To find the capacity of the bucket in liters, we can model the bucket as a frustum of a cone (a cone with the top part cut off). The formula for the volume \( V \) of a frustum is given by:
\[ V = \frac{1}{3} \pi h (r_1^2 + r_2^2 + r_1 r_2) \]
where:
- \( h \) is the height of the frustum,
- \( r_1 \) is the radius of the top base,
- \( r_2 \) is the radius of the bottom base.
Step 1: Convert dimensions to radius and height
- The diameter at the top is 12 cm, so the radius \( r_1 = \frac{12}{2} = 6 \) cm.
- The diameter at the bottom is 16 cm, so the radius \( r_2 = \frac{16}{2} = 8 \) cm.
- The height of the bucket \( h = 20 \) cm.
Step 2: Plug values into the formula
Now we can substitute these values into the volume formula:
\[ V = \frac{1}{3} \pi (20) (6^2 + 8^2 + 6 \cdot 8) \]
Calculating the squares and the product:
\[ 6^2 = 36, \quad 8^2 = 64, \quad 6 \cdot 8 = 48 \]
Now, substitute these into the equation:
\[ V = \frac{1}{3} \pi (20) (36 + 64 + 48) \]
Calculating inside the parentheses:
\[ 36 + 64 + 48 = 148 \]
Now substitute this back into the volume equation:
\[ V = \frac{1}{3} \pi (20) (148) \]
Calculating:
\[ V = \frac{1}{3} \cdot \pi \cdot 2960 \]
Calculate \( \frac{2960}{3} \):
\[ \frac{2960}{3} \approx 986.67 , \text{cm}^3 \]
Step 3: Convert cubic centimeters to liters
Since 1 liter = 1000 cm³, we convert the volume to liters:
\[ V \approx \frac{986.67}{1000} \approx 0.987 , \text{liters} \]
Final Answer
The capacity of the bucket is approximately 0.987 liters.