The yield/acre of wheat on a given parcel of land can be represented as the outcome of a random variable \(Y\) defined by \(Y=10 x^{1 / 3} e^{\varepsilon}\) for \(x \in[8,100]\), where

\(Y=\) wheat output in bushes/acre

\(x=\) pounds/acre of fertilizer applied

\(\varepsilon=\) is a random variable having the probability density function \(f(\varepsilon)=3 e^{-3 \varepsilon} I_{(0, \infty)}(\varepsilon)\).

(a) If fertilizer is applied at the rate of \(27 \mathrm{lb} / \mathrm{acre}\), what is the probability that greater than 50 bushels/acre of wheat will be produced?

(b) You sign a forward contract to sell your wheat for \(\$ 3.00 /\) bushel at harvest time. Fertilizer costs \(\$ 0.20 /\) \(1 \mathrm{~b}\). If you apply fertilizer at a rate of \(27 \mathrm{lb} / \mathrm{acre}\), what is your expected return above fertilizer cost, per acre?

(c) What is the variance of \(Y\) if fertilizer is applied at the rate of \(27 \mathrm{lb} / \mathrm{acre}\) ? Does the variance of \(Y\) change if a different level of fertilizer is applied? Why or why not?