/*! 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 17 Estimate the radius of a spheric... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 17

Estimate the radius of a spherical balloon that has a volume of \(4 \mathrm{ft}^{3}\).

Short Answer

Expert verified
Approx. 0.987 feet

Step by step solution

01

Recall the formula for the volume of a sphere

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

Insert the given volume into the formula

We are provided with the volume of the spherical balloon, which is \(4\, \mathrm{ft}^3\). Substitute \(V = 4\) into the formula: \[ 4 = \frac{4}{3} \pi r^3 \]
03

Solve for the radius

To isolate \(r^3\), multiply both sides of the equation by \(\frac{3}{4}\): \[ r^3 = \frac{3}{4\pi} \times 4 \]Simplify the equation: \[ r^3 = \frac{3}{\pi} \]Now, take the cube root of both sides: \[ r = \sqrt[3]{\frac{3}{\pi}} \]
04

Calculate the numerical value

Use a calculator to find the approximate value of \( \sqrt[3]{\frac{3}{\pi}} \): \[ r \approx \sqrt[3]{\frac{3}{3.14159}} \approx \sqrt[3]{0.95493} \approx 0.987 \mathrm{ft} \]

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Key Concepts

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

Volume of a Sphere
Understanding how to calculate the volume of a sphere is fundamental in geometry. The volume formula is \[ V = \frac{4}{3} \pi r^3 \] where \(V\) represents the volume, and \(r\) denotes the radius. This equation implies that the volume of a sphere is proportional to the cube of its radius.
  • Given a sphere's radius, you can determine its volume.
  • Likewise, if you know the volume, you can work backwards to find the radius.
In our exercise, the balloon has a volume of \( 4 \text{ft}^3 \). We must use the volume formula and solve for the radius.
Cube Roots
A cube root is the number that, when multiplied by itself twice, results in the original number. For instance, the cube root of 8 is 2 since \(2 \times 2 \times 2 = 8\). The cube root is denoted as \( \sqrt[3]{x} \).
To find the radius using the volume formula, we need to take the cube root of a fraction:
  • Start by isolating \(r^3\) in the volume formula.
  • This may require multiplication, division, and simplification steps.
  • Finally, take the cube root of the result to find \(r\).
Returning to our problem, we had \( r^3 = \frac{3}{\pi} \), so we took the cube root of \( \frac{3}{\pi} \) to estimate the radius.
Mathematical Formulas
Mathematical formulas like the volume of a sphere are essential tools. They provide a systematic way to problem-solve and find unknown values.
Here鈥檚 a quick breakdown of how we used a mathematical formula in our solution:
  • We plugged in the known volume into the volume formula of a sphere.
  • Next, we isolated the variable \( r^3 \) by multiplying both sides by the reciprocal of the coefficient.
  • Finally, we took the cube root to solve for \( r \).
This method shows the power of mathematical formulas in turning complex problems into manageable steps. By understanding the formula and the process, we effectively estimated the radius of the spherical balloon as approximately 0.987 feet.

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Most popular questions from this chapter

Without using a calculator, find two consecutive integers such that one is smaller and one is larger than each of the following (for example, \(3<\sqrt{11}<4\) ). Show your reasoning. a. \(\sqrt{13}\) b. \(\sqrt{22}\) c. \(\sqrt{40}\)

Rewrite in an equivalent form using logarithms: a. \(10^{4}=10.000\) b. \(10^{-2}=0.01\) c. \(10^{0}=1\) d. \(10^{-5}=0.00001\)

Evaluate the following without a calculator. a. Find the following values: i. \(\log 100\) ii. log 1000 iii. \(\log 10,000,000\) What is happening to the values of \(\log x\) as \(x\) gets larger? b. Find the following values: i. \(\log 0.1\) ii. \(\log 0.001\) iii. \(\log 0.00001\) What is happening to the values of \(\log x\) as \(x\) gets closer to \(0 ?\) c. What is \(\log 0 ?\) d. What is \(\log (-10) ?\) What do you know about \(\log x\) when \(x\) is any negative number?

Estimate the value of each of the following: a. \(\log 4000\) b. \(\log 5,000,000\) c. \(\log 0.0008\)

Estimate the length of a side, \(s,\) of a cube with volume, \(V,\) of 6 \(\mathrm{cm}^{3}\) (where \(V=s^{3}\) ).
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