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.
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