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.
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