Question

A cake recipe will make a round cake that is 6 inches in diameter and 2 inches high. If you use the same recipe but pour the batter into a round cake pan that is 8 inches in diameter, how tall will the cake be?

Answer 1

The cake will be 1.125 inches tall

Explanation:

The initial diameter of the cake, d₁ = 6 inches

The initial height of the cake, h₁ = 2 inches

The initial radius of the cake, r₁ = d₁/2 = 6/2 = 3 inches

r₁ = 3 inches

Because the round cake has a height, it will take the form of a cylinder

The initial volume of the cake, V₁ = π r₁²h₁

V₁ = π * 3² * 2

V₁ = 18π inch³

When the batter is poured into a round cake pan

the diameter, d₂ = 8 inches

r₂ = 8/2 4 inches

Since the cake is still produced using the same recipe, the volume of the cake does not change, i.e. V₂ = V₁ = 18π inch³

To get the height of the new cake

V₂ = π r₂²h₂

18π = π * 4² * h₂

h₂ = 18/16

h₂ = 1.125 inches

The new height of the cake will be 1.125 inches

Answer 2

`h \approx 1.125\,in`

Explanation:

The volume of the original cake is:

`V_(o) = (\pi)/(4)\cdot (6\,in)^(2)\cdot (2\,in)`

`V_(o) \approx 56.549\,in^(3)`

The new height on the new cake pan is:

`V_(o) = (\pi)/(4)\cdot (8\,in)^(2)\cdot h`

`h \approx 1.125\,in`