Question

Superbats Inc, manufactures two different types of wood baseball bats, the Homer-Hitter and the Big
Timber, The Homer-Hitter takes 8 hours to trim and turn on the lathe and 2 hours to finish. Each
Homer-Hitter sold makes a profit of $17. The Big Timber takes 5 hours to trim and turn on the lathe
and 5 hours to finish, and its profit is $29. The total time available for trimming and lathing is 80
hours. The total available time for finishing is 50 hours.

Answer 1

The optimal number of each type of baseball bat to be produced is

given by linear programming.

(a) The variables are;

• x is for the number of Homer-Hitter manufactured

• y is for the number of Big Timber manufactured

(b) The objective quantity is profit

• The equation for the objective quantity is; P = 17·x + 29·y

(c) The system of inequalities are;

• 8·x + 5·y ≤ 80, which gives; y ≤ 16 - 1.6·x

• 2·x + 5·y ≤ 50, which gives; y ≤ 10 - 0.4·x

• x ≥ 0

• y ≥ 0

(d) Please find attached the graph of the system of inequalities created with MS Excel;

• The vertex points are; $\underline{(0, \, 10), \, (5, \, 8), \, (10, \, 0), \, (0, 0)}$

(e) The number of each type to be produced to maximize profit are;

• 5 Homer-Hitter and 8 Big Timber

• The maximum profit is $317

Methods used for the linear programming

Time to trim and turn the Homer-Hitter = 8 hours

Time it takes to finish the Homer-Hitter = 2 hours

Profit each Homer-Hitter sold, makes = $17

Time to trim and turn the Big Timber = 5 hours

Time to finish the Big Timber = 5 hours

Profit each Big Timber sold, makes = $29

Total available time for trimming and lathing = 80 hours

Total time for finishing = 50 hours

 

(a) The variables to be used to describe the constraints in the situation are;

• Variable x represents the Homer-Hitter

• y represents the Big Timber

(b) A general objective of a manufacturing company is profit, and the objective equation is therefore;

• $\underline{P = 17 \cdot x + 29 \cdot y}$

(c) The system of inequalities which describes the given constraints are presented as follows;

• 8·x + 5·y ≤ 80

• 2·x + 5·y ≤ 50

• x ≥ 0, y ≥ 0

(d) The equations for the graph are therefore;

$y \leq \dfrac{80}{5} - \dfrac{8}{5} \cdot x = 16 - \dfrac{8}{5} \cdot x = 16 - 1.6 \cdot x$

$y \leq \dfrac{50}{5} - \dfrac{2}{5} \cdot x = 10 - \dfrac{2}{5} \cdot x = 10 - 0.4 \cdot x$

Which gives;

• y ≤ 16 - 1.6·x

• y ≤ 10 - 0.4·x

The vertices of the feasible region are;

• $\underline{ (0, \, 10), \, (5, \, 8), \, (10, \, 0), \, (0, \, 0)}$

• Please find attached the graph of the above system of inequalities created with MS Excel

(e) The profits at the margins of the feasible region are;

Profit at (0, 10), P = 17 × 0 + 29 × 10 = 290

Profit at (5, 8), P = 17 × 5 + 29 × 8 = 317

Profit at (10, 0), P = 17 × 10 + 29 × 0 = 170

Profit at (0, 0), P = 17 × 0 + 29 × 0 = 0

Therefore, the for maximum profit, the number of each type to be produced are;

• 5 Homer-Hitter and 8 Big Timber baseball bats

• The maximum profit is; $317

The above responses are based on the order and questions obtained from a similar online uploaded question.

Learn more about linear programming here:

https://brainly.com/question/15356519