Question

leah is writing a computer program that will calculate the ratio of the surface area of a cone to the volume of the cone so that the user can input the radius and height. which of the following formulas represent the ratio: select one: a. $\frac{9(2r^2 + h^2)}{r^2h^2}$ b. none of these c. $\frac{3(r + \sqrt{h^2 + r^2})}{rh}$ d. $\frac{3(\sqrt{h^2 + r^2})}{h}$ given the rational function $y = \frac{x + 9}{x - 1}$ vertical asymptote: horizontal asymptote:

Explanation:

Step1: Find cone surface area and volume

Surface area of cone: $A = \pi r(r + l)$ where $l = \sqrt{r^2 + h^2}$ (slant height). Volume of cone: $V = \frac{1}{3}\pi r^2 h$.

Step2: Calculate ratio $A/V$

$$\frac{A}{V} = \frac{\pi r(r + \sqrt{r^2 + h^2})}{\frac{1}{3}\pi r^2 h} = \frac{3(r + \sqrt{r^2 + h^2})}{rh}$$

Step3: Check vertical asymptote of $y=\frac{x+9}{x-1}$

Vertical asymptote at denominator zero: $x-1=0 \Rightarrow x=1$.

Step4: Check horizontal asymptote of $y=\frac{x+9}{x-1}$

Degrees of numerator and denominator equal (1), so asymptote is ratio of leading coefficients: $\frac{1}{1}=1$.

Answer:

First question: c. $\frac{3(r+\sqrt{h^2+r^2})}{rh}$
Second question vertical asymptote: 1
Second question horizontal asymptote: 1