Chapter 3.1, Problem 43E

To determine

To find: the number of bicycle and tricycle production in order to maximize the profit.

Answer to Problem 43E

50 bicycles and 75 tricycles

Explanation of Solution

Given information:

Let $x$ be the number of bicycle and $y$ be the number of tricycles.

Profit on $x$ = $6

Profit on $y$ = $4

Time required for $x$ production by department A = 3 hours

Time required for $x$ production by department B = 5 hours

Time required for $y$ production by department A = 4 hours

Time required for $y$ production by department B = 2 hours

Total time available for department A = 450 hours

Total time available for department B = 400 hours

Calculation:

Let $P\left(x,y\right)$ be the maximum profit. Thus, the equation for maximum profit is $P\left(x,y\right)=6x+4y$ .

The inequalities derived from the given information are as follows:

  $\begin{array}{c}3x+4y\le 450\\ 5x+2y\le 400\\ x\ge 0\left[\text{Assume}\right]\\ y\ge 0\left[\text{Assume}\right]\end{array}$

Use a graphing calculator to plot the inequalities as shown below.

  

The vertices are $\left(0,0\right),\left(0,112.5\right)\text{,}\left(50,75\right)\text{ and }\left(80,0\right)$ . Now test each vertex as shown below.

  $\begin{array}{c}P\left(0,0\right)=6\left(0\right)+4\left(0\right)\\ =\$0\\ P\left(0,112.5\right)=6\left(0\right)+4\left(112.5\right)\\ =\$450\\ P\left(50,75\right)=6\left(50\right)+4\left(75\right)\\ =\$600\\ P\left(80,0\right)=6\left(80\right)+4\left(0\right)\\ =\$480\end{array}$

Thus, the number of bicycle and tricycle production in order to maximize the profit is 50 and 75 respectively.