MATH SOLVE

3 months ago

Q:
# The annual consumption of beef per person is on the decline. It was 80 pounds per person per year in 1985 and about 67 pounds per person per year in 1996. Assuming consumption is decreasing according to the exponential-decay model, in what year (theoretically) will the yearly consumption of beef be 20 pounds per person.

Accepted Solution

A:

Answer:2071Step-by-step explanation:Since, the decline model follows exponential- decay modelthus, [tex]P = P_oe^{kt}[/tex]Here, P₀ is the initial consumptiont is the time in yearsP is the consumption after t yearsk is the decay constantnow,1985 is the base year, thus for year 1985; t = 0at t = 0, P = 80Therefore, [tex]80 = P_oe^{k(0)}[/tex]orP₀ = 80 poundsalso, in the year 1996 i,e t = 1996 - 1985 = 11 yearsP = 67 poundsthus,[tex]67 = 80e^{k(11)}[/tex]or0.8375 = [tex]e^{k(11)}[/tex]taking the log both sides, we get-0.177 = 11kork = - 0.01612Therefore,For P = 20 pounds per personwe have[tex]20 = 80e^{(-0.01612)(t)}[/tex]or0.25 = [tex]e^{(-0.01612)(t)}[/tex]taking natural log both the sides, we get-1.3863 = (- 0.01612 )(t)ort = 85.99 ≈ 86 yearsHence, the year will be 1985 + 86 = 2071