Question

a car enthusiast learns that a particular model of car, which costs $34,411 new, loses 25% of its value every year. how much will the car be worth in 6 years? if necessary, round your answer to the nearest cent.

Explanation:

Step1: Identify the formula for depreciation

The car loses 25% of its value each year, so it retains \(100\% - 25\%= 75\% = 0.75\) of its value each year. The formula for exponential depreciation is \(V = P(1 - r)^t\), where \(V\) is the final value, \(P\) is the initial principal (initial value), \(r\) is the rate of depreciation per period, and \(t\) is the number of periods. Here, \(P=\$34411\), \(r = 0.25\), and \(t = 6\).

Step2: Substitute the values into the formula

Substitute \(P = 34411\), \(r=0.25\), and \(t = 6\) into the formula \(V=P(1 - r)^t\). So we have \(V=34411\times(1 - 0.25)^6=34411\times(0.75)^6\).

Step3: Calculate \((0.75)^6\)

First, calculate \(0.75^6\): \(0.75\times0.75 = 0.5625\), \(0.5625\times0.75=0.421875\), \(0.421875\times0.75 = 0.31640625\), \(0.31640625\times0.75=0.2373046875\), \(0.2373046875\times0.75 = 0.177978515625\).

Step4: Calculate the final value

Multiply \(34411\) by \(0.177978515625\): \(34411\times0.177978515625\approx34411\times0.1779785\). Let's calculate \(34411\times0.1779785\):
\(34411\times0.1 = 3441.1\)
\(34411\times0.07=2408.77\)
\(34411\times0.007 = 240.877\)
\(34411\times0.0009 = 30.9699\)
\(34411\times0.0000785\approx34411\times0.00007 = 2.40877\), \(34411\times0.0000085\approx0.2924935\)
Adding these approximations: \(3441.1+2408.77 = 5849.87\); \(5849.87+240.877 = 6090.747\); \(6090.747+30.9699 = 6121.7169\); \(6121.7169+2.40877 = 6124.12567\); \(6124.12567+0.2924935\approx6124.418\). But a more accurate calculation is \(34411\times0.177978515625 = 34411\times\frac{177978515625}{1000000000000}\). Let's do the exact multiplication:
\(34411\times0.177978515625 = \frac{34411\times177978515625}{1000000000000}\)
\(34411\times177978515625 = 34411\times177978515625\)
We can also use a calculator for \(34411\times0.177978515625\approx34411\times0.1779785\approx6124.42\) (after more precise calculation: \(34411\times0.177978515625 = 34411\times\frac{3^6}{4^6}=34411\times\frac{729}{4096}\))
\(34411\times729 = 34411\times(700 + 29)=34411\times700+34411\times29 = 24087700+997919 = 25085619\)
Then \(\frac{25085619}{4096}\approx6124.42\) (rounded to the nearest cent)

Answer:

\(\$6124.42\)