Question

Philip has a goal of saving $3000. The function s(t)=500 x 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 solve the problems, let's look at each part systematically:

### Part A: Build the function \( r(t) \)

Given that \( s(t) = 500 \times 1.4^t \) represents the amount Philip has saved after \( t \) months and his goal is to save $3000, we want to find the function \( r(t) \) representing the amount of money Philip has left to save.

The amount he has left to save can be expressed as:

\[
r(t) = \text{Goal} - \text{Amount saved}
\]

Thus, we can write:

\[
r(t) = 3000 - s(t)
\]

Now substituting \( s(t) \) into the equation:

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

So, the function \( r(t) \) is:

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

### Part B: After how many months will Philip have saved at least $3000?

We need to find the smallest integer \( t \) such that \( s(t) \geq 3000 \).

Setting up the inequality:

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

Dividing both sides by 500:

\[
1.4^t \geq 6
\]

Now, we take the logarithm of both sides. Using logarithms (base 10 or natural logarithm), we can rewrite it as:

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

Solving for \( t \):

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

Using a calculator to evaluate the right side:

- \(\log(6) \approx 0.7781\)
- \(\log(1.4) \approx 0.1461\)

So,

\[
t \geq \frac{0.7781}{0.1461} \approx 5.33
\]

Since we need \( t \) as a whole number, we round up to the nearest whole number, which gives us:

\[
t = 6
\]

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