Question

Current Attempt in Progress With yearly inflation of 8%, prices are given by P=P 0 (1.08) where P 0​ is the price in dollars when t=0 and r is time in years. Suppose P 0​

=1. How fast (incents/year) are pricesrising when t=10? Round your answer to two decimal places. cents/year eTextbook and Media Hint Attempts: 0 of 3 used

Answer 1

The rate at which prices are rising after 10 years with an 8% annual inflation rate can be calculated by taking the time derivative of the price function P = P0(1.08)^t and evaluating it at t=10, converting the result from dollars/year to cents/year.

Step-by-step explanation:

The student has inquired about how fast prices are rising after 10 years with an annual inflation rate of 8%. To find out how fast the prices are rising, we need to calculate the rate of change of the price at t=10. Since the formula given is P=P0 (1.08)^t, where P0 is the initial price and is equal to 1 and t is the time in years, we can derive the rate of change by taking the derivative of P with respect to t.

First, let's find the derivative of P with respect to t:
dP/dt = P0 * ln(1.08) * (1.08)^t

Now, let's substitute t=10 into the derivative to find out the rate of change at that specific time:
dP/dt at t=10 = 1 * ln(1.08) * (1.08)^10

Upon calculating this, you'll find the rate of change (in dollars/year) which then can be converted into cents/year by multiplying by 100 (since there are 100 cents in a dollar). After computing this calculation, you would round the answer to two decimal places which provides the required rate in cents/year that prices are rising after 10 years.