Let us solve the provided questions step by step.

Question 1

The amount of money $M$ in the piggy bank after $t$ days is given by: $M(t) = 2t^3 + 2t + 10$

(a) How much money is in the piggy bank after 3 days?

Substitute $t = 3$ into $M(t)$: $M(3) = 2(3)^3 + 2(3) + 10$ $M(3) = 2(27) + 6 + 10 = 54 + 6 + 10 = 70$

Answer: The amount of money in the piggy bank after 3 days is 70 dollars.

(b) Determine the average rate of change of the amount of money in the piggy bank from day 1 to day 3.

The average rate of change formula is: $\text{Average rate of change} = \frac{M(3) - M(1)}{3 - 1}$

First, calculate $M(1)$: $M(1) = 2(1)^3 + 2(1) + 10 = 2(1) + 2 + 10 = 14$

Now, calculate the average rate of change: $\text{Average rate of change} = \frac{M(3) - M(1)}{3 - 1} = \frac{70 - 14}{2} = \frac{56}{2} = 28$

Answer: The average rate of change is 28 dollars per day.

(c) Estimate the instantaneous rate of change of the amount of money in the piggy bank after 2 days.

The instantaneous rate of change is the derivative of $M(t)$ at $t = 2$.

First, find $M'(t)$ (the derivative of $M(t)$): $M(t) = 2t^3 + 2t + 10$ $M'(t) = 6t^2 + 2$

Now, substitute $t = 2$ into $M'(t)$: $M'(2) = 6(2)^2 + 2 = 6(4) + 2 = 24 + 2 = 26$

Answer: The instantaneous rate of change at $t = 2$ is 26 dollars per day.

Question 2

The volume of the rectangular box is given as: $V(x) = x^3 + 16x^2 + 83x + 140 \, \text{cm}^3$

The box has dimensions:

• Length: $(x + 5)$,
• Width: $(x + 7)$,
• Height: unknown.

We need to determine the height.

To find the height, divide the volume $V(x)$ by the product of the length and width, $(x + 5)(x + 7)$.

First, expand $(x + 5)(x + 7)$: $(x + 5)(x + 7) = x^2 + 7x + 5x + 35 = x^2 + 12x + 35$

Now, divide $V(x)$ by $(x^2 + 12x + 35)$ using polynomial division.

Step 1: Divide the leading terms

$\frac{x^3}{x^2} = x$ Multiply $x$ by $x^2 + 12x + 35$: $x(x^2 + 12x + 35) = x^3 + 12x^2 + 35x$ Subtract from $V(x)$: $(x^3 + 16x^2 + 83x + 140) - (x^3 + 12x^2 + 35x) = 4x^2 + 48x + 140$

Step 2: Divide the leading terms again

$\frac{4x^2}{x^2} = 4$ Multiply $4$ by $x^2 + 12x + 35$: $4(x^2 + 12x + 35) = 4x^2 + 48x + 140$ Subtract: $(4x^2 + 48x + 140) - (4x^2 + 48x + 140) = 0$

Thus, the quotient is: $x + 4$

Answer: The height of the box is $x + 4 \, \text{cm}$.

Do you want more details or have any questions?

Related Questions:

1. How would you interpret the average rate of change physically in this context?
2. How is the instantaneous rate of change calculated at any arbitrary time $t$?
3. What would happen to the box's height if one of the dimensions changes?
4. What other methods could be used to calculate polynomial division (e.g., synthetic division)?
5. Can the height of the box ever be negative? Why or why not?

Tip: When solving problems involving rates of change, always check whether you’re asked for an average or an instantaneous rate. Use derivatives only for instantaneous rates.