Questions: 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)=0.15 e^(0.2 t) million dollars, where t is the number of years since 2000. For the 4-plex in 2010, how fast in millions of dollars was it increasing per year? Enter your answer to 3 decimal places. t=10 To find the change of price of the 4-plex per year in 2010, we take the derivative of p. Set up the derivative of 0.15 e^(0.20 t) using prime notation.

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)=0.15 e^(0.2 t) million dollars, where t is the number of years since 2000. For the 4-plex in 2010, how fast in millions of dollars was it increasing per year? Enter your answer to 3 decimal places. t=10 To find the change of price of the 4-plex per year in 2010, we take the derivative of p. Set up the derivative of 0.15 e^(0.20 t) using prime notation.
Transcript text: 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 $p(t)=0.15 e^{0.2 t}$ million dollars, where $t$ is the number of years since 2000. For the 4-plex in 2010, how fast in millions of dollars was it increasing per year? Enter your answer to 3 decimal places. $t=10$ To find the change of price of the 4-plex per year in 2010, we take the derivative of $p$. Set up the derivative of $0.15 e^{0.20 t}$ using prime notation.
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Solution

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Solution Steps

Step 1: Derivative Calculation

To determine how fast the price of the 4-plex is increasing per year, we first calculate the derivative of the price function p(t)=0.15e0.2t p(t) = 0.15 e^{0.2 t} . The derivative is given by:

p′(t)=0.03e0.2t p'(t) = 0.03 e^{0.2 t}

Step 2: Evaluate the Derivative at t=10 t = 10

Next, we evaluate the derivative at t=10 t = 10 (which corresponds to the year 2010):

p′(10)=0.03e0.2⋅10=0.03e2≈0.221671682967920 p'(10) = 0.03 e^{0.2 \cdot 10} = 0.03 e^{2} \approx 0.221671682967920

Final Answer

The rate of increase in the price of the 4-plex in 2010 is approximately 0.222 0.222 million dollars per year. Thus, we can express the final answer as:

0.222 \boxed{0.222}

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