/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 Find the volume of each sphere. ... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 8

Find the volume of each sphere. \(r=11\) in.

Short Answer

Expert verified
The volume of the sphere is approximately 5574.05 cubic inches.

Step by step solution

01

- Recall the formula for the volume of a sphere

The formula to find the volume of a sphere is \[ V = \frac{4}{3} \pi r^3 \]where \(V\) represents the volume and \(r\) is the radius of the sphere.
02

- Substitute the given radius into the formula

Given that the radius \(r = 11\) inches, substitute \(r\) into the formula: \[ V = \frac{4}{3} \pi (11)^3 \]
03

- Calculate the radius cubed

Calculate \(11^3\): \[ 11^3 = 11 \times 11 \times 11 = 1331 \]
04

- Multiply by \(\pi\) and by \(\frac{4}{3}\)

Substitute \(1331\) into the volume formula: \[ V = \frac{4}{3} \pi \times 1331 \]First, multiply \(\frac{4}{3}\) by 1331: \[ \frac{4}{3} \times 1331 = \frac{5324}{3} \approx 1774.67 \]Then, multiply by \(\pi\): \[ V \approx 1774.67 \pi \approx 1774.67 \times 3.14159 \approx 5574.05 \] Thus, the volume \(V\) is approximately 5574.05 cubic inches.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

sphere volume formula
The volume of a sphere formula is a fundamental concept in geometry. To find the volume of a sphere, you use the formula: The variables in the formula are: - - - typically uses the symbol understanding each part of the formula makes it easier to calculate the volume of any sphere.
radius cubed
One of the key aspects of the sphere volume formula is the term 'radius cubed.' This means you need to raise the radius to the power of three, symbolized as Typically, you calculate this as follows: - Start with the radius value, which is 11 inches for our example. - Then multiply the radius by itself three times: This means you: - First, do -Then, take the result (121) and multiply by the radius again: The final result tells you how many cubic units fit along the radius three times over, a crucial step in the volume calculation.
substitution in geometry
Substitution is a technique used in geometry to replace variables in a formula with actual values to solve for a specific measurement. For the sphere volume problem, we start with the general formula and replace the radius with our given value: With the radius ( inch in our example, you replace every instance of the variable 'r' in the formula with the number - Start with - Replace in every place a radius appears: . . This substitution leads you to: landmarks step is substituting correctly. By mastering substitution, you simplify complex geometric calculations into straightforward arithmetic.

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Most popular questions from this chapter

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