To find out how many months it will take Philip to achieve his savings goal of $3000, we can set the function \( r(t) = 3000 - 500(1.4^t) \) equal to 0, since at that point he has reached his goal.
We start with the equation:
\[ 3000 - 500(1.4^t) = 0 \]
Next, we will isolate \( 1.4^t \):
- Add \( 500(1.4^t) \) to both sides:
\[ 3000 = 500(1.4^t) \]
- Divide both sides by 500:
\[ \frac{3000}{500} = 1.4^t \]
Calculating the left side:
\[ 6 = 1.4^t \]
Now, we will take the logarithm of both sides to solve for \( t \). We can use the natural logarithm (or any logarithm). For this case, we will use the natural logarithm:
\[ \ln(6) = \ln(1.4^t) \]
By the property of logarithms, we can bring the exponent down:
\[ \ln(6) = t \cdot \ln(1.4) \]
Now, we solve for \( t \):
\[ t = \frac{\ln(6)}{\ln(1.4)} \]
Now we will calculate the values using a calculator:
- Calculate \( \ln(6) \):
\[ \ln(6) \approx 1.79176 \]
- Calculate \( \ln(1.4) \):
\[ \ln(1.4) \approx 0.3365 \]
Now plug these values into the equation for \( t \):
\[ t = \frac{1.79176}{0.3365} \approx 5.32 \]
Since \( t \) represents the number of months, and we typically round up since we can't have a fraction of a month in this context:
Thus, it will take Philip approximately 6 months to achieve his savings goal of $3000.