A school sports day water cooler is a cylindrical tank of radius 18 cm and height 55 cm, on a conical stand of the same radius and height 20 cm, as shown. W...
Question 1 Report
A school sports day water cooler is a cylindrical tank of radius 18 cm and height 55 cm, on a conical stand of the same radius and height 20 cm, as shown. Water drains out at 250 \(\text{cm}^3\) per minute once the event begins.
- Work out the volume of the cylindrical tank. (2)
- Work out the volume of the conical stand. (2)
- Work out the total volume of the whole unit, tank and stand together. (1)
- The tank starts full. Work out how long, in minutes to the nearest minute, it takes to empty. (2)
- The event lasts 75 minutes. State, with a reason, whether the tank needs refilling before it ends. (1)
Answer Details
The whole unit is a cylindrical tank sitting on a conical stand, so its total volume is the sum of the two separate solids' volumes; draining time then follows from dividing the tank's own volume by the constant drain rate.
(a) Substituting \(r=18\) and \(h=55\) into the cylinder volume formula:
\[\pi \times 18^2 \times 55 = 55983.2... \text{ cm}^3\] [2 marks]
(b) Substituting \(r=18\) and \(h=20\) into the cone volume formula:
\[\frac{1}{3} \times \pi \times 18^2 \times 20 = 6785.8... \text{ cm}^3\] [2 marks]
(c) Adding the two volumes gives the total volume of the whole unit:
\[55983.2 + 6785.8 = 62769.0 \text{ cm}^3\] [1 mark]
(d) Only the cylindrical tank holds water that drains away, so dividing its volume by the drain rate:
\[55983.2 \div 250 = 223.9... \approx 224 \text{ minutes}\] [2 marks]
(e) Since 224 minutes is longer than the 75-minute event, the tank will not need refilling before the event ends. [1 mark]
Exam tip: read carefully which part of a compound solid actually holds the draining liquid; the conical stand here is solid support, not part of the water capacity, so it plays no part in the draining time.
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