Question

The value of a particular item can be modeled by P(t) = P₀(a)ᵗ

where P is in dollars and t is the number of years since the item was purchased. Suppose the value of the item increases 8% each year and the item was purchased for $80.
(a) Write a formula for P(t) according to the model.
(b) How fast is the value of the item increasing when t = 10 years? Round your answer to two decimal places.

Answer 1

The formula for P(t) is P(t) = 80(1.08)ᵗ and the rate of increase in the value of the item at t = 10 years can be calculated using the derivative of this formula.

Step-by-step explanation:

(a) The formula for P(t) according to the model is P(t) = P₀(a)ᵗ. In this case, P₀ is the initial value of the item, which is $80, and a is the growth factor, which is 1.08 (since the value of the item increases by 8% each year). Therefore, the formula for P(t) is P(t) = 80(1.08)ᵗ.

(b) To find how fast the value of the item is increasing when t = 10 years, we need to find the derivative of P(t) with respect to t. Taking the derivative of P(t) = 80(1.08)ᵗ, we get dP/dt = 80(1.08)ᵗ * ln(1.08). Plugging in t = 10, we can calculate the value of dP/dt, which represents the rate of increase in the value of the item at t = 10 years.