Question

answer the following questions. write your answers in terms of π. give exact answers (not decimal approximations). (a) the square pyramid and the cone below both have a height of 7 m. the pyramid has a base length of 5√π m, and the base of the cone has a radius of 5 m. a plane parallel to the bases crosses both solids at 2 m from the top. the resulting cross - sections (shaded) have the same area. for each solid, the top portion (which has the highlighted cross section as its base) is similar to the entire solid. use this fact to find the areas of the cross sections. (b) the height of the pyramid is 7 m. find the volume of the pyramid.

Explanation:

Step1: Use similarity for the cone's cross - section

For a cone, if the height of the whole cone is $H = 7$m and the height of the smaller cone (top - part) is $h=2$m, and the radius of the whole cone is $R = 5$m. By the property of similar triangles (or similar cones), the ratio of the radii of two similar cones is equal to the ratio of their heights. Let the radius of the cross - section (radius of the smaller cone) be $r$. Then $\frac{r}{R}=\frac{h}{H}$. Substituting the values, we have $\frac{r}{5}=\frac{2}{7}$, so $r=\frac{10}{7}$m. The area of a circle (cross - section of the cone) is $A=\pi r^{2}$. So $A=\pi(\frac{10}{7})^{2}=\frac{100\pi}{49}LXB0m^{2}$.

Step2: Calculate the volume of the pyramid

The volume formula for a square pyramid is $V=\frac{1}{3}Bh$, where $B$ is the area of the base and $h$ is the height. The base of the pyramid has a side length $s = 5\sqrt{\pi}$m, so the area of the base $B=s^{2}=(5\sqrt{\pi})^{2}=25\piLXB0m^{3}$.

Answer:

(a) Area of cross - section of pyramid: $\frac{100\pi}{49}LXB0m^{2}$
(b) Volume of pyramid: $\frac{175\pi}{3}$$m^{3}$