Question

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.

Answer 1

Answer:

2071

Step-by-step explanation:

Since, the decline model follows exponential- decay model

thus,

$P = P_oe^{kt}$

Here,

P₀ is the initial consumption

t is the time in years

P is the consumption after t years

k is the decay constant

now,

1985 is the base year, thus for year 1985;  t = 0

at t = 0, P = 80

Therefore,

$80 = P_oe^{k(0)}$

or

P₀ = 80 pounds

also,

in the year 1996 i,e t = 1996 - 1985 = 11 years

P = 67 pounds

thus,

$67 = 80e^{k(11)}$

or

0.8375 = $e^{k(11)}$

taking the log both sides, we get

-0.177 = 11k

or

k = - 0.01612

Therefore,

For P = 20 pounds per person

we have

$20 = 80e^{(-0.01612)(t)}$

or

0.25 = $e^{(-0.01612)(t)}$

taking natural log both the sides, we get

-1.3863 = (- 0.01612 )(t)

or

t = 85.99 ≈ 86 years

Hence,

the year will be 1985 + 86 = 2071