Question

If we assume the population of Grand Rapids is growing at a rate of approximately 4% per decade, we can model the population function with the formula
P( t ) = 181843 ( 1.04 )^(t / 10).
Use this formula to compute the average rate of change of the population on the intervals [ 5 , 10 ] , [ 5 , 9 ] , [ 5 , 8 ] , [ 5 , 7 ] , and [ 5 , 6 ] .

Answer 1

Answer:

The average rate of change of the population on the intervals [ 5 , 10 ] , [ 5 , 9 ] , [ 5 , 8 ] , [ 5 , 7 ] , and [ 5 , 6 ]  are 734.504, 733.06, 731.62, 730.185 and 728.75 respectively.

Step-by-step explanation:

The given function is

$P(t)=181843(1.04)^{(\frac{t}{10})}$

where, P(t) is population after t years.

At t=5,

$P(5)=181843(1.04)^{(\frac{5}{10})}=185444.20$

At t=6,

$P(6)=181843(1.04)^{(\frac{6}{10})}=186172.95$

At t=7,

$P(7)=181843(1.04)^{(\frac{7}{10})}=186904.57$

At t=8,

$P(8)=181843(1.04)^{(\frac{8}{10})}=187639.06$

At t=9,

$P(9)=181843(1.04)^{(\frac{9}{10})}=188376.44$

At t=10,

$P(10)=181843(1.04)^{(\frac{10}{10})}=189116.72$

The rate of change of P(t) on the interval $[x_1,x_2]$ is

$m=\frac{P(x_2)-P(x_1)}{x_2-x_1}$

Using the above formula, the average rate of change of the population on the intervals [ 5 , 10 ] is

$m=\frac{P(10)-P(5)}{10-5}=\frac{189116.72-185444.20}{5}=734.504$

The average rate of change of the population on the intervals [ 5 , 9 ] is

$m=\frac{P(9)-P(5)}{9-5}=\frac{188376.44-185444.20}{4}=733.06$

The average rate of change of the population on the intervals [ 5 , 8 ] is

$m=\frac{P(8)-P(5)}{8-5}=\frac{187639.06-185444.20}{3}=731.62$

The average rate of change of the population on the intervals [ 5 , 7 ] is

$m=\frac{P(7)-P(5)}{7-5}=\frac{186904.57-185444.20}{2}=730.185$

The average rate of change of the population on the intervals [ 5 , 6 ] is

$m=\frac{P(6)-P(5)}{6-5}=\frac{186172.95-185444.20}{1}=728.75$

Therefore the average rate of change of the population on the intervals [ 5 , 10 ] , [ 5 , 9 ] , [ 5 , 8 ] , [ 5 , 7 ] , and [ 5 , 6 ]  are 734.504, 733.06, 731.62, 730.185 and 728.75 respectively.