Triangle ABC is a right triangle with leg lengths of 5 and 12 inches. Find the length of the hypotenuse. Round answer to the nearest hundredth if necessary.

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:

The simplification of the mathematical expression 12^log base 12 of 144 is 144, according to the properties of logarithms.

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.

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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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°

Option C is the correct anwser, Hope this helps :)

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.

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.

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?

Given that tan(x) is -12/5 in Quadrant II where sin is positive and cos is negative, we can use the Pythagorean theorem to find sin(x) and cos(x), which are then used to calculate sin(2x), cos(2x), and tan(2x).

Given tan(x) = -12/5, x is in Quadrant II. Since tangent is negative in Quadrant II, we know that sin(x) is positive and cos(x) is negative. Using Pythagorean theorem, we can find sin(x) and cos(x). sin(x) = sqrt(1 - cos²(x)) and cos(x) = -sqrt(1-sin²(x)).

The values for sin(x) and cos(x) can then be plugged into the formulas sin2x = 2sin(x) cos(x), cos2x = cos²(x)-sin²(x) and tan2x = sin2x / cos2x to find sin(2x), cos(2x), and tan(2x).

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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]

Answer:

743.41 feet.

Step-by-step explanation:

In this question it is given that height of the tower OS is 175 feet and angle of depression ∠ASB = 13°15'

angle of depression to the boat B will be equal to the angle of elevation "∠B"

Now  ∠ASB = 13°15' ≈ 13° + [tex](\frac{15}{60})[/tex]°

= 13° + (0.25)°

= 13.25°

Now tan ∠B = [tex]\frac{SO}{OB}[/tex]

 tan (13.25) =  [tex]\frac{175}{x}[/tex]

            x     =  [tex]\frac{175}{0.2354}[/tex]

                   = 743.41 feet.

Motorboat is 743.41 feet far from the shore.

Answer:

5

Step-by-step explanation:

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]

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

Answer:

29 watts

Step-by-step explanation:

239 ÷ 8 = 29.875

Using the exponential decay function for Pacific halibut, it is calculated that approximately 24.7% of the original cohort is still alive after 7 years.

To approximate the percentage of the original number of Pacific halibut still alive after 7 years, we use the exponential decay model provided, N(t) = N0e^-0.2t. Plugging in t = 7 years into the equation gives us:

N(7) = N0e^-0.2(7) = N0e^-1.4.

To find this percentage, we can multiply the value of e^-1.4 by 100%. First, we calculate e^-1.4 approximately using a calculator:

e^-1.4 ≈ 0.24660

Multiplying by 100 to get the percentage: 0.24660 × 100% ≈ 24.7%.

Therefore, approximately 24.7% of the original Pacific halibut cohort is still alive after 7 years.

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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