bikes not to exceed 5x number of motorcycles. can produce less than 100 motorcycles, or less than 250 bikes. total 300.

Question

the number of bicycles made cannot exceed 5 times the number of motorcycles. They lack the facilities to produce more than 100 motorcycles, or more than 250 bicycles, and the total production of motorcycles and bicycles cannot exceed 300. If $1220 on each motorcycle and $380 on each bicycle, how many of each must be made to maximize profit?

Answer 1

This is a type of problem called Linear Programming. You'll want to first define your variables:

 

x = number of motorcycles made

y = number of bikes made

 

 

 

Then, write equations that bound your problem:

 

0 ≤ x ≤ 100    (you can't make more than 100 or fewer than 0 motorcycles)

0 ≤ y ≤ 250    (you can't make more than 250 or fewer than 0 bikes)

x + y ≤ 300    (you can't make more than 300 vehicles total)

 

 

 

Now, plot those equations and find the region of intersection. The region should be some enclosed polygon where all the equations are true. So:

 

Draw vertical lines at x = 0 and x = 100. The first equation is true between those lines.

 

Draw horizontal lines at y = 0 and y = 250. The second equation is true between those lines.

 

Draw a diagonal line at y = -x + 300. The third equation is true below that line.

 

The enclosed region is a pentagon, with one side formed by each of the five lines.

 

 

 

Now, find the vertices of the region of intersection. Since yours is a pentagon, you'll want to find five vertices. Some are straightforward:

 

x = 0 and y = 0 intersect at (0, 0)

x = 0 and y = 250 intersect at (0, 250)

x = 100 and y = 0 intersect at (100, 0)

 

Some are slightly less straightforward:

 

y = -x + 300 and x = 100 intersect where y = -100 + 300. That's (100, 200)

y = -x + 300 and y = 250 intersect where 250 = -x + 300, or -x = -50, or x = 50. That's (50, 250)

 

So, the five points of intersection are (0, 0), (0, 250), (100, 0), (100, 200), and (50, 250).

 

 

 

Now, write an equation for profit, P:

 

P = 1220x + 380y

 

This equation will have its maximum value (and its minimum value) at one of those vertices. Plug each vertex in to figure out which gives you the maximum profit:

 

P = 1220(0) + 380(0) = 0 + 0 = $0

P = 1220(0) + 380(250) = 0 + 95000 = $95,000

P = 1220(100) + 380(0) = 122000 + 0 = $122,000

P = 1220(100) + 380(200) = 122000 + 76000 = $198,000

P = 1220(50) + 380(250) = 61000 + 95000 = $156,000

 

Your maximum profit is $198,000, at 100 motorcycles and 200 bikes.

Answer 2

Is the $1220 on each motorcycle and the $380 on each bicycle the profit made on each?????

If so, they should make 100 motorcycles and 200 bikes to maximize their profit which would be $198,000.