Sure, let’s go step by step to solve the given problem.
Given:
- The bucket is in the shape of a frustum.
- The bottom radius of the frustum [tex]\( r_1 = 18 \)[/tex] cm.
- The top radius of the frustum [tex]\( r_2 = 28 \)[/tex] cm.
- The height of the frustum [tex]\( h_f = 30 \)[/tex] cm.
We need to find the volume of the frustum first. The formula for the volume [tex]\( V \)[/tex] of a frustum is:
[tex]\[ V_{\text{frustum}} = \frac{1}{3} \pi h \left( R^2 + r^2 + Rr \right) \][/tex]
Substituting the given values:
[tex]\[ R = 28 \text{ cm} \][/tex]
[tex]\[ r = 18 \text{ cm} \][/tex]
[tex]\[ h = 30 \text{ cm} \][/tex]
[tex]\[ V_{\text{frustum}} = \frac{1}{3} \pi \times 30 \left( 28^2 + 18^2 + 28 \times 18 \right) \][/tex]
After calculating this expression, we get:
[tex]\[ V_{\text{frustum}} \approx 50642.47 \text{ cm}^3 \][/tex]
Next, we need to determine the height of water when the frustum-shaped water is poured into a cylindrical container with radius [tex]\( r_{\text{cylinder}} = 20 \text{ cm} \)[/tex].
The volume of a cylinder [tex]\( V \)[/tex] is calculated using the formula:
[tex]\[ V_{\text{cylinder}} = \pi r^2 h \][/tex]
Here, we need to solve for height [tex]\( h \)[/tex] given:
[tex]\[ V_{\text{cylinder}} = V_{\text{frustum}} \][/tex]
So,
[tex]\[ V_{\text{cylinder}} = \pi \times 20^2 \times h \][/tex]
Since the volume of the cylinder must equal the volume of the frustum:
[tex]\[ 50642.47 = \pi \times 400 \times h \][/tex]
Dividing both sides by [tex]\( \pi \times 400 \)[/tex], we get the height [tex]\( h \)[/tex] of the water in the cylinder:
[tex]\[ h = \frac{50642.47}{\pi \times 400} \][/tex]
[tex]\[ h \approx 40.30 \text{ cm} \][/tex]
So, the depth of the water in the cylindrical container is approximately 40.30 cm.
