Question

FOUNDATIONAL ALGEBRA II

LINEAR PROGRAMMING 1
SHE'S GOT THE "WRITE" STUFF
NAME
Jamie has just finished writing a research paper. She has hired a typist who will type the paper
on the computer for her. The typist charges $3.50 per page if no charts or graphs are used and
$8.00 per page if a chart or graph appears on the page. Jamie knows there will be at most 40
pages having no charts or graphs. There will be no more than 16 pages with charts or graphs,
and the paper will be 50 pages or less. What is the greatest possible cost to have the paper
types? How many pages with graphs and how many without graphs would cause the greatest
cost?
BATTER UP
BingBATaBoom.Inc. manufactures two different quality wood baseball bats, the Battlefield and
the Dingbat. The Battlefield takes 8 hours to trim and turn and 2 hours to finish it. It has a profit
of $17. The Dingbat takes 5 hours to trim and turn and 5 hours to finish, but its profit is $29.
The total time per day available for trimming and turning is 80 hours and for finishing is 50
hours. How many of each type of bat should be produced to have the maximum profit? What is
this maximum profit?

Answer 1

Greatest possible cost to have the paper typed = $247

• 34 pages with graphs
• 16 pages without graphs

Maximum profit = $317

• 5 Battlefield bats
• 8 Dingbats bats

Explanation:

SHE'S GOT THE "WRITE" STUFF

Let x be the number of pages without charts or graphs.

Let y be the number of pages with charts or graphs.

Given that the typist charges $3.50 per page if no charts or graphs are used, and $8.00 per page if a chart or graph appears on the page, the cost (C) to have the paper typed is given by the equation:

`C(x,y)=3.5x+8y`

To maximize the cost, we first need to determine any constraints based on the given information:

• The number of pages without charts or graphs (x) must be at most 40: x ≤ 40

• The number of pages with charts or graphs (y) must be at most 16: y ≤ 40

• The total number of pages must be 50 or less: x + y ≤ 50

• Since we cannot have a negative number of pages, it must be that: x ≥ 0 and y ≥ 0

Therefore, we have five constraints represented by the following system of inequalities:

`\begin{cases}x \leq 40\\y \leq 16\\x+y\leq50\\x\geq 0\\y\geq 0\end{cases}`

To maximize the cost of having the paper typed subject to the given constraints, we need to analyze the feasible region defined by the system of inequalities, which is the area where all the shaded regions defined by the inequalities overlap.

Upon graphing the inequalities (see attachment 1), the feasible region has five vertices (corner points):

• (0, 16)
• (34, 16)
• (40, 10)
• (40, 0)
• (0, 0)

Determine the value of C at each vertex by substituting the x and y values of the points into the equation for C:

`C(0, 16)=3.5(0)+8(16)=128`

`C(34, 16)=3.5(34)+8(16)=247`

`C(40, 10)=3.5(40)+8(10)=220`

`C(40, 0)=3.5(40)+8(0)=140`

`C(0, 0)=3.5(0)+8(0)=0`

So, the maximum value of C is $247 at vertex (34, 16).

This means that the greatest possible cost to have the paper typed is $247 where the paper has:

• 34 pages with graphs
• 16 pages without graphs

`\hrulefill`

BATTER UP

Let x be the number of Battlefield bats produced.

Let y be the number of Dingbat bats produced.

Given that the Battlefield has a profit of $17 and the Dingbat has a profit of $29, the total profit (P) is given by the equation:

`P(x,y)=17x+29y`

To maximize profit, we first need to determine any constraints based on the given information:

• The Battlefield takes 8 hours to trim and turn and the Dingbat takes 5 hours to trim and turn. The total time per day available for trimming and turning is 80 hours, so: 8x + 5y ≤ 80
• It takes 2 hours to finish the Battlefield and 5 hours to finish the Dingbat. The total time per day available for finishing is 50 hours, so: 2x + 5y ≤ 50
• Since we cannot produce a negative number of bats, it must be that: x ≥ 0 and y ≥ 0

Therefore, we have four constraints represented by the following system of inequalities:

`\begin{cases}8x + 5y \leq 80\\2x + 5y \leq 50\\x\geq 0\\y\geq 0\end{cases}`

To maximize the profit subject to the given constraints, we need to analyze the feasible region defined by the system of inequalities, which is the area where all the shaded regions defined by the inequalities overlap.

Upon graphing the inequalities (see attachment 2), the feasible region has four vertices (corner points):

• (0, 10)
• (5, 8)
• (10, 0)
• (0, 0)

Determine the value of P at each vertex by substituting the x and y values of the points into the equation for P:

`P(0, 10)=17(0)+29(10)=290`

`P(5, 8)=17(5)+29(8)=317`

`P(10, 0)=17(10)+29(0)=170`

`P(0, 0)=17(0)+29(0)=0`

So, the maximum value of P is $317 at vertex (5, 8).

This means that to maximize their profit, the company should produce:

• 5 Battlefield bats
• 8 Dingbats bats