Q. 64 - Calculus

Question

Express the volume, V, and surface area, S, of a right circular cone with radius r and height h as functions of two variables. What is the domain of each function?

Step-by-Step Solution

Answer

The domains of the functions, $$V(r, h)$$  and $$ S(r, h)$$  are $$ \frac{1}{3} \pi r^2 h$$  and $$\pi r^2+\pi r \sqrt{r^2+h^2}$$ respectively.

Step 1. Given Information

A right circular cone with a radius (r) and height (h) has a volume (V) and surface area (S).

Step 2. Explanation

The volume of a right circular cone is expressed using the equation, $$\frac{1}{3} \pi r^2 h$$  

So, the domain of the volume (V) of the right circular cone expressed as the function of two variables, r and h, is given as $$V(r, h)=\frac{1}{3} \pi r^2 h$$ 

The slant height (l) of the right circular cone is given as, $$l= \sqrt{r^2+h^2}$$ 

Next, the Curved Surface Area of the given cone is given as,

$$CSA=\pi rl$$ 

$$\Rightarrow CSA=\pi r \sqrt{r^2+h^2}$$  

Then, the total surface area, or simply surface area (S), can be given by the equation, $$S=\pi r^2+\pi r \sqrt{r^2+h^2}$$  

Hence, the domain of the surface area (S) of the right circular cone expressed as the function of two variables, r and h, is given as $$V(r, h)=\pi r^2+\pi r \sqrt{r^2+h^2}$$