To find the amount of time required for the loan to reach $36,000, we solve the equation 36,000 = 32,000(1.05)^t. The time is approximately 7.12 years. To find the doubling time, we solve the equation 64,000 = 32,000(1.05)^t. The doubling time is approximately 14.23 years.
To solve part (a), we need to find the value of t when the amount due, A(t), reaches $36,000. The equation for the amount due is given by A(t) = 32,000(1.05)^t. Plugging in the given value, we have:
$36,000 = 32,000(1.05)^t
Dividing both sides by 32,000, we get:
1.125 = (1.05)^t
To solve for t, we can take the logarithm of both sides:
log(1.125) = log((1.05)^t)
Using logarithm properties, we can bring the exponent down:
log(1.125) = t * log(1.05)
Calculating the values using a calculator, we find that t ≈ 7.12. Therefore, it will take approximately 7.12 years for the amount due to reach $36,000.
To solve part (b) and find the doubling time, we need to find the value of t when the amount due, A(t), is twice the original loan amount of $32,000. The equation for the doubling time is given by:
$64,000 = 32,000(1.05)^t
Dividing both sides by 32,000 and simplifying, we get:
2 = (1.05)^t
Using the same steps as before, we find that the doubling time is approximately 14.23 years.