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43. City urbanization City planners model the size of their city using1t

Question
43. City urbanization City planners model the size of their city using
1
t<50, where A
t2t20, for 0
50
the function A(t)
is measured in square miles and t is the number of years after 2010.
a. Compute A'(t). What units are associated with this derivative
and what does the derivative measure?
b. How fast will the city be growing when it reaches a size of
38 mi2?
c. Suppose the population density of the city remains constant
from year to year at 1000 people/mi2. Determine the growth
rate of the population in 2030.
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43. City urbanization City planners model the size of their city using 1 t<50, where A t2t20, for 0 50 the function A(t) is measured in square miles and t is the number of years after 2010. a. Compute A'(t). What units are associated with this derivative and what does the derivative measure? b. How fast will the city be growing when it reaches a size of 38 mi2? c. Suppose the population density of the city remains constant from year to year at 1000 people/mi2. Determine the growth rate of the population in 2030.

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Step 1

a)

Obtain the derivative of A(t) as shown below.

A'()4())
(4(t)
dt
1
-t2 2t20
50
dt
1
(2-)+2(1 )+ (0)
50
1
t+2
25
1
-t+2is the rate at which the city
25
The derivative A'(t)=-
grows per year is been calculated.
1
Thus, the derivative is A'(t)
:t+2 square miles/year
25
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A'()4()) (4(t) dt 1 -t2 2t20 50 dt 1 (2-)+2(1 )+ (0) 50 1 t+2 25 1 -t+2is the rate at which the city 25 The derivative A'(t)=- grows per year is been calculated. 1 Thus, the derivative is A'(t) :t+2 square miles/year 25

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Step 2

b)

Substitute A(t) = 38 in the given equation.

1
-t 2t 20 =38
50
1
-t 2t -18 0
50
2-100t900 0
(t-90)(-10)0
t 90,t 10
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1 -t 2t 20 =38 50 1 -t 2t -18 0 50 2-100t900 0 (t-90)(-10)0 t 90,t 10

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Step 3

The only one solution in the given do...

1
-t+2 as
Substitute t10 in A'(t) 25
A'(10)2(10)+2
2
+2
5
=-0.4 2
=1.6
Therefore, the growth of the city when it reaches a size of 38 mi2
is 1.6 square miles per year.
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1 -t+2 as Substitute t10 in A'(t) 25 A'(10)2(10)+2 2 +2 5 =-0.4 2 =1.6 Therefore, the growth of the city when it reaches a size of 38 mi2 is 1.6 square miles per year.

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