Question

Cones a and b are similar.
the surface area of cone a is 36cm2.
the surface area of cone b is 100cm2.
the volume of cone a is 162cm3.
work out the height,
h
, of cone a and the volume of cone b:

Answer 1

The height of cone A is 12 and the volume of cone B is 750

How to determine the height and the volume?

The given parameters are:

• Surface area of cone a is 36cm²
• Surface area of cone b is 100cm²
• Volume of cone a is 162cm³

Both cones are similar.

The similarity ratio is represented as:

$(\frac{V_A}{V_B})^\frac 13 = (\frac{A_A}{A_B})^\frac 12 = \frac{h_A}{h_B}$

Where:

• V represents volume
• A represents surface area
• h represents height

Remove the height expression in the equation $(\frac{V_A}{V_B})^\frac 13 = (\frac{A_A}{A_B})^\frac 12 = \frac{h_A}{h_B}$

So, we have:

$(\frac{V_A}{V_B})^\frac 13 = (\frac{A_A}{A_B})^\frac 12$

This gives

$(\frac{162}{V_B})^\frac 13 = (\frac{36}{100})^\frac 12$

Evaluate the exponent of 1/2

$(\frac{162}{V_B})^\frac 13 = \frac{6}{10}$

Take the cube of both sides

$\frac{162}{V_B} = \frac{216}{1000}$

Make VB the subject

$V_B = \frac{162 * 1000}{216}$

Evaluate

$V_B = 750$

Remove the area expression in the equation $(\frac{V_A}{V_B})^\frac 13 = (\frac{A_A}{A_B})^\frac 12 = \frac{h_A}{h_B}$

$(\frac{V_A}{V_B})^\frac 13 = \frac{h_A}{h_B}$

Substitute known values

$(\frac{162}{750})^\frac 13 = \frac{h_A}{20}$

Multiply both sides by 20

$h_A = 20 *(\frac{162}{750})^\frac 13$

Evaluate

$h_A= 12$

Hence, the height of cone A is 12 and the volume of cone B is 750

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