Let's solve the equation step by step.
Step 1: Understand the equation
The given equation is:
[tex]\[
\frac{8400}{5000} = \frac{5000}{5000} \left( \left(1 + \frac{0.075}{4}\right)^{4t} \right)
\][/tex]
Firstly, note that [tex]\(\frac{5000}{5000} = 1\)[/tex], so the equation simplifies to:
[tex]\[
\frac{8400}{5000} = \left(1 + \frac{0.075}{4}\right)^{4t}
\][/tex]
Step 2: Simplify the fraction on the left-hand side
[tex]\[
\frac{8400}{5000} = 1.68
\][/tex]
So the equation now is:
[tex]\[
1.68 = \left(1 + \frac{0.075}{4}\right)^{4t}
\][/tex]
Step 3: Simplify the base on the right-hand side
Calculate the base:
[tex]\[
1 + \frac{0.075}{4} = 1 + 0.01875 = 1.01875
\][/tex]
Thus, the equation becomes:
[tex]\[
1.68 = (1.01875)^{4t}
\][/tex]
Step 4: Solve for [tex]\(4t\)[/tex]
To isolate [tex]\(t\)[/tex], take the natural logarithm (log) of both sides:
[tex]\[
\log(1.68) = \log((1.01875)^{4t})
\][/tex]
Using the logarithm power rule [tex]\(\log(a^b) = b \cdot \log(a)\)[/tex], this simplifies to:
[tex]\[
\log(1.68) = 4t \cdot \log(1.01875)
\][/tex]
Step 5: Isolate [tex]\(t\)[/tex]
Solve for [tex]\(t\)[/tex] by dividing both sides by [tex]\(4 \cdot \log(1.01875)\)[/tex]:
[tex]\[
t = \frac{\log(1.68)}{4 \cdot \log(1.01875)}
\][/tex]
Step 6: Numerical Calculation
Using the known values of logarithms:
[tex]\[
\log(1.68) \approx 0.51879
\][/tex]
[tex]\[
\log(1.01875) \approx 0.018576
\][/tex]
Substitute these values into the equation:
[tex]\[
t = \frac{0.51879}{4 \cdot 0.018576}
\][/tex]
Now, perform the division:
[tex]\[
t = \frac{0.51879}{0.074304} \approx 6.9819
\][/tex]
So, the value of [tex]\(t\)[/tex] is approximately [tex]\(6.98\)[/tex].
Thus, the detailed step-by-step solution:
1. Simplified [tex]\(\frac{8400}{5000}\)[/tex] to 1.68.
2. Calculated the base on the right-hand side as [tex]\(1.01875\)[/tex].
3. Took the logarithm of both sides.
4. Divided by the coefficient to isolate [tex]\(t\)[/tex].
5. Found [tex]\(t \approx 6.98\)[/tex].
