Chapter 3.4, Problem 27E

Refer to Fig. 6. As the lines of constant profit were lowered, the final line had slope $-\frac{8}{7}$ and contained the optimal vertex of the feasible set. Figure 15 shows that, as long as the slope of the final line is between -2 and -1, the optimal solution would still be at the same vertex.

(a) Suppose that the profit per chair is changed from $80 to $ A, but that the profit per sofa remains at $70. Show that the lines of constant profit will each have slope $-\frac{A}{70}$.

(b) Show that, when $-2\le -\frac{A}{70}\le -1$, then $70\le A\le 140$. Conclude that the interval $\left[70,140\right]$ is the range of optimality for the profit per chair.

(c) Use similar reasoning to show that the range of optimality for the profit per sofa is $\left[40,80\right]$