Chapter 3.4, Problem 15E

a)

To determine

To find: system of inequalities to represent the given situations.

Explanation of Solution

Given Information

Let, ‘ c’ be the number of chairs and ‘ t’ be the number of tables

According to question,

Carpentry and finishing time per chair is 3 hrs and 0.5 hr respectively and for table it is 2 hrs and 1 hr respectively.

Formula Used for Graph

$\frac{x}{a}+\frac{y}{b}=1$

where, ‘ c’ and ‘ b’ are x and y intercepts respectively.

Calculation Maximum carpentry time is 108 hours

So,

  $3c+2t\le 108$  ...... (i)

and, maximum finishing time is 20 hours

So,

  $0.5c+t\le 20$  ...... (ii)

Now, we will plot equation (i) and (ii) graphically, to find out the feasible region and also calculate the maximum profit.

For plotting the graph, we will simplify the equations as:

  $3c+2t\le 108$

  ⇒ $\frac{3c}{108}+\frac{2t}{108}=1$

  ⇒ $\frac{c}{36}+\frac{t}{54}=1$  ...... (iii)

This equation is of the form,

  $\frac{x}{a}+\frac{y}{b}=1$

where, ‘ c’ and ‘ b’ are x and y intercepts respectively.

  $[a_1=36,b_1=54]$

Now,

  $0.5t+t=20$

  $\frac{0.5c}{20}+\frac{t}{20}=1$

  $\frac{c}{40}+\frac{t}{20}=1$  ...... (iv)

  $[a_2=40,b_2=20]$

Now we will plot equations (iii) and (iv) graphically.

b)

To determine

To draw: showing the feasible region

Explanation of Solution

Graph

  

c)

To determine

To find: required number of tables and chairs to maximize the profit and maximum profit.

Answer to Problem 15E

Maximum Profit = $1265

Explanation of Solution

Given Information

Let, ‘ c’ be the number of chairs and ‘t’ be the number of tables

Point ‘ P’ represents the required number of tables and chairs to maximize the profit.

On solving the two equations, the required point of intersection, ‘ P’ is got.

The required solution of P is (34, 3)

So, maximum chairs will be 34and tables will be 3

Now, maximum profit = (34 × $35) + (3 × $25)

  = 1190 + 75

  = $1265