A family's total savings, in pounds, after \(n\) weeks are modelled by \(T = n^2 + 3n\). They want to know how many weeks it takes to reach total savings of...
Question 1 Report
A family's total savings, in pounds, after \(n\) weeks are modelled by \(T = n^2 + 3n\). They want to know how many weeks it takes to reach total savings of \(\pounds 180\).
- Use the savings model to show that \(n\) satisfies \(n^2 + 3n - 180 = 0\). (1)
- Solve the equation by factorising. (2)
- State which solution fits the context, giving a reason. (1)
Answer Details
This question builds a quadratic equation from a savings target, solves it by factorising, and selects the root that fits a time context.
Setting the savings model equal to the target amount:
\[n^2 + 3n = 180\]which rearranges to \(n^2 + 3n - 180 = 0\), as required [1 mark].
Looking for two numbers that multiply to \(-180\) and add to \(3\): these are \(15\) and \(-12\), so:
\[(n-12)(n+15) = 0\]giving \(n = 12\) or \(n = -15\) [2 marks].
The number of weeks cannot be negative, so the value that fits the context is \(n = 12\) [1 mark].
Checking: after \(12\) weeks, \(T = 12^2 + 3(12) = 144+36 = 180\), confirming the target is reached at exactly \(n=12\) weeks.
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