Question

In t years, the population of a certain city grows from 600,000 to a size P given by P(t) = 600,000+ 9000t².

a) Find the growth rate, 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).

Answer 1

Explanation:

a) To find the growth rate, we need to take the derivative of the population function P(t) with respect to time t, which will give us the rate of change of population:

P(t) = 600,000 + 9,000t²

Now, take the derivative:

dP/dt = d/dt [600,000 + 9,000t²]

dP/dt = 0 + 18,000t

So, the growth rate, dP/dt, is 18,000t.

b) To find the population after 15 years (t = 15), substitute t = 15 into the population function:

P(15) = 600,000 + 9,000(15)²

P(15) = 600,000 + 9,000(225)

P(15) = 600,000 + 2,025,000

P(15) = 2,625,000

The population after 15 years is 2,625,000.

c) To find the growth rate at t = 15, substitute t = 15 into the growth rate formula we found in part (a):

dP/dt = 18,000t

dP/dt at t = 15 = 18,000(15)

dP/dt at t = 15 = 270,000

So, the growth rate at t = 15 is 270,000.

d) The answer to part (c) means that at t = 15 years, the population is growing at a rate of 270,000 people per year. This represents the rate at which the population is increasing at that specific point in time. In other words, for each additional year at t = 15, the population is growing by 270,000 people.