To find out how long it will take for the investment to triple, we need to solve for $t$ in the equation shown in your work.
Here is the step-by-step solution:
1. Set up the equation
You correctly simplified the problem to:
$$3 = (1.01375)^{4t}$$
This means we are looking for the time $t$ when the growth factor equals 3.
2. Use logarithms to solve for the exponent
To get the variable $t$ out of the exponent position, we take the logarithm of both sides. You can use the natural log ($\ln$) or log base 10 ($\log$). The result will be the same. Let's use $\ln$:
$$\ln(3) = \ln((1.01375)^{4t})$$
Using the power rule for logarithms ($\ln(x^y) = y \cdot \ln(x)$), we can bring the exponent down:
$$\ln(3) = 4t \cdot \ln(1.01375)$$
3. Isolate $t$
Now, divide both sides by $4 \cdot \ln(1.01375)$ to get $t$ by itself:
$$t = \frac{\ln(3)}{4 \cdot \ln(1.01375)}$$
4. Calculate the value
Now we plug the numbers into a calculator:
* $\ln(3) \approx 1.098612$
* $\ln(1.01375) \approx 0.013657$
* $4 \cdot 0.013657 \approx 0.054628$
Now, divide the top by the bottom:
$$t \approx \frac{1.098612}{0.054628} \approx 20.1107...$$
5. Round to the nearest hundredth
The question asks to round to the hundredth place (two decimal places).
The third decimal digit is 0, so we do not round up.
$$t \approx 20.11$$
Final Answer:
20.11
Parent Tip: Review the logic above to help your child master the concept of exponential and logarithmic word problems worksheet.
