Question

Philip has a goal of saving \$3000. The function s(t) = 500 * 1.4^t represents the amount of money Philip has saved after t months.
A. Build a function r(t) to represent the amount of money Philip has left to save at t months.

B. After how many months will Philip have saved at least \$3000? Enter your answer as a whole number.

Answer 1

To answer your questions, we'll break them into parts.

### A. Build a function \( r(t) \)
To find the amount of money Philip has left to save at time \( t \), we need to subtract the amount he has saved \( s(t) \) from his goal of saving \$3000.

Given:
- \( s(t) = 500 \cdot 1.4^t \)
- Goal = \$3000

The function \( r(t) \) representing the amount he has left to save is:

\[
r(t) = 3000 - s(t) = 3000 - 500 \cdot 1.4^t
\]

### B. After how many months will Philip have saved at least \$3000?
We need to solve the inequality:

\[
s(t) \geq 3000
\]

Substituting for \( s(t) \):

\[
500 \cdot 1.4^t \geq 3000
\]

Dividing both sides by 500:

\[
1.4^t \geq 6
\]

Now, we take the logarithm of both sides. We'll use the natural logarithm (ln) but you can use log base 10 or any logarithmic base:

\[
\ln(1.4^t) \geq \ln(6)
\]

Using the power rule for logarithms:

\[
t \cdot \ln(1.4) \geq \ln(6)
\]

Now, solve for \( t \):

\[
t \geq \frac{\ln(6)}{\ln(1.4)}
\]

Now, we can calculate \( \ln(6) \) and \( \ln(1.4) \):

- \( \ln(6) \approx 1.79176 \)
- \( \ln(1.4) \approx 0.3365 \)

Now substitute these values into the equation:

\[
t \geq \frac{1.79176}{0.3365} \approx 5.32
\]

Since \( t \) must be a whole number, we round up to the next whole number:

\[
t \geq 6
\]

Thus, Philip will have saved at least \$3000 after **6 months**.

### Summary of Answers:
A. \( r(t) = 3000 - 500 \cdot 1.4^t \)

B. Philip will have saved at least \$3000 after **6 months**.