Question

A

Answer 1

By "similar cone" do they mean it has the same angle in the tip?

Answer 2

Ok, I'll assume it is. 1 sec while I write this all out!

Answer 3

Since it's a similar cone the angles of the triangle inside are the same. Therefore the triangle sides are in a ratio.

\[\frac{radius1}{radius2} = \frac{slantheight1}{slantheight2} = \frac{height1}{height2} \]
\[\frac{5}{R} = \frac{20}{24}\]

Use cross multiplication to find R, which is the radius of the second cone.

The lateral area of a cone is the curved area of a cone (NOT the base). Its formula (where l = the slant height) is:
\[\pi*r*l\]
For example: The lateral area of the diagram is:
\[\pi*5*20 = 100\pi\]

The lateral area of the cone that you want to solve for is:
\[\pi*R*24\]

Answer 4

Here's a nice simple way.

You have that your new side length is 24, and the cones are similar. Look at the fraction \[\frac{24}{20}=\frac{6}{5}\]Since surface area is related to the square, of the side length, your new surface area should be\[A'=A\cdot\left(\frac{6}{5}\right)^2\]where \(A\) is the old surface area. Can you tell me what the surface area of the original cone is?

Answer 5

Wired's way also works, it just requires more steps.

Answer 6

The formula for the lateral area is \(A=\pi\cdot r\cdot s\), where \(r\) is the radius, and \(s\) is the length of the diagonal. Using \(r=5\) and \(s=20\) can you tell me what \(A\) is?

Answer 7

Close enough. However let's leave it in exact form for now. That means that \[A=100\pi\]Now, can you simplify this \[100\pi\cdot\;\frac{36}{25}?\]

Answer 8

@KingGeorge where did you get that formula from:
\[A \prime = A * (\frac{5}{6})^{2}\]

Answer 9

The surface area is directly proportional to the square of the side lengths. The ratio of the side lengths is \[\frac{24}{20}=\frac{6}{5}\]so the surface area must be proportional to \[\left(\frac{6}{5}\right)^2\]Hence, \[A'=A\cdot\left(\frac{6}{5}\right)^2\]