Question

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

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

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 \(V_{8} = \frac{4}{3} \pi 8^3\).

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.

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.