Question

an anthill has a volume of 8792 mm³ of dirt. its radius is 20 mm. how far does an ant have to crawl to get from the base of the cone to the top of the hill? (this is the slant height, s, of the cone.) use 3.14 for π, and round your answer to the nearest mm if needed. what is the height of the cone? the can be used to find the slant height, s. what is the slant height of the cone?

Explanation:

Step1: Find the height of the cone

The volume formula of a cone is \(V=\frac{1}{3}\pi r^{2}h\). Given \(V = 8792\space mm^{3}\), \(r = 20\space mm\), and \(\pi=3.14\).
Substitute the values into the formula: \(8792=\frac{1}{3}\times3.14\times20^{2}\times h\).
First, calculate \(20^{2}=400\), then \(\frac{1}{3}\times3.14\times400=\frac{1256}{3}\).
So the equation becomes \(8792=\frac{1256}{3}h\).
Solve for \(h\): \(h=\frac{8792\times3}{1256}\).
\(h = 21\space mm\).

Step2: Use the Pythagorean theorem to find the slant height

The Pythagorean theorem for a cone is \(s=\sqrt{r^{2}+h^{2}}\) (where \(s\) is the slant height, \(r\) is the radius, and \(h\) is the height).
Substitute \(r = 20\space mm\) and \(h = 21\space mm\) into the formula: \(s=\sqrt{20^{2}+21^{2}}\).
Calculate \(20^{2}=400\) and \(21^{2}=441\). Then \(r^{2}+h^{2}=400 + 441=841\).
So \(s=\sqrt{841}=29\space mm\).

Answer:

The height of the cone is \(21\space mm\). The Pythagorean theorem (\(s=\sqrt{r^{2}+h^{2}}\)) can be used to find the slant height. The slant height of the cone is \(29\space mm\).