Question

Two cones are similar. The surface area of the larger cone is 65π square inches. The surface area of the smaller cone is 41.6π square inches. The radius of the smaller cone is 6.4 inches. What is the radius of the larger cone?


8 inches
10 inches
11.52 inches
14.4 inches

Answer 1

Step $1$

Find the scale factor

we know that

$surface\ area\ of\ the\ larger\ cone=(scale\ factor^{2})* surface\ area\ of\ the\ smaller\ cone$

$(scale\ factor^{2})=surface\ area\ of\ the\ larger\ cone/ surface\ area\ of\ the\ smaller\ cone$

in this problem we have

$surface\ area\ of\ the\ larger\ cone=65\pi\ in^{2} \\surface\ area\ of\ the\ smaller\ cone=41.6\pi\ in^{2}$

Substitute the values

$(scale\ factor^{2})=\frac{65\pi }{41.6\pi } =1.5625\\ scale\ factor=1.25$

Step $2$

Find the radius of the larger cone

we know that

$the\ radius\ of\ the\ larger\ cone=(scale\ factor)* the\ radius\ of\ the\ smaller\ cone$

in this problem we have

$the\ radius\ of\ the\ smaller\ cone=6.4\ in$

substitute

$the\ radius\ of\ the\ larger\ cone=1.25* 6.4=8\ in$

therefore

the answer is

the radius of the larger cone is $8\ in$

Answer 2

Using the concept of similarity, the radius of the larger cone with a surface area of 65π comes to be 8cm.

What is the total surface area of a cone with radius r, slant height l?

The total surface area of a cone with radius r, slant height l is $\pi rl+\pi r^{2}$.

The surface area of the smaller cone = 41.6π square inches

Suppose the radius and height of smaller and bigger cones are (r,h) and (R, H) respectively.

Suppose the slant height of the smaller and bigger cones are l and L respectively.

So, $\pi rl+\pi r^{2} =41.6\pi$

$r(l+r)=41.6$

Given r=6.4 inches

So, l=0.1 inch

$\pi RL+\pi R^{2} =65\pi$

$RL+R^{2} =65$

$R(L+R)=65$......(1)

Since cones are similar

So, $\frac{l}{r} =\frac{L}{R}$

$So, \frac{L}{R}=\frac{1}{64}$

Suppose L=x

R=64x

Put L=x & R=64x in (1)

So, $64x(65x)=65$

$x^{2} =\frac{1}{64}$

$x=\frac{1}{8}$

So, R=64x = 64*1/8 =8cm

Thus, the radius of the larger cone comes to be 8cm.

To get more about cones visit:

https://brainly.com/question/1082469