Question

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\sqrt{\pi} 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. area = m² area = m² (b) the height of the pyramid is 7 m. find the volume of the pyramid. volume of pyramid = m³

Explanation:

Step1: Find cross - section area for the cone

For a cone, if the height of the whole cone is $H = 7$m and the height of the top - part (formed by the cross - section) is $h=2$m, and the radius of the base of the whole cone is $R = 5$m. Since the top - part is similar to the whole cone, the ratio of the radii of similar cones is equal to the ratio of their heights. Let the radius of the cross - section 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 $A=\pi r^{2}$. Substituting $r = \frac{10}{7}$m, we get $A=\pi\times(\frac{10}{7})^{2}=\frac{100\pi}{49}$m². Since the cross - section areas of the cone and the pyramid are the same, the area of the cross - section of the pyramid is also $\frac{100\pi}{49}$m².

Step2: Find volume of the pyramid

The volume formula for a square - based pyramid is $V=\frac{1}{3}Bh$, where $B$ is the base area and $h$ is the height. The base of the pyramid has a side length $s = 5\sqrt{\pi}$m, so the base area $B=s^{2}=(5\sqrt{\pi})^{2}=25\pi$m². The height $h = 7$m. Then $V=\frac{1}{3}\times25\pi\times7=\frac{175\pi}{3}$m³.

Answer:

Area of cross - section of pyramid: $\frac{100\pi}{49}$m²
Area of cross - section of cone: $\frac{100\pi}{49}$m²
Volume of pyramid: $\frac{175\pi}{3}$m³