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

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.