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Chapter 1: Problem 112

Volume Use the Intermediate Value Theorem to show that for all spheres with radii in the interval \([5,8],\) there is one with a volume of 1500 cubic centimeters.

Short Answer

Expert verified
With the Intermediate Value Theorem, it can be proven that there exists a radius size between 5 cm and 8 cm for which the volume of the sphere is 1500 cubic cm.

Step by step solution

01

Calculate the sphere volumes for the radius interval ends

First calculate the volume of spheres with radii 5 and 8 cm. Use the volume formula for a sphere, which is \(V = \frac{4}{3} \pi r^3\). So if \(r = 5 cm\), the volume is \(V_{5} = \frac{4}{3} \pi 5^3\) and if \(r = 8 cm\), the volume is \(V_{8} = \frac{4}{3} \pi 8^3\).
02

Check if 1500 cubic centimeters is between these two volumes

Verify that 1500 cubic cm is a volume that falls between \(V_{5}\) and \(V_{8}\). If so, continue with the next step. If not, there's a miscalculation or misunderstanding of the problem.
03

Use the Intermediate Value Theorem

Here use the Intermediate Value Theorem, which states that if a function is continuous on the closed interval [a, b], then it takes on every value between f(a) and f(b). Note that the volume function for a sphere is continuous for all radii greater than 0, and thus, by the Intermediate Value Theorem, there is indeed one radius between 5 cm and 8 cm that will give a sphere volume of 1500 cubic cm.

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Most popular questions from this chapter

Testing for Continuity In Exercises \(77-84\) , describe the interval(s) on which the function is continuous. $$ f(x)=\sec \frac{\pi x}{4} $$

Testing for Continuity In Exercises \(77-84\) , describe the interval(s) on which the function is continuous. $$ f(x)=\cos \frac{1}{x} $$

Using the Intermediate Value Theorem In Exercises \(91-94,\) use the Intermediate Value Theorem and a graphing utility to approximate the zero of the function in the interval \([0,1] .\) Repeatedly "zoom in" on the graph of the function to approximate the zero accurate to two decimal places. Use the zero or root feature of the graphing utility to approximate the zero accurate to four decimal places. $$ f(x)=x^{4}-x^{2}+3 x-1 $$

Finding Discontinuities In Exercises \(73-76\) , use a graphing utility to graph the function. Use the graph to determine any \(x\) -values at which the function is not continuous. $$ g(x)=\left\\{\begin{array}{ll}{x^{2}-3 x,} & {x>4} \\ {2 x-5,} & {x \leq 4}\end{array}\right. $$

Removable and Nonremovable Discontinuities In Exercises \(35-60,\) find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? \ $$ f(x)=\frac{|x+7|}{x+7} $$
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