Question

question 6
a cone and a cylinder have the same radius and height. the volume of the cone is 100π cubic feet. what is the volume of the cylinder?

question 7
a right cone has a base with a diameter of 10 units. the volume of the cone is 100π cubic units. what is the length of a segment drawn from the apex to the edge of the circular base?

question 8
a right pyramid has a square base with sides of length 10 units. each segment connecting the apex to a midpoint of a side of the base has length 13 units. what is the volume of the pyramid?

Explanation:

Question 6

Step1: Recall the volume formulas

The volume formula for a cone is \(V_{cone}=\frac{1}{3}\pi r^{2}h\), and for a cylinder is \(V_{cylinder}=\pi r^{2}h\).

Step2: Relate the volumes

Since the cone and cylinder have the same \(r\) and \(h\), and \(V_{cone} = 100\pi=\frac{1}{3}\pi r^{2}h\), then \(V_{cylinder}=\pi r^{2}h = 3\times V_{cone}\)

Step3: Calculate the cylinder's volume

\(V_{cylinder}=3\times100\pi = 300\pi\)

Question 7

Step1: Find the radius

Given diameter \(d = 10\), radius \(r=\frac{d}{2}=5\)

Step2: Use the cone volume formula to find height

\(V=\frac{1}{3}\pi r^{2}h\), so \(100\pi=\frac{1}{3}\pi\times5^{2}\times h\). Solving for \(h\):

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Step3: Use the Pythagorean theorem

The length \(l\) from the apex to the edge of the base (slant height) is \(l=\sqrt{r^{2}+h^{2}}=\sqrt{5^{2}+12^{2}}=\sqrt{25 + 144}=\sqrt{169}=13\)

Question 8

Step1: Find the height of the pyramid

Let the height of the pyramid be \(h\). The distance from the center of the base to the mid - point of a side is \(5\) (since side length \(a = 10\)). Using the Pythagorean theorem \(13^{2}=h^{2}+5^{2}\), so \(h=\sqrt{13^{2}-5^{2}}=\sqrt{169 - 25}=\sqrt{144}=12\)

Step2: Calculate the volume of the pyramid

The volume formula for a square - based pyramid is \(V=\frac{1}{3}a^{2}h\). Substituting \(a = 10\) and \(h = 12\):

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Answer:

Question 6: \(300\)
Question 7: \(13\)
Question 8: \(400\)