new car is purchased for $$ \$ 11,000 $$ and over time its value depreciates by one half every 5.5 years. Write a function showing the alue of the car after $$ t $$ years, where the arnual decay rate can be found from a constant in the function. Round all coefficients in the unction to four decimal places. Also, determine the percentage rate of decay per year, to the neorest hundredth of a pencent.

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We start with the information that the car’s value halves every 5.5 years. This means that when t = 5.5:

V(5.5) = 11000 · a^(5.5) = 11000/2

Dividing both sides by 11000 gives:

a^(5.5) = 1/2

To solve for the annual decay factor a, take the 5.5th root (or raise both sides to the 1/5.5 power):

a = (1/2)^(1/5.5)  
    = 2^(–1/5.5)

Now, we want a numerical value for a rounded to four decimal places. Compute the exponent first:

1/5.5 ≈ 0.1818

So:

a ≈ 2^(–0.1818)

Another way to express this is by using the natural exponential (since 2^(–0.1818) = e^(–0.1818·ln2) ). Since ln2 ≈ 0.6931, then:

a ≈ e^(–0.1818 × 0.6931)
    ≈ e^(–0.1260)
    ≈ 0.8812

Thus, the annual decay factor is about 0.8812, meaning that each year the value is about 88.12% of the previous year’s value.

The function for the value of the car after t years is therefore:

V(t) = 11000 · (0.8812)^t

Next, to find the annual percentage rate of decay, notice that if the value each year is multiplied by 0.8812 then the percentage loss per year is:

1 – 0.8812 = 0.1188, or approximately 11.88%

Therefore:

• The value function is: V(t) = 11000 · (0.8812)^t  
• The annual percentage rate of decay is: ≈11.88%

All coefficients have been rounded to four decimal places as required, and the percentage is rounded to the nearest hundredth of a percent.
The car's value after $$ t $$ years is given by the function: $$ V(t) = 11000 \times (0.8812)^t $$ Each year, the car depreciates by approximately **11.88%**.

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