MHB How Do You Calculate the Slant Height of a Cone?

[gazparkin]

Hello,

Could anyone help me understand the steps on the below questions?

A cone has a total surface area of 300π cm² and a radius of 10 cm. What is its slant height?


A cone has a slant height of 20 cm and a curved surface area of 330 cm2. What is the circumference of its base? I'd really like to know what steps I need to take to get to the answer on these.

Thank you in advance :-)

 

[skeeter]

Hello,

Could anyone help me understand the steps on the below questions?

A cone has a total surface area of 300π cm² and a radius of 10 cm. What is its slant height?


A cone has a slant height of 20 cm and a curved surface area of 330 cm2. What is the circumference of its base? I'd really like to know what steps I need to take to get to the answer on these.

(1) total surface area = lateral surface area + base area

$A_T = \pi r s + \pi r^2$, where $s$ is the slant height and $r$ is the base radius

solving for $s$ $\implies s = \dfrac{A_T - \pi r^2}{\pi r}$

(2) assuming "curved surface area" is the lateral surface area ...

$A_L = \pi r s \implies r = \dfrac{A_L}{\pi s}$

use the formula for a circle's circumference to finish

 

[HallsofIvy]

Suppose a cone (minus the circular bottom) has radius r and slant height s. Cut a slit along the slant and flatten it (Unlike a sphere a cone can be flattened. it is a "developable surface."). It will form part of a circle with radius h. That entire circle has radius h so area \pi h^2 and circumference 2\pi h. But the base of the cone had radius r so circumference 2\pi r. The cone is only \frac{2\pi r}{2\pi h}= \frac{r}{h} of the entire circle so has area \frac{r}{h}\pi h^2= \pi rh.