Suppose that the future price pt of a certain item is given

Suppose that the future price pt of a certain item is given by the following exponential function. In this function, pt is measured in dollars and t is the number of years from today. =pt15001.026t Find the initial price of the item. $ Does the function represent growth or decay?growthdecayBy what percent does the price change each year? %

Solution

The question is not too clear. It is, therefore, being attempted on the basis of best possible interpretation.

We have been given that the future price p_t of a certain item, t years from now is given by p_t = p₀^15001.026t where p₀ is the initial price. Then, the initial price of the item will be p₀ = p_t^1/15001.026t . Also, the function will represent growth if p₀ > 1 and decay if p₀ < 1( the prices are in $). After 1 year from today, the price of the item will be p₁ = p₀^12001.026 ( t = 1). Hence the change in price (in percentage) is [(p₁ – p₀)/ p₀] * 100 = [(p₀^15001.026   - p₀)/p₀] * 100 = [100p₀(p₀^15000.026    -1)/p₀] = 100(p₀^15000.026    -1) %