Cold drinks at the club pool come in two similar cone shaped cups, 9 cm and 12 cm tall. Write down the scale factor from the small cup to the large cup. (1)...
Question 1 Report
Cold drinks at the club pool come in two similar cone shaped cups, 9 cm and 12 cm tall.
- Write down the scale factor from the small cup to the large cup. (1)
- The small cup has curved surface area 189 cm². Find that of the large cup. (2)
- The large cup holds 640 ml. Find what the small cup holds. (2)
Answer Details
Between similar solids, lengths scale by \(k\), surface areas by \(k^2\) and volumes by \(k^3\).
(a) The scale factor from the smaller cup to the larger is \(\frac{12}{9} = \frac{4}{3}\). [1]
(b)
- The area scale factor is \(\left(\frac{4}{3}\right)^2 = \frac{16}{9}\). [1]
- Curved surface area of the larger cup \(= 189 \times \frac{16}{9} = 21 \times 16 = 336\) cm\(^2\). [1]
(c)
- Going from the larger cup to the smaller reverses the length factor to \(\frac{3}{4}\), so the volume factor is \(\left(\frac{3}{4}\right)^3 = \frac{27}{64}\). [1]
- Capacity of the smaller cup \(= 640 \times \frac{27}{64} = 10 \times 27 = 270\) ml. [1]
Part (c) runs in the opposite direction to parts (a) and (b), so the scale factor must be inverted before cubing; using \(\left(\frac{4}{3}\right)^3\) would enlarge rather than reduce and give 1517 ml, more than the larger cup itself.
Working with the fractions rather than decimals keeps the arithmetic exact, since \(\frac{16}{9}\) and \(\frac{27}{64}\) both cancel neatly against the given figures.
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