A: To build the function \( r(t) \) that represents the amount of money Philip has left to save at \( t \) months, we need to subtract the amount he has saved \( s(t) \) from his goal of $3000.
Since the function for the amount of money he has saved is given by:
\[ s(t) = 500 \times 1.4^t \]
The function \( r(t) \) representing the amount he has left to save can be expressed as:
\[ r(t) = 3000 - s(t) = 3000 - (500 \times 1.4^t) \]
Thus,
\[ r(t) = 3000 - 500 \times 1.4^t \]
B: To find the number of months \( t \) after which Philip will have saved at least $3000, we need to solve the inequality:
\[ s(t) \geq 3000 \]
Substituting the function for \( s(t) \):
\[ 500 \times 1.4^t \geq 3000 \]
Now, we can divide both sides by 500:
\[ 1.4^t \geq 6 \]
Next, we take the logarithm of both sides to solve for \( t \):
\[ \log(1.4^t) \geq \log(6) \]
Using the power rule of logarithms:
\[ t \log(1.4) \geq \log(6) \]
Now, we can solve for \( t \):
\[ t \geq \frac{\log(6)}{\log(1.4)} \]
Using approximate values for the logarithms (using base 10):
\[ \log(6) \approx 0.7781 \quad \text{and} \quad \log(1.4) \approx 0.1461 \]
Calculating \( t \):
\[ t \geq \frac{0.7781}{0.1461} \approx 5.32 \]
Since we are looking for the smallest whole number of months, we round up to the nearest whole number:
\[ t \geq 6 \]
Therefore, after 6 months, Philip will have saved at least $3000.