A bucket is in the form of a frustum of a cone and holds 28.490 litres of water. The radii of the top and bottom are 28 cm and 21 cm, respectively. Find the height of the bucket.
15 cm
Step 1: Write down the known values.
Step 2: Convert volume into cubic centimetres (SI unit for volume).
We know \(1\,\text{litre} = 1000\,\text{cm}^3\).
So, \(28.490\,\text{litres} = 28.490 \times 1000 = 28490\,\text{cm}^3\).
Step 3: Recall the formula for the volume of a frustum of a cone.
\[ V = \dfrac{1}{3}\pi h \big(R^2 + r^2 + Rr\big) \]
Step 4: Substitute the values of \(R\) and \(r\).
\(R^2 = 28^2 = 784\)
\(r^2 = 21^2 = 441\)
\(Rr = 28 \times 21 = 588\)
So, \(R^2 + r^2 + Rr = 784 + 441 + 588 = 1813\).
Step 5: Put values in the volume formula.
\(28490 = \dfrac{1}{3}\pi h (1813)\)
Step 6: Simplify the equation.
Multiply denominator first: \(\dfrac{1}{3} \times 1813 = 604.33\).
So, equation becomes: \(28490 = 604.33 \pi h\).
Step 7: Use \(\pi \approx 3.1416\).
\(604.33 \pi \approx 604.33 \times 3.1416 = 1899.56\).
So, \(28490 = 1899.56 h\).
Step 8: Solve for \(h\).
\(h = \dfrac{28490}{1899.56} \approx 15.0\,\text{cm}\).
Final Answer: The height of the bucket is 15 cm.