Population Growth in Clark County

Question

 A certain county in Nevada is one of the fastest-growing metropolitan areas in the United States. The population of this county from 1970 through 2010 is approximated by the function

P(t) = 40,775t³ −  81,394t² + 234,633t + 273,288    (0 ≤ t ≤ 4) 

where t is measured in decades, with t = 0

 corresponding to the beginning of 1970.

(a)

Was the population of the county always increasing or decreasing over the time period in question. Hint: Show that P'(t) > 0

 for all t in the interval (0, 4).

increasingdecreasing     varies

(b)

Find when the population is increasing or decreasing at the slowest pace. Hint: Find the inflection point of P in the interval (0, 4). (Round you answer to two decimal places.)

t = 

Answer 1

(a) Differentiate the population function:

P(t) = 40,775t^3 - 81,394t^2 + 234,633t + 273,288

So:

P'(t) = 122,325t^2 - 162,788t + 234,633

To determine whether the population is increasing or decreasing, check whether P'(t) is positive.

This quadratic opens upward because the coefficient of t^2 is positive. Its discriminant is:

b^2 - 4ac

= (-162,788)^2 - 4(122,325)(234,633)

< 0

Since the discriminant is negative and the quadratic opens upward, P'(t) is always positive.

Therefore, the population is always:

increasing

(b) The population is increasing at the slowest pace when P'(t) is at its minimum. This occurs where:

P''(t) = 0

Differentiate again:

P''(t) = 244,650t - 162,788

Set equal to zero:

244,650t - 162,788 = 0

t = 162,788 / 244,650

t ≈ 0.67

So the population was increasing at its slowest pace at:

t = 0.67

Since t is measured in decades after the beginning of 1970, this is about 6.7 years after 1970, or approximately 1976.7.