1. **State the problem:** Kyle invests money in a simple savings fund where the amount increases at a constant rate. We know the amount after 3 years is 16500 and after 6 years is 18000. We want to find: a) A rule linking amount $A$ and time $t$. b) The initial investment (when $t=0$). c) How long until the amount reaches 20000. d) The amount after 12.5 years. 2. **Formula and rules:** Since the amount increases at a constant rate, this is a linear relationship: $$A = mt + c$$ where $m$ is the rate of increase per year and $c$ is the initial amount. 3. **Find $m$ and $c$ using given points:** Given points: $(3,16500)$ and $(6,18000)$. Calculate slope $m$: $$m = \frac{18000 - 16500}{6 - 3} = \frac{1500}{3} = 500$$ 4. **Find $c$ by substituting one point:** Using $(3,16500)$: $$16500 = 500 \times 3 + c$$ $$16500 = 1500 + c$$ $$c = 16500 - 1500 = 15000$$ 5. **Rule linking $A$ and $t$:** $$\boxed{A = 500t + 15000}$$ 6. **Initial investment (when $t=0$):** $$A = 500 \times 0 + 15000 = 15000$$ So, Kyle initially invested 15000. 7. **Time to reach 20000:** Set $A=20000$: $$20000 = 500t + 15000$$ Subtract 15000: $$20000 - 15000 = 500t$$ $$5000 = 500t$$ $$\cancel{5000} = \cancel{500}t$$ $$t = \frac{5000}{500} = 10$$ Kyle must wait 10 years. 8. **Value after 12.5 years:** $$A = 500 \times 12.5 + 15000 = 6250 + 15000 = 21250$$ The investment will be 21250 after 12.5 years.