Answer:
A. 36 miles
B. 30 miles
Step-by-step explanation:
Let x miles be the number of miles of highway and y miles be the number of miles of highway on bridges the city can build.
Part A:
For every mile of highway on land, the city needs two truckloads of asphalt and one gallon of paint, so for x miles the city needs 2x truckloads of asphalt and x gallons of paint.
For every mile of highway on bridges, the city needs one truckload of asphalt and three gallons of paint, so for y miles the city needs y trackloads of asphalt and 3y gallons of paint.
The city has 50 truckloads of asphalt, thus
[tex]2x+y\le50[/tex]
The city has 80 gallons of paint, so
[tex]x+3y\le80[/tex]
You get the system of two inequalities:
[tex]\left\{\begin{array}{l}2x+y\le 50\\ \\x+3y\le 80\end{array}\right.[/tex]
Plot the solution region on the coordinate plane (see first attached diagram).
The total number of miles is [tex]x+y[/tex]. Its maximum value is at intersection point of lines [tex]2x+y=50[/tex] and [tex]x+3y=80[/tex]. Find the coordinates of this point:
[tex]y=50-2x\\ \\x+3(50-2x)=80\\ \\x+150-6x=80\\ \\-5x=-70\\ \\x=14\\ \\y=22\\ \\x+y=36[/tex]
Part B:
Suppose the city is on an island and must build at least 25 miles of highway on bridges, then the city needs
[tex]25\cdot 1=25[/tex] trackloads of asphalt
[tex]25\cdot 3=75[/tex] gallons of paint
Then
[tex]50-25=25[/tex] trackloads of asphalt and
[tex]80-75=5[/tex] gallons of paint are left.
Hence,
[tex]2x+y\le 25\\ \\x+3y\le 5[/tex]
Plot the solution region on the coordinate plane (see second attached diagram). The total number of miles is [tex]x+y[/tex]. Its maximum value is at point (5,0), hence [tex]x+y=5[/tex] and the total number of miles is
[tex]25+5=30[/tex] miles.
