Question

The value of a mountain bike y (in dollars) can be approximated by the model y=200(0.75)t, where t is the number of years since the bike was new. Would this table show growth or a decline (decay)? How do you know? Identify the annual percent increase or decrease in the value of the bike. Estimate when the value of the bike will be $50.​

Answer 1

Explanation:

To determine if the table shows growth or decay, we can look at the coefficient of the exponential function y = 200(0.75)^t. The coefficient is 0.75, which is less than 1. This means that the value of the bike decreases over time, so the table shows decay.

To identify the annual percent increase or decrease in the value of the bike, we can compare the value of the bike after one year to its initial value. Plugging in t = 1 into the model, we get:

y = 200(0.75)^1

y = 150

So after one year, the value of the bike is $150. This represents a decrease of $50 from its initial value of $200. To find the percent decrease, we can use the formula:

percent decrease = (amount of decrease / initial value) x 100%

Plugging in the values, we get:

percent decrease = (50 / 200) x 100%

percent decrease = 25%

Therefore, the value of the bike decreases by 25% each year.

To estimate when the value of the bike will be $50, we can set y = 50 in the model and solve for t:

50 = 200(0.75)^t

0.25 = 0.75^t

log(0.25) = t log(0.75)

t = log(0.25) / log(0.75)

t ≈ 4.2

Therefore, the value of the bike will be $50 approximately 4.2 years after it was new.