Question

6 a circular cake tin has radius 20 cm and height 7 cm. when cake mix was added to the tin its height was 2 cm. after the cake was cooked it rose to 1.5 cm below the top of the tin.
a sketch these two situations.
b find the volume of the cake mix.
c find the volume of the cooked cake.
d what was the percentage increase in the volume of the cake while it cooked?

Explanation:

Part b:

Step1: Recall the formula for the volume of a cylinder

The volume \( V \) of a cylinder is given by \( V=\pi r^{2}h \), where \( r \) is the radius and \( h \) is the height.

Step2: Identify the values for cake mix

The radius \( r = 20\space\text{cm}\) and the height of the cake mix \( h = 2\space\text{cm}\).

Step3: Substitute the values into the formula

Substitute \( r = 20 \) and \( h = 2 \) into \( V=\pi r^{2}h \):
\( V=\pi\times(20)^{2}\times 2=\pi\times400\times 2 = 800\pi\space\text{cm}^3\approx800\times3.14 = 2512\space\text{cm}^3\)

Step1: Find the height of the cooked cake

The height of the tin is \( 7\space\text{cm}\) and the cooked cake is \( 1.5\space\text{cm}\) below the top. So the height of the cooked cake \( h=7 - 1.5=5.5\space\text{cm}\)

Step2: Use the cylinder volume formula

The radius \( r = 20\space\text{cm}\) and height \( h = 5.5\space\text{cm}\). Using \( V=\pi r^{2}h\)

Step3: Substitute the values

\( V=\pi\times(20)^{2}\times5.5=\pi\times400\times5.5 = 2200\pi\space\text{cm}^3\approx2200\times3.14=6908\space\text{cm}^3\)

Step1: Find the increase in volume

First, find the difference between the volume of the cooked cake (\( V_{cooked}\)) and the volume of the cake mix (\( V_{mix}\)). From part b, \( V_{mix}=800\pi\) and from part c, \( V_{cooked}=2200\pi\). The increase in volume \( \Delta V=V_{cooked}-V_{mix}=2200\pi - 800\pi=1400\pi\)

Step2: Calculate the percentage increase

The formula for percentage increase is \( \text{Percentage Increase}=\frac{\Delta V}{V_{mix}}\times 100\%\)
Substitute \( \Delta V = 1400\pi\) and \( V_{mix}=800\pi\) into the formula:
\( \text{Percentage Increase}=\frac{1400\pi}{800\pi}\times 100\%=\frac{1400}{800}\times 100\% = 1.75\times 100\%=175\%\)

Answer:

The volume of the cake mix is \( 800\pi\space\text{cm}^3\) (or approximately \( 2512\space\text{cm}^3\))

Part c: