17. The Colosseum in Rome was the largest amphitheater built in the Roman Empire. The interior of the Colosseum is a perfect circle. The roof of this enormous amphitheater was about 600 feet in diameter. If the center of the Colosseum is assumed to be at origin, then write an equation for the circle that represents the roof of the Colosseum.

Answer:

[tex]\displaystyle y' = -4e^{-4x}[/tex]

General Formulas and Concepts:

Calculus

Differentiation

Derivative Property [Multiplied Constant]:                                                          [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]

Basic Power Rule:

Derivative Rule [Chain Rule]:                                                                                 [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle y = e^{-4x}[/tex]

Step 2: Differentiate

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation

The derivative of y = e^-4x is -4e^-4x.

To find the derivative of y = e-4x, we can use the power rule for derivatives. The power rule states that if we have a function of the form y = axb, then the derivative is given by dy/dx = abxb-1. Applying this rule to the given function, we have dy/dx = -4e-4x. Therefore, the derivative of y = e-4x is -4e-4x.

Answer:

Step-by-step explanation:

[tex]y=-\dfrac{3}{4}x+3\ and\ y=-12\\\\\text{Substitute the value of y to the first equation:}\\\\-12=-\dfrac{3}{4}x+3\qquad\text{multiply both sides by 4}\\\\-48=-3x+12\qquad\text{subtract 12 from both sides}\\\\-60=-3x\qquad\text{divide both sides  by (-3)}\\\\20=x\to x=20[/tex]

Answer:

x = 20, y = -12

Step-by-step explanation:

y= -3/4x+3 and y=-12

x = 20, y = -12

Answer:

  64 : 27

Step-by-step explanation:

When using scale factors

Length = scale factor

Area = scale factor squared

Volume = scale factor cubed

The ratio is 4:3

The ratio of the volume is 4^3 : 3^3

                                            64 : 27

Answer:

Part 1) The quadratic equation is [tex]x^{2}-5x-50=0[/tex]

Part 2) The length of rectangle is 10 in and the width is 5 in

Step-by-step explanation:

Part 1)

Find the quadratic equation

Let

x -----> the length of rectangle

y ----> the width of rectangle

we know that

The area of rectangle is equal to

[tex]A=xy[/tex]

[tex]A=50\ in^{2}[/tex]

so

[tex]50=xy[/tex] -----> equation A

[tex]x=y+5[/tex]

[tex]y=x-5[/tex] -----> equation B

substitute equation B in equation A

[tex]50=x(x-5)\\50=x^{2} -5x\\ x^{2}-5x-50=0[/tex]

Part 2) Find the length of the rectangle

[tex]x^{2}-5x-50=0[/tex]

Solve the quadratic equation by graphing

The solution is [tex]x=10\ in[/tex]

see the attached figure

Find the value of y

[tex]y=10-5=5\ in[/tex]

therefore

The length of rectangle is 10 in and the width is 5 in

Answer:

Quadratic equation: [tex]x^{2} +5x-50=0[/tex]

Length of the rectangle: 10 inches.

Step-by-step explanation:

In order to solve this you just have to factorize the equation to solve the different values for X:

[tex]x^{2} +5x-50=0\\(x+10)(x-5)=0[/tex]

So the only possible answer for the problem would be 5, so if the width is equal to X and the length is x+5 then the length of the rectangle would be 5+5 and that would be 10.

Answer:

34.54

Step-by-step explanation:

C= 2pir

11/2= 5.5

5.5* pi=17.27

17.27*2=34.54

Answer:

The circumference of the circle is [tex]\pi * 11[/tex] inches, which is about [tex]34.54[/tex] inches.

Step-by-step explanation:

The circumference of a circle when you only have the diameter ([tex]11[/tex] inches in this case) can be calculated with the formula [tex]C=\pi d[/tex] where [tex]C[/tex] represents the circumference and [tex]d[/tex] represents the diameter.

Plug in the value for the diameter, which is [tex]11[/tex] inches, to get [tex]C=\pi * 11[/tex].

[tex]\pi * 11[/tex] inches is the final exact answer, but you can estimate the answer by replacing [tex]\pi[/tex] with [tex]3.14[/tex] to get [tex]C=3.14 * 11[/tex].

This can be simplified to get [tex]C=34.54[/tex], but this is only an estimate.

Answer:

Constant of variation (k) = -18

Step-by-step explanation:

We are given that y varies inversely as x and we are to find the constant of variation (lets assume its [tex] k [/tex] if [tex] y = 2 [/tex] and [tex] x = - 9 [/tex].

[tex] y [/tex] ∝ [tex] \frac { 1 } { x } [/tex]

Changing this inverse proportionality to equality to get:

[tex] y = \frac { k } { x } [/tex]

Substituting the given values:

[tex] 2 = \frac { k } { -9 } [/tex]

[tex]k=-9 \times 2[/tex]

k = -18

Answer:

The function rule is y = (2/3)x - 5.

Step-by-step explanation:

Let's assume for now that the function is a linear one.  Then y = mx + b.

As we move from (0, -5) to (3, -3), x increases by 3 and y increases by 2.

Thus, the slope of this line is m = rise / run = 2/3.

Then our y = mx + b becomes  y = (2/3)x + b, or -5 = (2/3)(0) + b.

Thus, b = -5, and the equation is y = (2/3)x - 5.

Now let's check this result.  When x = 6, does the equation predict y = -1?

y = (2/3)(6) - 5 = 4 - 5 = -1.  YES

The function rule is y = (2/3)x - 5.

Answer: Option  C

(C ) 96

Step-by-step explanation:

The correct answer would be C, which is 96.

                         You're Welcome!

Answer:

p = 3

Step-by-step explanation:

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :  

                    p-(3000*(1/1000))=0  

Step by step solution :

Step  1  :

             1  

Simplify   ————

           1000

Equation at the end of step  1  :

                1  

 p -  (3000 • ————)  = 0  

              1000

Step  2  :

Equation at the end of step  2  :

 p - 3  = 0  

Step  3  :

Solving a Single Variable Equation :

3.1      Solve  :    p-3 = 0  

Add  3  to both sides of the equation :  

                     p = 3

Please mark brainliest and have a great day!

Answer:

y=50

Step-by-step explanation:

Since the y value doesn't change and it is a line

y=50

Answer:

[tex]y= 0x + 50[/tex]

Step-by-step explanation:

Since, the slope intercept form of a line is,

y = mx + c,

Where, m is the slope of the line.

Also, the equation of a line passes through [tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex] is,

[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]

Thus, the equation of the line passes through (-25, 50) and (25, 50) is,

[tex]y-50= \frac{50-50}{25+25}(x+25)[/tex]

[tex]y-50=0[/tex]

[tex]\implies y = 0x + 50[/tex]

Which is the required equation.

Answer:

Step-by-step explanation:

We have a square in the base and four triangles on the lateral surface.

The formula of an area of a square:

[tex]A_{\square}=s^2[/tex]

s - side

We have s = 9mm. Susbtitute:

[tex]A_{\square}=9^2=81\ mm^2[/tex]

The formula of an area of a triangle:

[tex]A_{\triangle}=\dfrac{bh}{2}[/tex]

b - base

h - height

We have b = 9 mm and h = 12 mm. Substitute:

[tex]A_{\triangle}=\dfrac{(9)(12)}{2}=54\ mm^2[/tex]

The Surface Area:

[tex]S.A.=A_{\square}+4A_{\triangle}[/tex]

Substitute:

[tex]S.A.=81+4(54)=297\ mm^2[/tex]

Answer:

278

Step-by-step explanation:

Answer:

He shrunk .6 cm each year

Step-by-step explanation:

To find the decrease per year, we take the total decrease and divide by the number of years

2.4 cm/ 4 years

.6 cm/ year

He shrunk .6 cm each year

Answer:

[tex]A=24\ un^2.[/tex]

Step-by-step explanation:

Plot points A(0,0), B(6,0), C(2,4) and D(8,4) on the coordinate plane (see attached diagram). The segment CE is the height of the parallelogram ABDC.

The area of the parallelogram is

[tex]A=\text{Base}\cdot \text{Height}[/tex]

Base= AB

Height =CE

So,

[tex]AB=\sqrt{(6-0)^2+(0-0)^2}=\sqrt{36+0}=\sqrt{36}=6\\ \\CE=\sqrt{(2-2)^2+(4-0)^2}=\sqrt{0+16}=\sqrt{16}=4[/tex]

Hence, the area of the parallelogram is

I'll tell you how to do it for any polygon in the cartesian plane with the vertices listed in order.

First we have to list the vertices in order so each pair is a side:

(0,0) (6,0) (8,4) (2,4)

Now for each side (a,b)(c,d) we calculate the cross product ad-bc

(0,0)(6,0)   0(0)-0(6)=0

(6,0)(8,4)   6(4)-0(8)=24

(8,4)(2,4)   8(4)-4(2) = 24

(2,4)(0,0)   2(0)-4(0)=0

We add up the cross products, and take half the absolute value of the sum for the area:

Area = (1/2) | 0 + 24 + 24 + 0 | = 24

Answer: 24

Answer:

f(x) +g(x)  = -1.3 - 1.75x

Step-by-step explanation:

f(x) +g(x) is simply obtained by adding the two given functions, f(x) and g(x). We are given that;

F(x)= 3.7-2x and g(x)= 0.25x-5

f(x) +g(x)  = 3.7-2x + (0.25x-5)

f(x) +g(x)  = 3.7 -5 + 0.25x - 2x

f(x) +g(x)  = -1.3 - 1.75x

For this case we have the following functions:

[tex]f (x) = 3.7-2x\\g (x) = 0.25x-5[/tex]

We must find[tex]f (x) + g (x):[/tex]

We have to:

[tex]f (x) + g (x) = 3.7-2x + (0.25x-5)\\f (x) + g (x) = 3.7-2x + 0.25x-5[/tex]

We add similar terms:

[tex]f (x) + g (x) = - 1.75x-1.3[/tex]

Thus, the result is:[tex]-1.75x-1.3[/tex]

Answer:

[tex]-1.75x-1.3[/tex]

Answer:

cost of system is $175 and cost of the games is $525

Step-by-step explanation:

Let us take the cost of the system to be X.The games cost 3 times as much as the system and are therefore given 3X. The total cost of the system and the games is $700.Therefore,we form the equation 3X+X=$700.Meaning that 4X=$700 and X is equal to $175.The cost of the system is X therefore it is $175 and the cost of the games is 3X and is therefore $525.

The cost of the video game system is $175, while the cost of the games is $525,.

The student is asking to find the cost of a video game system and the games sold with it, given that the total cost is $700 and that the cost of the games is three times as much as the cost of the system.

Let's define the cost of the system as 'x'.

The cost of the games would then be '3x'.

The total cost is given as $700, so we can write the equation as:

x + 3x = 700

Combining like terms, we get:

4x = 700

x = 700 / 4

x = $175

So, the cost of the system is $175,

and the cost of the games is three times that, which is:

3 × $175 = $525

Answer:

Step-by-step explanation:

I don't think it really does grow by a constant rate. As he puts money in, the amount he puts in becomes less significant to the total.

Suppose he starts at 100 dollars.

After week one, he puts in 100 + 15 = 115 dollars.

The 15 dollars represents an increase of  15/100

After the second week, he puts in another 15 dollars. He has 115 in there already.

(15/115) * 100% = 13.04%

After the third week, he puts in another 15 dollars. (15/130 ) * 100% = 11.53

And so one

Answer:

amplitude = 3

Step-by-step explanation:

given a cosine function in standard form, that is

y = acosx, then

amplitude = | a |

Hence for f(x) = 3cosx, amplitude = | 3 | = 3

Final answer:

The amplitude of the function f(x) = 3cosx is 3, as the amplitude is the coefficient in front of the cosine function.

Explanation:

The amplitude of a trigonometric function like f(x) = 3cosx can be found by looking at the coefficient in front of the cosine function. In this case, the function f(x) = 3cosx has an amplitude of 3. This is because the amplitude of cosx or sinx is the absolute value of the coefficient multiplied by the cosine or sine function, which dictates how much the wave's peak or valley deviates from the center line of the graph. For example, if the function was f(x) = A cosx, the amplitude would be the absolute value of A. Since there is no negative sign or other function manipulating the 3 in f(x) = 3cosx, the amplitude is simply 3.

Answer:

11.6%

Step-by-step explanation:

On the first selection, there are 20 red pencils out of 42 total pencils.

On the second selection, there are 10 green pencils out of 41 pencils left over.

So the probability is:

(20/42) (10/41)

100/861

≈11.6%

Answer:

hyperbola

Step-by-step explanation:

hyperbola

Ax^2+By^2+Cx+Dy+E=0

A=B you probably have a circle

A and B have same sign but A isn't B you probably have an ellipse

A and B are opposite in sign you probably have an hyperbola

If either A or B=0 (but not both) then you have a parabola

ANSWER

Hyperbola

EXPLANATION

The given conic has equation:

[tex] {x}^{2} - {y}^{2} = 16[/tex]

We divide through by 16.

[tex] \frac{ {x}^{2} }{16} - \frac{ {y}^{2} }{16} = \frac{16}{16} [/tex]

We simplify the right hand side to get

[tex] \frac{ {x}^{2} }{16} - \frac{ {y}^{2} }{16} = 1[/tex]

Or

[tex] \frac{ {x}^{2} }{ {4}^{2} } - \frac{ {y}^{2} }{ {4}^{2} } = 1[/tex]

This is a hyperbola, that has its vertex at the origin because the quadratic terms have different signs. One is positive and the other is negative.

Answer:

[tex]\large\boxed{a)\ 21,\ 24,\ 29}\\\boxed{b)\ \text{five terms}}[/tex]

Step-by-step explanation:

[tex]a_n=n^2+20\\\\a)\\\\\text{Substitute}\ n=1,\ n=2\ \text{and}\ n=3\ \text{to the }\ a_n:\\\\a_1=1^2+20=1+20=21\\\\a_2=2^2+20=4+20=24\\\\a_3=3^2+20=9+20=29[/tex]

[tex]b)\\\\\text{You have to solve the inequality:}\\\\n^2+20<50\qquad\text{subtract 20 from both sides}\\\\n^2<30\to n<\sqrt{30}\to n\leq5\\\\a_1,\ a_2,\ a_3,\ a_4\ \text{and}\ a_5\ \text{are less than 50.}[/tex]

a) The first three terms of the sequence are 21, 24 and 29.

b) The no. of terms less than 50 are 6.

An arithmetic sequence is a set of numbers in which every no. next to the previous number has the same common difference

d = aₙ - aₙ₋₁ = aₙ₋₁ - aₙ ₋₂.

In a geometric sequence numbers are written in the same constant ratio(r).

It means every next number is a multiple of a common constant and the previous number.

r = aₙ/aₙ₋₁ = aₙ-₁/aₙ₋₂.

Given, The nth term of a sequence is n² + 20.

The first three terms we'll get by replacing 1, 2, and 3 in place of n.

The first term is (1)² + 20 = 21.

The second term is (2)² + 20 = 24.

The third term is (3)² + 20 = 29.

No. of terms that are less than 50 are

n² + 20 < 50.

n² < 30.

n < ± √30.

n < 6 approx.

learn more about sequences here :

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Answer:

5 1/8 - 2 7/8 =

3 -6/8 =

2 2/8 =

2 1/4 =

2.25

Step-by-step explanation:

Answer:

D. :)

Step-by-step explanation:

Answer:

The length of AB is [tex]\sqrt{74}\ units[/tex]

Step-by-step explanation:

we know that

The formula to calculate the distance between two points is equal to

[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]

[tex]A(2,-6)\\B(7,1)[/tex]  

substitute

[tex]d=\sqrt{(1+6)^{2}+(7-2)^{2}}[/tex]

[tex]d=\sqrt{(7)^{2}+(5)^{2}}[/tex]

[tex]d=\sqrt{49+25}[/tex]

[tex]AB=\sqrt{74}\ units[/tex]

Answer:

Up to 39 shortbread cookies.

Step-by-step explanation:

Start by considering: how many cookies can Ramon make if

Assume that flour runs out before the other two ingredients. How many cookies can Ramon make?

It takes 1 1/2 = 3/2 cups of flour to bake fifteen cookies. 5 cups of flour is available. How many batches of fifteen cookies will that 5 cups of flour make?

[tex]\displaystyle \frac{5}{3/2}\right = \frac{10}{3}[/tex].

That's

[tex]\displaystyle 5\times \frac{10}{3} = 50\;\text{cookies}[/tex].

Similarly, assume that sugar runs out before the other two ingredients. How many cookies can Ramon make?

It takes one cup of sugar to bake fifteen cookies. 4 cups of sugar is available. That 4 cups of sugar will make up to four batches of fifteen cookies. That's 60 cookies.

Assume that butter runs out before the other two ingredient. How many cookies can Ramon make?

It takes 1 1/4 = 5/4 cups of butter to bake fifteen cookies. 13/4 cups of butter is available. That 13/4 cups of butter will make up to 13/5 batches of fifteen cookies. That's

[tex]\displaystyle 15 \times \frac{13}{5} = 39\;\text{cookies}[/tex].

These three numbers differ. How many cookies will these materials actually make? The ingredient that will make the smallest number of cookies will run out before other ingredients. In this case, butter runs out first. These materials will make up to 39 shortbread cookies.

First you must distribute the 2 to the numbers inside the parentheses, which would be 1 and -x...

(2 * 1) + (2 * -x) > 2x

2 + (-2x) > 2x

2 - 2x >2x

Add 2x to both sides (what you do on one side you must do to the other). Since 2x is being subtracted, addition (the opposite of subtraction) will cancel it out (make it zero) from the left side and bring it over to the right side.

2 - 2x + 2x > 2x + 2x

2 + 0 > 4x

2 > 4x

Next divide 4 to both sides to finish isolating x. Since 4 is being multiplied by x, division (the opposite of multiplication) will cancel 4 out (in this case it will make 4 one) from the right side and bring it over to the left side.

2/4 > 4x / 4

1/2 > x

or

x < 1/2

Hope this helped!

~Just a girl in love with Shawn Mendes

Answer:

x<1/2

Step-by-step explanation:

Expand.

 ↓

2(1-x)=2-2x

2-2x>2x

First, subtract by 2 from both sides of equation.

2-2x-2>2x-2

Simplify.

-2x>2x-2

Then subtract by 2x from both sides of equation.

-2x-2x>2x-2-2x

Simplify.

-4x>-2

Multiply by -1 from both sides of equation.

(-4)(-1)<(-2)(-1)

Simplify.

4x<2

Divide by 4 from both sides of equation.

4x/4<2/4

Simplify, to find the answer.

2/4=1/2

X<1/2 is the correct answer.

Answer:

The answer is 337.5.

Step-by-step explanation:

Srry Im late. :(

Answer: C. 337.5

Hope this helps!

Answer:

[tex]m + \frac{1}{3} [/tex]

Step-by-step explanation:

[tex]1. \: 1 - \frac{2}{3}+m \\ 2. \: m + (1 - \frac{2}{3}) \\ m + \frac{1}{3} [/tex]

Answer:

29/6

Step-by-step explanation:

1+ -2/3 - (-m)

1 - 2/3 + m

1/3 + m

Let m = 9/2

1/3 +9/2

Get a common denominator of 6

1/3 *2/2 = 2/6

9/2 *3/3 = 27/6

2/6 + 27/6

29/6

Answer:

[tex]4x(2x + 3)[/tex]

Step-by-step explanation:

We have the expression

[tex]8x^2 + 12x[/tex]

and we must factor it

Note that the expression has no independent term

Then we can factor the expression by taking the variable 4x as a common factor

[tex]8x^2 + 12x[/tex]

[tex]4x(2x + 3)[/tex]

Finally the factored form of [tex]8x^2 + 12x[/tex] is [tex]4x(2x + 3)[/tex]

Answer: 4x(2x+3).

Step-by-step explanation: To factor a number means to break it up into numbers that can be multiplied together to get the original number. In the given problem, we can factorize the expression by taking out a common factor, in this case 4x:

[tex]8x^{2} +12x=[/tex]

[tex]4x(2x+3)[/tex]

as we can see, if we multiply 4x*(2x+3) we obtain the original expression.