

Step-by-step explanation:

The given hypothesis are

Null hypothesis: H0:μ= 1.7

Alternative hypothesis: H1:μ≠ 1.7

The alternative hypothesis demonstrates that  mean number of children are not 1.7 in 2000. This means that mean number of children can be greater than 1.7 or mean number of children can be less than 1.7. Thus, the given alternative hypothesis indicates the two tailed test.

Final answer:

To find the probability of selecting one ace and one of either a ten, jack, queen, or king from a deck with replacement, calculate the probability for each draw and account for both possible orders of drawing these cards the final probability is 2 * ((4/52) * (16/52)) to account for both orders.

Explanation:

To calculate the probability that one card is an ace and the other is either a ten, jack, queen, or king when two cards are selected randomly from an ordinary deck, we approach this problem considering the combinations and probabilities of each event happening.

First, let's establish the total possibilities. Since we are replacing the card back into the deck after the first draw, the number of possibilities for each draw remains 52. So, the total number of ways to draw any two cards (with replacement) is 52 * 52.

The probability of drawing an ace (P(Ace)) on the first draw is 4/52 because there are 4 aces in a deck of 52 cards.

The probability of drawing a ten, jack, queen, or king (P(10/J/Q/K)) on the second draw is 16/52, as there are 4 of each of these cards in the deck, totaling 16 cards.

To find the probability of both events happening, you multiply the probabilities of each event: P(Ace) * P(10/J/Q/K). This gives you (4/52) * (16/52).

However, the order in which the ace and the 10/J/Q/K are drawn matters, as the ace could also be drawn second. This means we need to calculate the probability again with the ace being drawn second and the 10/J/Q/K card drawn first, which is the same calculation. Therefore, the final probability is 2 * ((4/52) * (16/52)) to account for both orders.

To calculate the probability of drawing an ace followed by a ten, jack, queen, or king, we find the probabilities of each event separately and multiply them since the events are independent, assuming the card is replaced after the draw.

The probability of selecting one ace and another card which is a ten, jack, queen, or king involves understanding the composition of a standard deck and calculating the odds of each draw.

A standard deck has 52 cards with 4 aces and 16 face cards (including the tens). The chance of drawing an ace (event A) is therefore 4/52. For the second draw (event B), with the card being replaced after the first draw, the odds of drawing either a ten, jack, queen, or king from any suit remain identical, since we're back to having a full deck.

To find the overall probability, we calculate the probability of both events occurring, which involves multiplying the probabilities of each individual event, assuming they are independent. This is represented by: P(A and B) = P(A) * P(B).

For the specific example of drawing a four (event A) then a five (event B) when replacing the card back into the deck after each draw, the probability of event A is 1/13 since there are four fours in the deck and the probability of event B is also 1/13 for the same reason. Therefore, the combined probability is (1/13)*(1/13), which simplifies to 1/169.

Answer:

Unimodal Distribution

Mode = 6        

Step-by-step explanation:

We are given the data of hours that 9 students spend on the computer on a typical day:

1, 6, 7, 6, 8, 11, 6, 12, 15

Sorted data: 1, 6, 6, 6, 7, 8, 11, 12, 15

Mode is the most frequent observation in the given data.

Here, 6 repeats itself most frequently that is 3 times. No other observation repeats itself three times.

Thus, the mode of the data is 6.

Since, there is a single mode for the given data, the distribution of students is unimodal.

Unimodal Distribution

Mode = 6        

The sorted data is 1, 6, 6, 6, 7, 8, 11, 12, 15

Since 6 repeats 3 times so the mode should be 6

Also, there should be the single mode so the type of distribution is unimodal  

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Answer:

Step-by-step explanation:

The given simultaneous equations are expressed as

y = 2x + 7 - - - - - - - - - -1

y = -2x - 5 - - - - - - - - - - 2

We would apply the method of substitution.

The first step is to equate equation 1 to equation 2. It becomes

2x + 7 = - 2x - 5

Next step is to add 2x to the left hand side and the right hand side of the equation. It becomes

2x + 2x + 7 = - 2x - 2x - 5

4x + 7 = - 5

Next step is to subtract 7 from the left hand side and the right hand side of the equation. It becomes

4x + 7 - 7 = - 5 - 7

4x = - 12

Next step is to divide the left hand side and the right hand side of the equation by 4. It becomes

4x/4 = -12/4

x = - 3

Substituting x = - 3 into equation 1, it becomes

y = 2 × - 3 + 7 = - 6 + 7

y = 1

Answer:

Step-by-step explanation:

Let X be the time for any customer to call at any time to make reservation in Nite Time Inn.

Given that X is exponential with mean = 4 minutes

We are to find the probability

(a) 3 minutes or less.

=[tex]P(X\leq 3)=1-e^{-3/4} =0.5276[/tex]

(b) 4 minutes or less.

[tex]=P(X\leq 4)\\=1-e^{-4/4} =0.6321[/tex]

(c) 5 minutes or less.

[tex]=P(X\leq 5)\\=1-e^{-5/4} =0.7135[/tex]

(d) longer than 5 minutes.

=1-P(X≤5) = 0.2865

Answer:

d = (max distance / max time)t + 0.

Step-by-step explanation:

Earthquake travels 55km of rock in 25 seconds, the relationship between d, the distance traveled in km, and t, the time elapsed in seconds is:

d = (max distance / max time)t + 0

d = (55 km / 25 sec) * t

  = 2.2t

Answer:

2.2km/secs

Step-by-step explanation:

Speed = distance /time

= 55km/25secs

= 2.2km/secs

Answer:

Option B) No. The standard deviation cannot be negative.

Step-by-step explanation:

We are given the following in the question:

[tex]\mu = 2.0\\\sigma = -3.5[/tex]

The above results are not possible.

Option B) No. The standard deviation cannot be negative.

Final answer:

The results from Excel indicating that the standard deviation of a discrete probability distribution is negative are incorrect, as B. standard deviation cannot be negative.

Explanation:

When analyzing a discrete probability distribution in Excel, we expect certain characteristics from the mean (mu) and standard deviation (sigma). The mean represents the expected value or average of the distribution, while the standard deviation measures how much the values in the distribution vary from the mean. In the context of a probability distribution, a standard deviation cannot be negative as it represents the square root of variance, which is a squared value and therefore nonnegative.

Given the Excel results showing mu equals 2.0 and sigma equals negative 3.5, we can immediately identify an error. A negative standard deviation does not make sense in the real world since it is a measure of dispersion. Therefore, the correct answer to whether these results can be correct is: B. No. The standard deviation cannot be negative.