Final answer:
The crop size that results in maximum revenue is 11.31 billion bushels and the price per bushel is $3.53.
Step-by-step explanation:
Let's first calculate the revenue for the 4-billion-bushel corn crop at the given price of $2.4/bushel:
Revenue = price x crop size = $2.4/bushel x 4 billion bushels = $9.6 billion.
Now, let's consider the rule of thumb given by the commodity broker. If the crop is reduced by x percent, the price increases by 10x cents.
To find the crop size that results in maximum revenue, we need to find the point where the revenue is maximized. We can do this by finding the derivative of the revenue function with respect to the crop size, and setting it equal to zero:
Revenue' = (2.4 + 0.1x) x - 2.4 = 0
Simplifying, we get:
0.1x^2 + 2.4x - 9.6 = 0
Solving this quadratic equation, we find that x = 11.31 or x = -21.31. Since the percentage cannot be negative and the crop size cannot be negative either, we can ignore the negative value.
Therefore, the crop size that results in maximum revenue is 11.31 percent smaller than the original size.
To find the price per bushel at this crop size, we can substitute the value of x into the rule of thumb:
New price per bushel = $2.4/bushel + (10 x 11.31) cents = $2.4 + 1.131 = $3.531.
Therefore, the crop size that results in maximum revenue is 11.31 billion bushels and the price per bushel is $3.53.