Final answer:
Using the Law of Cosines, the length of the third side of the obtuse triangle is x = sqrt(52) or x = 2sqrt(13).
Explanation:
To find the length of the third side of an obtuse triangle with sides of lengths 4 and 10 and an area of 12, you can use the Law of Cosines. The Law of Cosines states that in a triangle with sides a, b, and c, and angle C opposite side c, the equation c^2 = a^2 + b^2 - 2ab * cos(C) can be used to find the length of side c.
In this case, let's assume that the third side of the triangle has length x. We can set up the equation as x^2 = 4^2 + 10^2 - 2*4*10 * cos(C), where C is the angle opposite side x.
Simplifying the equation, we get x^2 = 116 - 80*cos(C).
Since we know that the area of the triangle is 12, we can use the formula Area = 0.5 * a * b * sin(C) to find the value of sin(C). Plugging in the given values, we get 12 = 0.5 * 4 * 10 * sin(C). Solving for sin(C), we find sin(C) = 0.6.
Using this value of sin(C), we can then find the value of cos(C) by using the trigonometric identity sin^2(C) + cos^2(C) = 1. Substituting sin(C) = 0.6, we get 0.6^2 + cos^2(C) = 1. Solving for cos(C), we find cos(C) = 0.8.
Now, we can substitute the value of cos(C) into the equation x^2 = 116 - 80*cos(C), giving us x^2 = 116 - 80*0.8. Simplifying further, we get x^2 = 52, which means that the length of the third side is either x = sqrt(52) or x = -sqrt(52).
However, since we are dealing with the length of a side, it cannot be negative. Therefore, the length of the third side is x = sqrt(52), which simplifies to x = 2sqrt(13).
When you see a traffic light turn red, you apply the brakes until you come to a stop. Suppose your initial speed was 11.2 m/s, and you come to rest in 34.7 m. How much time does this take? Assume constant deceleration.
...?
A = bh/2
A' = b'h/2 + bh'/2
b' = (A' - bh'/2)(2/h) can someone explain this. its steps to the derivative of bh/2.
Answer:
To find A' they used the rule of multiplication, which is:
the derivative of a product of two terms is the first term times the derivative of the second term plus the second term times the derivative of the first.
To find b' they just isolated b'
Step-by-step explanation:
What type of polynomial is 2x2-6x+4? ...?
f the graph of y=ax+b/x+c has a horizontal asymptote y=2 and a vertical asymptote x=-3 then a+c=?
a)-3
b)-1
c)0
d)1
E)5 ...?
Answer: 5
Step-by-step explanation:
Lim x-> a (ax+b)/x+c = a you know that a=2
Since the limit of x ->-3 equals infinity, you know that x =3.
Thus a+c = 5
distance can be represented by absolute value.true or false.?
Solve for x.
log6(2x + 3) = 3 ...?
The value of x that satisfies the original equation is 106.5. To solve the given logarithmic equation, convert it to its exponential form and simplify. Subtract 3 from both sides, and finally divide by 2 to find that x equals 106.5.
Explanation:To solve for x in the equation log6(2x + 3) = 3, we can convert the logarithmic equation into its exponential form. This means that we interpret the equation as 6 raised to the power of 3 equals 2x + 3:
63 = 2x + 3
The next step is to simplify. 6 to the power of 3 is 216:
216 = 2x + 3
We can now solve for x by first subtracting 3 from both sides:
216 - 3 = 2x
213 = 2x
And then dividing by 2:
x = 213 / 2
x = 106.5
Thus, the value of x that satisfies the original equation is 106.5.
6375/10000 in simplest form
A box contains 13 transistors, 3 of which are defective. if 3 are selected at random, find the probability that
a. All are defective
b. None are defective
a. Probability all defective: [tex]\( \frac{1}{286} \)[/tex]. b. Probability none defective: [tex]\( \frac{60}{143} \)[/tex].
To solve this problem, we can use the concept of probability.
a. Probability that all selected transistors are defective:
When selecting the first transistor, the probability of choosing a defective one is [tex]\( \frac{3}{13} \)[/tex], as there are 3 defective transistors out of 13 total.
After the first defective transistor is chosen, there are 2 defective transistors left out of 12 total transistors.
So, the probability of choosing a second defective transistor given that the first one was defective is [tex]\( \frac{2}{12} \).[/tex]
Similarly, for the third selection, the probability of choosing a defective transistor given that the first two were defective is [tex]\( \frac{1}{11} \)[/tex].
To find the probability that all three selected transistors are defective, we multiply the individual probabilities:
P(All defective) = [tex]\frac{3}{13} \times \frac{2}{12} \times \frac{1}{11}[/tex]
P(All defective) = [tex]\frac{3 \times 2 \times 1}{13 \times 12 \times 11}[/tex]
P(All defective) = [tex]\frac{6}{1716}[/tex]
P(All defective) = [tex]\frac{1}{286}[/tex]
So, the probability that all selected transistors are defective is [tex]\( \frac{1}{286} \).[/tex]
b. Probability that none of the selected transistors are defective:
This is essentially the complement of the event that all selected transistors are defective. Since there are 3 defective transistors out of 13, the remaining 10 transistors are not defective.
So, to find the probability that none are defective, we select 3 out of the 10 non-defective transistors.
P(None defective) = Number of ways to choose 3 non-defective transistors/Total number of ways to choose 3 transistors
P(None defective) = [tex]\frac{{\binom{10}{3}}}{{\binom{13}{3}}}[/tex]
P(None defective) = [tex]\frac{{120}}{{286}}[/tex]
P(None defective) = [tex]\frac{{60}}{{143}}[/tex]
So, the probability that none of the selected transistors are defective is [tex]\( \frac{{60}}{{143}} \).[/tex]
Which algebraic expression means “three more than a number squared”?
a) 2n+3
b) 2n-3
c) n^2+3
d) n^2-3
Option C is the correct anwser, Hope this helps :)
What is the product of 1.5 and 2.8 is 4.2?
The product of 1.5 and 2.8 is correctly calculated as 4.2, in accordance with the rules of significant figures.
Explanation:The statement "1.5 and 2.8 is 4.2" seems to imply a multiplication between 1.5 and 2.8 resulting in 4.2. However, this is a misconception. The actual product of multiplying 1.5 by 2.8 is 4.2. To confirm this, you multiply the two numbers: 1.5 × 2.8 equals 4.2.
In significant figures, if the numbers 1.5 and 2.8 were part of a calculation with inexact numbers, the answer would be limited by the smallest number of significant figures present in the inexact numbers. In this case, as both 1.5 and 2.8 have two significant figures, the product is correctly expressed with two significant figures as 4.2.
What is the value of the expression when a = 5 and b = 2?
a + 2b −3
6
9
11
24
On a coordinate grid, both point (−4, −1) and (2, 6) point are reflected across the y-axis. what are the coordinates of the reflected points?
Let f and g be two functions whose second derivatives are defined. Then (f · g) '' = f · g '' + f '' · g. true or false?
The statement (f · g)'' = f · g'' + f'' · g is false. The second derivative of a product of two functions is given by a different formula.
The statement (f · g)'' = f · g'' + f'' · g is false.
The second derivative of a product of two functions, f(x) and g(x), is given by (f · g)'' = f''g + 2f'g' + fg''.
To see why the given statement is false, consider the example where f(x) = x and g(x) = x^2. The left-hand side is 0, but the right-hand side is not zero, so the statement does not hold true.
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The formula for the area of a square can be written as A = s^2 and means “area equals side-squared”. Explain how the formula for the area of a square can be derived from the formula for the area of a rectangle.
The area of a square, represented as A = s², is derived from the formula for the area of a rectangle (A = w × h) where the width and height are equal in a square, hence the side (s) is squared (s × s).
The formula for the area of a square is derived from the area of a rectangle because a square is a special type of rectangle where all sides are equal.
The formula for the area of a rectangle is A = w × h, where w is the width and h is the height. In the case of a square, the width and height are the same, which we call the side length s. Therefore, the formula A = s imes s, or A = s², naturally follows for finding the area of a square.
To visualize this, consider placing a square on graph paper. Each side of the square touches the same number of squares on the graph paper because the sides are of equal length.
When you multiply the side length by itself, you are effectively counting the total number of squares inside the square, which gives the area in square units. This visual method confirms that the area of a square is indeed the side length squared.
What is the factored form of the expression?
s^4 – 16
A. (s - 2)^2(s + 2)^2
B. (s - 2)(s + 2)
C. (s - i)(s + i)(s - 2)(s + 2)
D. (s - 2i)(s + 2i)(s - 2)(s + 2)
A company has tow electric motors consume varying amounts of power. The power consumed by each motor is a function of the time (t in minutes) for which it runs. The cost of power (in $) to run one motor is given by the function Ca(t)=t^2-2t+5. The cost of running the second motor is given by Cb(t)=3t+2. Which gives the total cost of running both motors?
C(t)=3t^3-6t^2+15t
C(t)=2t^2-4t+10
C(t)=t^2+t+7
C(t)=3t^3+6t^2-15t
Answer:
Option (c) is correct.
The total cost of running both motors is [tex]t^2+t+7[/tex]
Step-by-step explanation:
Given : The cost of power (in $) to run one motor is given by the function [tex]C_a(t)=t^2-2t+5[/tex] and The cost of running the second motor is given by [tex]C_b(t)=3t+2[/tex]
We have to find the total cost of running both motors.
Since we are given the cost to run each motors so, total cost will be the sum of running both motors.
Let C(t) be the total cost of running both motors.
[tex]C(t)=C_a(t)+C_b(t)[/tex]
Substitute,
[tex]C_a(t)=t^2-2t+5[/tex]
and [tex]C_b(t)=3t+2[/tex]
We get,
[tex]C(t)=t^2-2t+5+3t+2[/tex]
Simplify, we get,
[tex]C(t)=t^2+t+7[/tex]
Thus, The total cost of running both motors is [tex]t^2+t+7[/tex]
Which relationship is always true for the angles r, x, y, and z of triangle ABC?
x + z = y
180 degrees − x = r
x + y + z = 180 degrees
x + y + z = 90 degrees
Answer with explanation:
In the problem , given about triangle, four angles are given , r,x ,y and z.
→→One of the three, must be exterior angle of triangle ABC, and other three, must be interior angles of Δ ABC.
So, two properties of triangle that we will use in this problem
1. Angle sum property of triangle: the sum of three angles of triangle is 180°.
2.Exterior angle property: Exterior angle of a triangle is equal to sum of two interior opposite angles.
So, if x, y and z are interior angles , and r is exterior angle
then
Option 3: x + y + z = 180° is always true.
either, r= y + z
and ,r +x=180°
f(x) = sin(x^2 - 2)
find points of discontinuity, if any ...?
The revenue from selling x shirts is r(x) = 11x.
The cost of buying x shirts is c(x) = 6x + 20.
The profit from selling x shirts is p(x) = r(x) – c(x).
What is p(x)?
A. p(x) = 5x + 20
B. p(x) = 17x + 20
C. p(x) = 17x – 20
D. p(x) = 5x – 20
HELP!!give me the correct answer!! thank you
Answer:
Correct answer is: D p(x)=5x-20: Apex
Step-by-step explanation:
The value of the profit function is equal to 5x - 20. The correct option is D.
Given that:
Revenue function: r(x) = 11x
Cost function: c(x) = 6x + 20
To find the profit function, we need to subtract the cost function from the revenue function:
p(x) = r(x) - c(x)
Substituting the given revenue and cost functions and calculated as:
p(x) = 11x - (6x + 20)
Simplifying above expression as:
p(x) = 11x - 6x - 20
Combining like terms as:
p(x) = 5x - 20
Therefore, the value of the profit function is equal to 5x - 20. The correct option is D.
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Simplify each expression:
12^log base 12 of 144
The simplification of the mathematical expression 12^log base 12 of 144 is 144, according to the properties of logarithms.
Explanation:To simplify 12^log base 12 of 144, you need to understand the definition of logarithms. A logarithms in base b of a number x is the power to which we have to raise b to get x. In other words, if y = log base b of x, then b^y = x.
Applying this to our example, since log base 12 of 144 is 2 (because 12^2 = 144), then 12^log base 12 of 144 = 12^2 = 144.
Therefore, the simplification of the expression 12^log base 12 of 144 is 144.
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To simplify the expression 12^log base 12 of 144, evaluate the logarithm, substitute the value back into the expression, and perform exponentiation.
Explanation:In essence, the logarithm with base 12 and argument 144 essentially asks, "To what power must 12 be raised to obtain 144?" The answer, as established, is 2. Consequently, when we raise 12 to the power of 2, we retrieve the original value of 144. This simplification showcases the inherent relationship between logarithms and exponentiation, providing a clearer understanding of the expression's evaluation.
To simplify the expression 12^log12(144), we need to evaluate the logarithm first. Since the base of the logarithm is the same as the base of the exponentiation, we can simplify it to just log(144). Using a calculator, we find that log(144) = 2.158. Now we can substitute this value back into the original expression: 12^2.158. Evaluating the exponentiation gives us the final answer: 205.08.
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A computer uses 239 watts per hour. How many watts would it use if it is on for 8 days?
Answer:
29 watts
Step-by-step explanation:
239 ÷ 8 = 29.875
Evaluate the following expression.
153^0
can someone help me please Write 243 in exponential form ...?
The exponential form of 243 is: 243 = 3⁵.
To write 243 in exponential form, you need to express it as a power of a base number.
The most common base used for this purpose is 3.
243 can be written as: 243 = 3⁵
Here's the reasoning:
[tex]\[ 3 \times 3 = 9 \][/tex]
[tex]\[ 9 \times 3 = 27 \][/tex]
[tex]\[ 27 \times 3 = 81 \][/tex]
[tex]\[ 81 \times 3 = 243 \][/tex]
So, 243 is 3 raised to the power of 5.
Therefore, 243 In exponential form is: [tex]\[ 243 = 3^5 \][/tex]
What is the value of x, when 10(x + 2) = 5(x + 8)?
The value of ( x ) is 4.
To find the value of ( x ) in the equation [tex]\( 10(x + 2) = 5(x + 8) \)[/tex], we'll follow these steps:
[tex]\[ 10(x + 2) = 5(x + 8) \][/tex]
[tex]\[ 10x + 20 = 5x + 40 \][/tex]
[tex]\[ 10x - 5x + 20 = 5x - 5x + 40 \][/tex]
[tex]\[ 5x + 20 = 40 \][/tex]
[tex]\[ 5x + 20 - 20 = 40 - 20 \][/tex]
[tex]\[ 5x = 20 \][/tex]
[tex]\[ \frac{5x}{5} = \frac{20}{5} \][/tex]
[tex]\[ x = 4 \][/tex]
Therefore, [tex]\( x = 4 \)[/tex] is the solution.
Mrs.ulrich has 3 times as many markers as colored pencils.the total number of markers and colored pencils is 84.how many markers does mrs.ulrich have
Rewrite (-9)^-2 without an exponent
jake goes to the grocery store and buys 3 apple, 2 cans of soup, and 1 box of cereal. the apples cost $0.89 each; the soup costs $2.98 per can; and the box of cereal costs $4.99.
Answer:
c=(3 x 0.89)+(2.98)+4.99
Step-by-step explanation:
48 is how many percent of 70
What is the value of f(-2)
To calculate the value of f(-2), a function definition f(x) is required. The information provided does not include this, making it impossible to determine the value of f(-2) without further details.
Explanation:In the context given, it appears that the question f(-2) refers to finding the value of a function f when the variable is -2. However, the provided information does not include a function or formula for f(x) that can be evaluated at x = -2. The details given seem to be related to various mathematical and scientific principles such as lens formula, chemical reactions, and quadratic equations, but none of these can be directly linked to a function f(x) to evaluate f(-2). To answer the question about the value of f(-2), the explicit function definition f(x) is necessary. Without this definition, it is impossible to determine the value of f(-2).
The value of f(2) for the function f(x) = 2x^2+1 is 9. This is calculated by substituting '2' into the function, squaring it, multiplying by 2, and then adding 1.
To find the value of f(2) for the function f(x) = 2x^2+1, you simply substitute '2' for every instance of 'x' in the function's formula. Here's the step-by-step calculation:
Replace 'x' with '2': 2(2)^2 + 1
Multiply '2' by '4': 8 + 1
Add '1' to '8': 9
Therefore, the value of f(2) is 9.
The probable question maybe:
What is the value of f(2) if f(x) = 2x^2+1?
Which of the following shows the correct factorization of x3 - 5x2 - 14x?
A. x(x + 7)(x + 2)
B. x(x - 7)(x - 2)
C. x(x + 7)(x - 2)
D. x(x - 7)(x + 2)