Plant growth: The amount of growth of plants in an ungrazed pasture is a function of the amount of plant biomass already present and the amount of rainfall.9 For a pasture in the arid zone of Australia, the formula
Y = −55.12 − 0.01535N − 0.00056N2 + 3.946R
gives an approximation of the growth. Here R is the amount of rainfall, in millimeters, over a 3-month period; N is the plant biomass, in kilograms per hectare, at the beginning of that period; and Y is the growth, in kilograms per hectare, of the biomass over that period. (For comparison, 100 millimeters is about 3.9 inches, and 100 kilograms per hectare is about 89 pounds per acre.) For this exercise, assume that the amount of plant biomass initially present is 400 kilograms per hectare, so N = 400.
a. Find a formula for the growth Y as a function of the amount R of rainfall.
b. Make a graph of Y versus R. Include values of R from 40 to 160 millimeters.
c. What happens to Y as R increases? Explain your answer in practical terms.
d. How much growth will there be over a 3-month period if initially there are 400 kilograms per hectare of plant biomass and the amount of rainfall is 100 millimeters?