Answer :
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].
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].