An equation for the depreciation of a car is given by y = A(1 – r)t , where y = current value of the car, A = original cost, r = rate of depreciation, and t = time, in years. The value of a car is half what it originally cost. The rate of depreciation is 10%. Approximately how old is the car?

Question

An equation for the depreciation of a car is given by y = A(1 – r)t , where y = current value of the car, A = original cost, r = rate of depreciation, and t = time, in years. The value of a car is half what it originally cost. The rate of depreciation is 10%. Approximately how old is the car?

Answer 1

just solve for t in

(1-0.10)^t = 0.5
.9^t = .5
t = log 0.5 / log 0.9

Answer 2

To determine the age of the car, we need to find the value of "t" in the equation y = A(1 - r)t.

We know that the current value of the car, represented by "y," is half of its original cost (A). So, we can substitute y = A/2 into the equation:

A/2 = A(1 - 0.10)t

Next, we can simplify the equation:

1/2 = (1 - 0.10)t [dividing both sides by A]

1/2 = 0.9t [simplifying the right side]

To isolate "t," we can take the logarithm of both sides of the equation. Let's use the natural logarithm (ln):

ln(1/2) = ln(0.9t)

Now, we can solve for "t" by dividing both sides by ln(0.9):

t = ln(1/2) / ln(0.9)

Using a calculator, we can evaluate the right side and find:

t ≈ 6.579

Therefore, the car is approximately 6.579 years old.

Answer 3

To find how old the car is, we can use the equation: y = A(1 - r)^t, where y is the current value of the car, A is the original cost, r is the rate of depreciation, and t is the time in years.

In this case, the value of the car, y, is given as half of what it originally cost, A. So we can say y = 0.5A.

We are also given that the rate of depreciation, r, is 10%. This can be written as a decimal, so r = 0.10.

Substituting the given values into the equation, we have:
0.5A = A(1 - 0.10)^t

Simplifying, we have:
0.5 = (1 - 0.10)^t

Now, let's solve for t. We can take the natural logarithm of both sides of the equation to isolate t:

ln(0.5) = ln[(1 - 0.10)^t]

Using properties of logarithms, we can bring the power t in front of the logarithm:

ln(0.5) = t * ln(1 - 0.10)

Now we can divide both sides of the equation by ln(1 - 0.10) to solve for t:

t = ln(0.5) / ln(1 - 0.10)

Using a calculator, we can find the approximate value of t:

t ≈ 6.931 / -0.0953

t ≈ -72.65

Since time cannot be negative, we can ignore the negative value and conclude that the car is approximately 73 years old.