Cross-Section Math Example 5

Solution

1. 1
   Step 1: The cut is at height $8$ cm, leaving $12 - 8 = 4$ cm from the apex. The cross-section radius scales as $r_{cs} = 6 \times \dfrac{4}{12} = 2$ cm.
2. 2
   Step 2: Area of cross-section $= \pi(2)^2 = 4\pi$ cm$^2$. Area of base $= \pi(6)^2 = 36\pi$ cm$^2$.
3. 3
   Step 3: Ratio $= \dfrac{4\pi}{36\pi} = \dfrac{1}{9}$. Note: $\left(\dfrac{2}{6}\right)^2 = \dfrac{1}{9}$ — ratios of areas equal square of ratio of radii.