Let's solve each question step by step.

Question 2:

A business deposits $40,000 in a fund that pays 5% interest, compounded continuously. Determine the balance in this account after 30 years.

The formula for continuous compounding is: $A = P e^{rt}$

Where:

• $P = 40,000$ (initial deposit)
• $r = 0.05$ (interest rate, 5%)
• $t = 30$ years
• $e$ is the base of the natural logarithm (approximately 2.71828)

Substitute the values: $A = 40,000 \cdot e^{0.05 \cdot 30}$

First, calculate the exponent: $0.05 \cdot 30 = 1.5$

Now calculate $e^{1.5}$: $e^{1.5} \approx 4.481689$

Finally, calculate the balance: $A = 40,000 \cdot 4.481689 \approx 179,267.56$

Answer for Question 2: The balance after 30 years is $179,267.56.

Question 3:

The cost of goods/services is modeled as: $C(t) = P \cdot (1.05)^t$

Where:

• $P = 23.91$ (present cost of the oil change)
• $t = 15$ years
• $1.05$ is the inflation rate (5% per year)

Substitute the values: $C(15) = 23.91 \cdot (1.05)^{15}$

First, calculate $(1.05)^{15}$: $(1.05)^{15} \approx 2.07893$

Now calculate the cost: $C(15) = 23.91 \cdot 2.07893 \approx 49.65$

Answer for Question 3: The estimated cost of the oil change after 15 years is $49.65.

Let me know if you need further clarifications or detailed steps!

Related Questions:

1. How is the formula for continuous compounding derived?
2. What would the balance be in Question 2 if the interest rate were 6% instead of 5%?
3. In Question 3, how does changing the inflation rate to 4% affect the final cost?
4. What is the difference between continuous compounding and annual compounding in terms of growth?
5. How can the exponential growth formula $C(t) = P(1 + r)^t$ be used to approximate inflation over shorter time periods?

Tip:

For continuous compounding, remember that the exponential growth is faster than standard compounding as time increases! Always use the exact value of $e$ (or a calculator) for accurate results.