Question #136818
Mr. Trenga plans to start a new business called River Explorers, which will rent canoes and kayaks
to people to travel 10 miles down the Clarion River in Cook Forest State Park. He has $45,000 to
purchase new boats. He can buy the canoes for $600 each and the kayaks for $750 each. His facility
can hold up to 65 boats. The canoes will rent for $25 a day, and the kayaks will rent for $30 a day. How
many canoes and how many kayaks should he buy to earn the most revenue? What is the maximum
revenue
1
Expert's answer
2020-10-06T16:24:50-0400

 Let, number of cаnoes bought = xx

Number of kayaks bought = yy

The optimization function is:

Z=25x+30yZ = 25x + 30y

Constraints are:

600x+750y45000600*x + 750*y \leq 45000

x+y65x + y \leq 65

Since the number of boat can`t be negative,

x0,y0x \geq 0,\\ y \geq 0


The linear programming problem is:

 Z=25x+30yZ = 25*x + 30*y

subject to:

600x+750y45000600*x + 750*y \leq 45000

x+y65x + y \leq 65

y0y \geq 0

x0;x \geq 0;


Graph:



First equation:

600x+750y=45000600*x + 750*y = 45000

Divide both parts by 150

4x+5y=3004*x + 5*y = 300


Second equation:

x+y=65x + y = 65


Let's solve the system of equations:

{4x+5y=300x+y65\begin{cases} 4*x + 5*y = 300 \\ x + y \leq 65 \end{cases}


From the first equation we express x:

4x+5y=3004x=3005y(4x)/4=(3005y)/4x=75(5y)/44*x+5*y= 300\\ 4*x = 300-5*y\\ (4*x)/4 = (300-5*y)/4\\ x=75-(5*y)/4


Substitute the found x into the second equation x+y=65x + y = 65 ;


We get:

y+(75(5y)/4)=6575y/4=65y/4=6575y/410y=40y + (75 - (5*y)/4) =65\\ 75-y/4 = 65\\ -y/4=65-75\\ y/4 - 10\\ y = 40


Substitute the found y into the equation:

x=75(540)/4x=7550x=25x=75-(5*40)/4\\ x = 75-50\\ x = 25


On solving these we get:

x=x = number of canoes =25= 25

y=y = number of kayaks =40= 40


Z =(25×25)+(30×40)=1825= ( 25 × 25 ) + ( 30 × 40 ) = 1825


Answer:

Mr. Trenga should buy 25 canoes and 40 kayaks to earn the most revenue.

Maximum revenue is $1825.


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