A flower pot in the form of a frustum of a cone. The diameters of the top and bottom ends are 64cm and 48cm respectively. If the slanting height is 15m, calculate (a) volume (b) TSA

The problem describes a flower pot in the form of a frustum of a cone. We are given the diameters of the top and bottom ends and the slanting height. We need to calculate its volume and total surface area.

First, let's list the given values and convert them to consistent units.
Diameter of the top end ($D_1$) = $64 \text{ cm}$
Diameter of the bottom end ($D_2$) = $48 \text{ cm}$
Slanting height ($L$) = $15 \text{ m}$

It is highly probable that the slanting height of $15 \text{ m}$ is a typo and should be $15 \text{ cm}$ for a "flower pot" to have realistic dimensions. We will proceed with the assumption that $L = 15 \text{ cm}$. If $L = 15 \text{ m}$ was intended, the calculations would be similar but result in extremely large values.

Step 1: Convert given diameters to radii and ensure consistent units.
Radius of the top end ($R_1$) = $\frac{D_1}{2} = \frac{64 \text{ cm}}{2} = 32 \text{ cm}$
Radius of the bottom end ($R_2$) = $\frac{D_2}{2} = \frac{48 \text{ cm}}{2} = 24 \text{ cm}$
Slanting height ($L$) = $15 \text{ cm}$ (Assumption: $15 \text{ m}$ was a typo and should be $15 \text{ cm}$)

Step 2: Calculate the perpendicular height ($h$) of the frustum.
The relationship between the slant height, perpendicular height, and radii is given by the Pythagorean theorem: