Calculating the missing dimension?

Question

1) a right cone has a surface area 12 m${^2}$ and radius 1.3 m

here is my answers:

1) $s$ = 1.28

(photo of how i got my answer)

textbook answer: 1.64 m

how did they get that??

Answer 1

The formula to calculate the surface area of a cone is:

$$SA=\pi r(r+\sqrt{h^2+r^2})$$

We know the $SA$ and $r$. Now solve for $h$.

$$12=\pi(1.3)(1.3+\sqrt{h^2+(1.3)^2})$$

$$\frac{12}{1.3\pi}-1.3=\sqrt{h^2+(1.3)^2}\text{ moved stuff we already know to left side}$$

$$(\frac{12}{1.3\pi}-1.3)^2=h^2\text{ squared both sides}$$

$$\sqrt{\frac{12}{1.3\pi}-1.3}=h\text{ take square root to solve for $h$}$$

$h\approx 1.28$

I'm guessing they made a mistake.

Answer 2

Your error is in thinking that they want the vertical height of the cone. For a cone, $s$ usually represents the slant height. So the first thing you want to is to express the total surface are as

$$S=\pi r s+\pi r^2$$

where $\pi r s$ is the lateral area. Then

$$ s=\frac{S-\pi r^2}{\pi r}\approx1.638 $$

as in your text!