Chapter 1.1, Problem 24E

Reminder Round all answers to two decimal places unless otherwise indicated.

Note Some of the formulas below use the special number $e$, which was presented in the Prologue.

Yield Response to Several Growth Factors This is a continuation of Exercise 23. If more than one nutrient is considered, the formula for percentage of maximum yield is a bit more complex. For three nutrients, the formula is

$Y\left(b,c,d\right)=\left(1-{0.5}^b\right)\left(1-{0.5}^c\right)\left(1-{0.5}^d\right)$.

a. Express using functional notation the percentage of maximum yield produced from $1$ baule of the first nutrient, $2$ baules of the third nutrient, and $3$ baules of the third nutrient, and then calculate that value.

b. One baule of nitrogen is $223$ pounds per acre, $1$ baule of phosphorus is $45$ per acre, and $1$ baule of potassium is $76$ pounds per acre. What percentage of maximum yield will be obtained from $200$ pounds of nitrogen per acre, $100$ pounds of phosphorus per acre, and $150$ pounds of potassium per acre?

23. Mitscherlich’s Equation An important agriculture problem is to determine how a quantity of nutrient, such as nitrogen, affects the growth of plants. We consider the situation wherein sufficient quantities of all but one nutrient are present. One boule of a nutrient is the amount needed to produce $50\%$ of maximum possible yield. In $1990$, E.A. Mitscherlich’s proposed the following relation, which is known as Mitscherlich’s Equation:

$Y=-1-{0.5}^b$.

Here $b$ is the number of baules of nutrient applied, and $Y$ is the percentage (as a decimal) of maximum yield produced.

a. Verify that the formula predicts that $50\%$ of maximum yield will be produced if $1$ baule of nutrient is applied.

b. Use functional notation to express the percentage of maximum yield produced by $3$ baule of nutrient, and calculate the value.

c. The exact value of a baule depends on the nutrient in question. For nitrogen, $1$ baule is $223$ pounds per acre. What percentage of maximum yield will be produced if $500$ pounds of nitrogen per acre is present?