A savings app offers two plans, each starting with a \(\pounds150\) bonus. Under Plan A, the total saved after \(y\) years is \(A(y) = 200y+150\) pounds. Un...
Question 1 Report
A savings app offers two plans, each starting with a \(\pounds150\) bonus. Under Plan A, the total saved after \(y\) years is \(A(y) = 200y+150\) pounds. Under Plan B, a growth bonus means the total is \(B(y) = 15y^2+120y+150\) pounds.
- Find the positive value of \(y\) for which the two plans give the same total. (3)
- State, with supporting figures, which plan gives more money after \(8\) years. (2)
Answer Details
(a) The two plans give the same total when \(A(y)=B(y)\): \(15y^{2}+120y+150=200y+150\). Subtracting \(200y+150\) from both sides gives \(15y^{2}-80y=0\). [1 mark] Factorising, \(5y(3y-16)=0\), so \(y=0\) or \(y=\dfrac{16}{3}\). [1 mark] Since \(y=0\) is the starting moment (both plans trivially agree there), the positive value required is \(y=\dfrac{16}{3}\approx5.33\) years. [1 mark]
(b) At \(y=8\): \(A(8)=200(8)+150=1750\), and \(B(8)=15(8)^{2}+120(8)+150=15(64)+960+150=960+960+150=2070\). Since \(2070 > 1750\), Plan B gives more money after \(8\) years. [2 marks]
Plan B's total grows quadratically because of its \(15y^{2}\) growth-bonus term, so although it starts level with Plan A at \(y=0\), it eventually overtakes Plan A permanently once \(y\) passes \(\dfrac{16}{3}\) years.
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