As part of household budgeting, a family models their heating cost, \(C\) pounds per month, as directly proportional to the square of the thermostat setting...
Question 1 Report
As part of household budgeting, a family models their heating cost, \(C\) pounds per month, as directly proportional to the square of the thermostat setting, \(t\) degrees above a fixed baseline. At \(t = 4\), \(C = £19.20\).
- Find the constant of proportionality. (2)
- Work out the cost when \(t = 6\). (2)
- The family currently pays \(£28.80\) per month. Without finding \(t\), show that raising the thermostat setting by \(50\%\) makes the new monthly cost exceed \(£60\). (2)
Answer Details
"Directly proportional to the square of \(t\)" means \( C = kt^2 \) for a fixed constant \(k\), so a given percentage increase in the thermostat setting causes a larger percentage increase in cost.
(a) Substituting \( t = 4 \), \( C = 19.20 \) gives \( 19.20 = k \times 4^2 = 16k \), so \( k = 19.20 \div 16 = 1.2 \) [2 marks].
(b) Using this constant with \( t = 6 \) gives \( C = 1.2 \times 6^2 = 1.2 \times 36 = £43.20 \) [2 marks].
(c) Raising the thermostat setting by 50% multiplies \( t \) by \( 1.5 \). Since \( C \) is proportional to \( t^2 \), the cost is multiplied by \( 1.5^2 = 2.25 \), regardless of the actual value of \( t \) [1 mark]. Applying this to the current cost of £28.80 gives a new cost of \( 28.80 \times 2.25 = £64.80 \), which is greater than £60, as required [1 mark].
This method avoids ever finding the actual thermostat setting: because cost is proportional to the square of the setting, any percentage change in the setting can be converted straight into the corresponding multiplier for cost, here \( 1.5^2 = 2.25 \).
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