Question

Suppose we express the amount of land under cultivation as the product of four factors:

Land = (land/food) x (food/kcal) X (kcal/person) X (population)
The annual growth rates for each factor are (1) the land required to grow a unit of food, - 1 percent (due to greater productivity per unit of land); (2) the amount of food grown per calorie of food eaten by a human, +0.5 percent (because with affluence, people consume more animal products, which greatly reduces the efficiency of land use); (3) the per capita calorie consumption, +0.1 percent; and (4) the size of the population, + 1.5 percent. At these rates, how long would it take to double the amount of cultivated land needed? At that time, how much less land would be required to grown a unit of food?

Answer 1

t = 63.01338

46.74%

Step-by-step explanation:

Using the exponential growth equation :

P = Po * exp(rt)

Rate factors are :

land required to grow a unit of food, = - 1%

amount of food grown/calorie of food = +0.5%

size of the population = 1.5 percent

the per capita calorie consumption, +0.1%

Σ growth rates = - 1 + 0.5 + 1.5 + 0.1 = 1.1% = 0.011

Time taken for Population to double ;

Population (P) = 2 * initial population (Po)

P = 2Po

P = Po * exp(rt)

Substitute 2Po for P

2Po = Po * exp(rt)

2 = exp (rt)

Take the In of both sides

In(2) = rt

0.6931471 = 0.011 * t

t = 0.6931471/ 0.011

t = 63.01338

At that time, how much less land would be required to grown a unit of food?

100 - exp(rt)

r = growth rate of land required to grow a unit of food, = - 1% = - 0.01

[1 - exp(-0.01 * 63.01338)]

[1 - 0.5325205]

= (0.4674795)

= (0.4674) * 100%

= 46.74%