During the past 10 years, the amount of money, M (in billions of dollars), spent in North America by car dealerships advertising their product can be modeled by the equation M = 1.2e0.25t + 0.13. In what year was about $19 billion (M = 19) spent by car dealerships in advertising?

Question

Mathematics

Noodle

3 November, 13:40

During the past 10 years, the amount of money, M (in billions of dollars), spent in North America by car dealerships advertising their product can be modeled by the equation M = 1.2e0.25t + 0.13. In what year was about $19 billion (M = 19) spent by car dealerships in advertising?

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Brennan · Answer 1

First, you have to formulate an equation from the problem statement. Since, the equation is already given for you, you could skip this step. Then, substitute M=19 in the equation to determine t in years:

19 = 1.2e^0.25t + 0.13

19 - 0.13 = 1.2e^0.25t

18.87 = 1.2e^0.25t

18.87/1.2 = e^0.5t

15.725 = e^0.5t

Take ln on both sides:

ln 15.725 = lne^0.5t

Applying logarithmic identities:

ln 15.725 = 0.5t

t = ln 15.725/0.5

t = 5.51 years

Therefore, the year where the amount of money in $19 billion is in between the 5th and 6th year.