Answer:
The depth is 5.33cm
Step-by-step explanation:
Given
[tex]D =20cm[/tex] --- diameter at open end
[tex]d = 12cm[/tex] --- diameter at bottom
[tex]H=16cm[/tex] -- depth
[tex]d_c = 28cm[/tex] --- diameter of the cylinder
Required
The depth the bucket will fill the cylinder
First, calculate the radii at the ends of the bucket
[tex]R=D/2 =20cm/2 = 10cm[/tex]
[tex]r=d/2 =12cm/2 = 6cm[/tex]
The volume of the bucket (frustum) is:
[tex]V = \frac{1}{3} \pi H(R^2 + Rr + r^2)[/tex]
[tex]V = \frac{1}{3} *\pi * 16 * (10^2 + 10*6 + 6^2)[/tex]
[tex]V = \frac{1}{3} *\pi * 16 * 196[/tex]
[tex]V = 1045.33\pi cm^3[/tex]
The volume of a cylinder is:
[tex]V = \pi r_c^2h_c[/tex]
Where:
[tex]r_c = d_c/2 = 28cm/2 =14cm[/tex]
So, we have:
[tex]1045.33\pi = \pi * 14^2 * h_c[/tex]
[tex]1045.33\pi = \pi * 196 * h_c[/tex]
[tex]1045.33\pi = 196\pi * h_c[/tex]
Make h the subject
[tex]h_c = \frac{1045.33\pi}{ 196\pi}[/tex]
[tex]h_c = \frac{1045.33}{ 196}[/tex]
[tex]h_c = 5.33[/tex]
