A real estate investor researches 4-plex apartment complexes in a city and finds that from 2000 to 2012, the average price of the 4-plex is approximated by p(t) = 0.15e0.20t million dollars, where t is the number of years since 2000. For the 4-plex in 2010, how fast in dollars was it increasing per year? Enter your answer without the dollar sign. (Ex: 456,000 would be entered for $456,000)

221,672

Explanation:

To find the rate of increase, find differential of p(t)

p(t)= 0.15e^(0.2t)

dp/dt= 0.03e^(0.2t)

at t=10

dp/dt= 0.03e^(0.2×10)

dp/dt= 0.03e^2

dp/dt= 0.221672 million dollars

dp/dt= 221,672

221,672

Explanation:

This can be calculated by first finding the derivative of  p(t) = 0.15e^0.20t. This done by differentiating  the p(t) with respect to t as follows:

Given p(t) = 0.15e^0.20

p'(t) = (0.15e^0.20t) × (0.20) = 0.03e^(0.20t) (1)

Number of years, i.e. t, is the difference between 2010 and 2000. Therefore,

t = 2010 - 2000 = 10

Substituting t = 10 into equation (1), we have:

p'(t) = 0.03e^(0.20 × 10) = 0.03e^2 (2)

Since e^2 = 7.38905609893065, we substitute it into equation (2) as follows:

p'(t) = 0.03 × 7.38905609893065 = 0.22167168296792  

Since it is in million dollars, we multiply it by 1,000,000 as follows:

p'(t) = 0.22167168296792 × 1,000,000

p'(t) = 221,672 approximately

Therefore, the 4-plex in 2010 was increasing by 221,672 dollars per year.

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