The consumer price index (CPI) measures how prices have changed for consumers. With 1995 as a reference of 100, a year with CPI=150 indicates that consumer costs in that year were 1.5 times the 1995 costs. With labor data from a country for selected years from 1995 and projected to 2050, the rate of change of the CPI can be modeled by dC/dt= 0.009t^2-0.096t+4.85dollars per year, where t=0 represents 1990.(a) Find the function that models C(t), if the CPI was 175 in 2010.(b) What does the model from part (a) predict for the consumer costsin 2040?

Question
Asked Jul 26, 2019

The consumer price index (CPI) measures how prices have changed for consumers. With 1995 as a reference of 100, a year with CPI=150 indicates that consumer costs in that year were 1.5 times the 1995 costs. With labor data from a country for selected years from 1995 and projected to 2050, the rate of change of the CPI can be modeled by 

dC/dt= 0.009t^2-0.096t+4.85

dollars per year, where t=0 represents 1990.

(a) Find the function that models C(t), if the CPI was 175 in 2010.

(b) What does the model from part (a) predict for the consumer costsin 2040?

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Step 1

Given:

The rate of change of CPI in dollars per year is dC/dt= 0.009t2 − 0.096t + 4.85, where t represents the year and t = 0 represents the year 1990.

Compute C(t) as follows.

Integrate the given equation on both sides.

e0.0090.096 + 4.85
dt
0.009f 0.096t
3
2
C(t) 0.00330.04842 4.85t + c
(1)
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e0.0090.096 + 4.85 dt 0.009f 0.096t 3 2 C(t) 0.00330.04842 4.85t + c (1)

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Step 2

Since t =0 represents the year 1990, the value of t will be 20 that represents the year 2010.

So, the value of C(20) is 175.

Substitute t = 20 and C(20) = 175  in equation (1).

C(20) 0.003 (20)-0.048 (20)+4.85(20)+ c
175 24-19.2 +97 + c
C1 73.2
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C(20) 0.003 (20)-0.048 (20)+4.85(20)+ c 175 24-19.2 +97 + c C1 73.2

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Step 3

Substitute c1 = 73.2 in equation (1).

C(t) = 0.003t3 – 0.048t2 + 4.85t + 73.2     &n...

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