Question

in t years, the population of a certain city grows from 400,000 to a size p given by ( p(t) = 400,000 + 1000t^2 )
a) find the growth rate, ( \frac{dp}{dt} )
b) find the population after 15 yr
c) find the growth rate at ( t = 15 )
d) explain the meaning of the answer to part (c)

a) ( \frac{dp}{dt} = 2000t )
b) the population after 15 yr is 625000
(simplify your answer )
c) the growth rate at ( t = 15 ) is 30000
(simplify your answer )
d) what is the meaning of the answer to part (c)?
( \bigcirc ) a. the growth rate tells the rate at which the population is growing at time ( t = 15 ).
( \bigcirc ) b. the growth rate tells the difference between the rate of growth at the beginning of ( t = 0 ) and ( t = 15 )
( \bigcirc ) c. the growth rate tells the average growth from time ( t = 0 ) and ( t = 15 )
( \bigcirc ) d. the growth rate tells the rate at which the population is growing at time ( t = 15 - 1 )

Explanation:

Part d)

The growth rate \(\frac{dP}{dt}\) at a specific time \(t\) represents the instantaneous rate of change of the population at that time. For part (c), we found the growth rate at \(t = 15\), which means we are finding how fast the population is growing exactly at \(t=15\) years.

• Option A says the growth rate tells the rate at which the population is growing at time \(t = 15\), which matches the definition of the derivative (instantaneous rate of change) at \(t = 15\).
• Option B is incorrect because the growth rate at \(t = 15\) does not relate to the difference between growth rates at \(t = 0\) and \(t = 15\).
• Option C is incorrect as the average growth from \(t = 0\) to \(t = 15\) would be calculated differently (using the average rate of change formula \(\frac{P(15)-P(0)}{15 - 0}\)), not the derivative at \(t = 15\).
• Option D is incorrect as \(t=15 - 1=14\) is not related to the growth rate at \(t = 15\).

Answer:

A. The growth rate tells the rate at which the population is growing at time \(t = 15\)