When finding the length of the hypotenuse of a right triangle why might you approximate the answer

Answer:

[tex]f^{-1}=4(x+2)[/tex]

Step-by-step explanation:

[tex]f(x)=\frac{x}{4} -2[/tex]

To find inverse function , replace f(x) with y

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

Swap the variables x and y

[tex]x=\frac{y}{4} -2[/tex], nbw we solve for y

Add 2 on both sides

[tex]x+2=\frac{y}{4}[/tex]

Now to eliminate 4, we multiply 4 on both sides

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

Replace y with f^-1

[tex]f^{-1}=4(x+2)[/tex]

To estimate the error in the hypotenuse length of a right triangle using differentials, we use the trigonometric identity sin(θ) = opposite/hypotenuse, finding the derivative, and then apply the angle error to get an error estimate of
±0.31 cm in the hypotenuse length.

The question asks to use differentials to estimate the error in computing the length of the hypotenuse of a right triangle when one side is 9 cm and the opposite angle is 30° with a possible error of ±1°. To calculate the hypotenuse (c) when dealing with problems involving right triangles, you can use the trigonometric identity sin(θ) = opposite/hypotenuse, which can be rearranged to c = opposite/sin(θ). The differential of c with respect to θ is dc/dθ = -opposite * cos(θ)/sin2(θ). Given that the opposite side is 9 cm and the angle θ is 30°, we can find dc/dθ and then multiply by the possible angle error of ±1° to estimate the error in the hypotenuse length.

Plugging in the given values and converting the angle to radians (since the derivative is taken with respect to radians), we find

dc/dθ ≈ -9 cm × cos(30°)/sin2(30°) ≈ -18 cm.

The possible change in angle is ±1°, which in radians is approximately ±0.0175 radians, thus

Estimated error in hypotenuse ≈ -18 cm × ±0.0175 ≈ ±0.315 cm.

Therefore, the estimated error in the length of the hypotenuse, rounded to two decimal places, is ±0.31 cm.

To calculate the volume of a solid bounded by two functions, set up an integral in polar coordinates. The volume is found by integrating the difference of the two volumes described by the functions (50 - sqrt(x²+y²) - sqrt(x²+y²)) over r, θ, and z within their appropriate limits.

The subject of the question is mathematics, more specifically, integral calculus in spherical or polar coordinates. Calculating the volume of a solid bounded by two functions can be done by taking the difference of the two function volumes.

The integrals, using polar coordinates (r and θ), are set up as follows:

Volume = ∫∫∫(50 - sqrt(x²+y²) - sqrt(x²+y²)) r dr dθ dz,

where the limits for r are [0, sqrt(50)], for θ are [0, 2π], and for z are [0, 50]. Also, note that 'x²+y²' is replaced with 'r²' because in polar coordinates, 'r' represents the square root of x^2 plus y^2.

This integral, when solved, gives the volume of the ice cream cone-shaped solid.

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The volume of the ice cream shaped solid can be found by converting the limits of the cone to polar coordinates and performing a double integral. The limits of the cone are represented by the expressions z = r and z = sqrt(50 - r^2). The volume integral will be ∫ _(theta=0 to 2*pi) ∫ _(r=0 to sqrt(50)) [(sqrt(50 - r^2)) - r] r dr dθ.

To evaluate the volume of the solid resembling an ice cream cone, we can use the method of double integration in polar coordinates. Because we're dealing with a cone, polar coordinates are suitable as they're related to circular symmetry.

The limits of the cone are given by the functions z = sqrt(x^2 + y^2) and z = sqrt(50 - x^2 - y^2). In polar coordinates these become z = r and z = sqrt(50 - r^2).

To evaluate the volume of the cone, we need to subtract the volume of the lower cone (given by z = r) from the volume of the upper cone (given by z = sqrt(50 - r^2)). We convert the values to polar coordinates and use double integration as follows:

V = ∫ _(theta=0 to 2*pi) ∫ _(r=0 to sqrt(50)) [(sqrt(50 - r^2)) - r] r dr dθ

This integral, once solved, will give the volume of the ice cream shaped solid.

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Final answer:

The second partial derivatives of v = e⁵x * ey are ∂²v/∂x² = 25e⁵x * ey and ∂²v/∂x² = 5e⁵x * e^y.

Explanation:

To find the second partial derivatives of the function v = e⁵x * ey, we need to differentiate twice with respect to x and y. Let's start with the first partial derivative with respect to x:

∂v/∂x = ∂(e⁵x * ey)/∂x

= 5e⁵x * ey

Next, we differentiate the first partial derivative with respect to x to find the second partial derivative with respect to x:

∂²v/∂x² = ∂(5e⁵x * ey)/∂x

= 25e⁵x * ey

Similarly, the second partial derivative with respect to y is:

∂²v/∂x² = ∂(5e⁵x * ey)/∂y

= 5e⁵x * e^y

The solution to the given-equation represented by "3(4x + 6) = 9x + 12" is : (b) x = -2.

To find the solution to the equation 3(4x + 6) = 9x + 12, we simplify and solve for x,

First, we distribute the 3 to both terms,

12x + 18 = 9x + 12

Next, we separate variable-term by subtracting 9x from both sides:

12x - 9x + 18 = 9x - 9x + 12,

Simplifying further:

We get,

3x + 18 = 12,

Next, we separate "x" by subtracting 18 from both sides,

3x + 18 - 18 = 12 - 18,

3x = -6,

(3x) / 3 = (-6) / 3,

x = -2

Therefore, the correct answer is (b) x = -2.

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The given question is incomplete, the complete question is

What is the solution to the equation 3(4x + 6) = 9x + 12?

(a) x = 2,

(b) x = -2,

(c) x = 10,

(d) x = -10.

Answer:

77%

Step-by-step explanation:

2:9 means for every 2 men there are 9 women

36/9 =4

4*2 = 8

 there are 8 men

The missing side length of the second triangle is 8.

To find the missing side length of the second triangle, we can use the property that similar triangles have proportional side lengths.

Given:

First triangle: ( 3, 4, 5 )

Second triangle: ( 6, x, 10 )

Since the triangles are similar, the ratios of corresponding sides are equal:

[tex]\[ \frac{6}{3} = \frac{x}{4} = \frac{10}{5} \][/tex]

From the first ratio, [tex]\( \frac{6}{3} = \frac{x}{4} \)[/tex], cross multiply:

[tex]\[ 6 \times 4 = 3 \times x \][/tex]

[tex]\[ 24 = 3x \][/tex]

Divide both sides by 3 to solve for ( x ):

[tex]\[ x = \frac{24}{3} \][/tex]

[tex]\[ x = 8 \][/tex]

So, the missing side length of the second triangle is 8.

Here's a detailed calculation step-by-step:

1. Set up the ratios of corresponding sides: [tex]\( \frac{6}{3} = \frac{x}{4} = \frac{10}{5} \).[/tex]

2. Use the first ratio to find [tex]\( x \): \( 6 \times 4 = 3 \times x \)[/tex].

3. Solve for [tex]\( x \): \( 24 = 3x \)[/tex].

4. Divide both sides by 3: [tex]\( x = \frac{24}{3} = 8 \)[/tex].

5. Therefore, the missing side length of the second triangle is 8.

Complete Question:

Two triangles have side lengths 3, 4, 5 and 6, _____, 10, respectively. The triangles are similar to each other. Find the missing side of the triangle.

The unit price of Starbucks Coffee, when 0.5 lb costs $6.48, is calculated by dividing the total cost by the pounds, resulting in $12.96 per pound.

If 0.5 lb of Starbucks Coffee costs $6.48, to find the unit price for Starbucks Coffee, we perform a simple division. You want to know how much 1 lb would cost, given that 0.5 lb costs $6.48. To do this, you divide the total cost by the number of pounds.

$6.48 ÷ 0.5 = $12.96 per pound

This means that the unit price of Starbucks Coffee, when given a price of $6.48 for a half-pound, is $12.96 per pound. This calculation is crucial in understanding how to break down costs into more manageable, standard units, making price comparisons easier and more straightforward.