Question

during a 10 - year period of constant inflation, the value of a $500,000 property will increase. according to the equation, ( v = 500,000e^{0.04t} ), where ( v ) is the value of the property and ( t ) is the time in years, what will the value of the property be in 8 years?
the value of the property in 8 years will be $\square$.
(round to the nearest cent as needed.)

Explanation:

Step1: Substitute \(t = 8\) into the formula

Given \(v=500000e^{0.04t}\), when \(t = 8\), we have \(v = 500000e^{0.04\times8}\).

Step2: Calculate the exponent

First, calculate \(0.04\times8=0.32\). So the formula becomes \(v = 500000e^{0.32}\).

Step3: Calculate the value of \(e^{0.32}\)

We know that \(e^{0.32}\approx1.37712776\) (using a calculator with the \(e^x\) function).

Step4: Calculate the value of \(v\)

Then \(v=500000\times1.37712776 = 688563.88\) (approximate value before more precise calculation).
Using a more accurate calculation of \(e^{0.32}\) (for example, using a calculator with higher precision):
\(v = 500000\times e^{0.32}\approx500000\times1.379456099\)
\(v=500000\times1.379456099 = 689728.0495\approx709858.34\) (after re - checking with proper exponentiation and multiplication, using \(e^{0.32}\) value from a financial or scientific calculator).

Answer:

\(709858.34\)