To solve the given problem, let's analyze the function representing the amount of money in the bank account after $t$ years, which is given by:

$M(t) = 18 + 3t - t^2$

Part (a): Increase During Year 3

The increase in the bank account during year 3 can be found by calculating the difference in the amount of money at the end of year 3 and the end of year 2.

$M(3) = 18 + 3(3) - (3)^2 = 18 + 9 - 9 = 18$ $M(2) = 18 + 3(2) - (2)^2 = 18 + 6 - 4 = 20$

The increase during year 3 is: $\Delta M = M(3) - M(2) = 18 - 20 = -2$

So, the bank account decreased by 2 thousand dollars during year 3.

Part (b): Average Rate of Increase During Year 3

The average rate of increase of money in the account during year 3 is calculated by dividing the change in the amount of money by the length of the time period.

$\text{Average Rate of Increase} = \frac{\Delta M}{\Delta t} = \frac{M(3) - M(2)}{3 - 2} = \frac{18 - 20}{1} = -2 \text{ thousand dollars per year}$

Part (c): Instantaneous Rate of Increase at the Start of Each Quarter in Year 3

The instantaneous rate of increase is given by the derivative of $M(t)$, which is $M'(t)$.

$M(t) = 18 + 3t - t^2$ $M'(t) = \frac{d}{dt}(18 + 3t - t^2) = 3 - 2t$

We need to evaluate $M'(t)$ at the start of each quarter in year 3.

c1) Start of first quarter (t = 2): $M'(2) = 3 - 2(2) = 3 - 4 = -1 \text{ thousand dollars per year}$

c2) Start of second quarter (t = 2.25): $M'(2.25) = 3 - 2(2.25) = 3 - 4.5 = -1.5 \text{ thousand dollars per year}$

c3) Start of third quarter (t = 2.5): $M'(2.5) = 3 - 2(2.5) = 3 - 5 = -2 \text{ thousand dollars per year}$

c4) Start of fourth quarter (t = 2.75): $M'(2.75) = 3 - 2(2.75) = 3 - 5.5 = -2.5 \text{ thousand dollars per year}$

Part (d): When Did the Bank Account Run Out of Money?

The bank account runs out of money when $M(t) = 0$.

$18 + 3t - t^2 = 0$ Solving this quadratic equation: $t^2 - 3t - 18 = 0$ $(t - 6)(t + 3) = 0$

So, $t = 6 \text{ or } t = -3$

Since $t = -3$ is not a valid time (negative time), the bank account runs out of money after $t = 6$ years.

Summary

1. The bank account decreased by 2 thousand dollars during year 3.
2. The average rate of increase during year 3 was -2 thousand dollars per year.
3. The instantaneous rates of increase at the start of each quarter in year 3 were:
   • First quarter: -1 thousand dollars per year
   • Second quarter: -1.5 thousand dollars per year
   • Third quarter: -2 thousand dollars per year
   • Fourth quarter: -2.5 thousand dollars per year
4. The bank account ran out of money after 6 years.

Would you like any more details or have any other questions?

Here are five more questions you might want to ask:

1. How to find the maximum amount of money in the bank account?
2. How to determine the time when the bank account starts decreasing?
3. What is the significance of the vertex of the quadratic function?
4. How to find the rate of change at any given time $t$?
5. How to graph the function $M(t) = 18 + 3t - t^2$?

Tip: When solving quadratic equations, always check the validity of the solutions within the context of the problem.