find the time required for an investment of 5000 dollars to grow to 8600 dollars at an interest rate of 7.5…

Explanation:

Step1: Identify the compound interest formula

$$A = P \left(1 + \frac{r}{n}\right)^{nt}$$

Step2: List the given values

$A = 8600$, $P = 5000$, $r = 0.075$, $n = 4$

Step3: Substitute values into the formula

$$8600 = 5000 \left(1 + \frac{0.075}{4}\right)^{4t}$$

Step4: Simplify the equation

$$1.72 = (1.01875)^{4t}$$

Step5: Apply logarithms to solve for $t$

$$\ln(1.72) = 4t \ln(1.01875)$$

Step6: Isolate the variable $t$

$$t = \frac{\ln(1.72)}{4 \ln(1.01875)}$$

Step7: Calculate the final value

$$t \approx 7.31$$

Answer:

7.31