Please help this question is from my unit on linear programing and I'm having trouble completing it.

A company manufactures bicycles and tricycles. The total number of frames the company manufactures cannot exceed 80 per month. It takes the company 1 hour to assemble a bicycle and 2 hours to assemble a tricycle. The assembly machine is only available for 100 hours each month. If the company makes a profit of $50 on each bicycle and $70 on each tricycle, determine the number of bicycles and tricycles that will maximize profits each month. Indicate the maximum profit the company can make each month.
This is what I have so far: x=#of bicycles y=#of tricycles
x>=0 y>=0 P=50x +70y
Now I don't know what to do. Any help would be greatly appreciated. Thanks

You need

b+t </= 80 (frames limit)

b + 2 t </= 100 (machine hour limit)

graph those t on x axis, b on y axis

they cross at
(80-t)+2 t = 100
t = 20
b = 60
look at resulting quadrilateral including axes, must be inside it

so we have three realistic (t,b) points
(0,80)
(20,60)
(50,0)
check
p = 70 t + 50 b
at those three points
which gives max p?

To solve this problem, you can use the concept of linear programming, which involves identifying constraints and optimizing an objective function.

Let's break down the problem and identify the constraints and the objective function.

1. Constraints:
a) The total number of frames manufactured cannot exceed 80 per month:
This constraint can be expressed as: x + y ≤ 80, where x is the number of bicycles and y is the number of tricycles.

b) The assembly machine is only available for 100 hours each month:
This constraint can be expressed as: x + 2y ≤ 100, where x is the number of bicycles and y is the number of tricycles.

2. Objective function:
The profit can be calculated by multiplying the number of bicycles (x) with $50, and the number of tricycles (y) with $70. So the objective function is:
P = 50x + 70y, where P is the profit in dollars.

Now that we have the constraints and the objective function, we can find the solution by graphing the constraints and identifying the feasible region (the area where all the constraints are satisfied). The corner point within the feasible region that maximizes the objective function will give us the solution.

To graph the constraints, plot the lines x + y = 80 and x + 2y = 100 on a coordinate plane.

Solving these two equations will give us the coordinates of the intersection points, which will be the corners of the feasible region.

x + y = 80:
By substituting x = 0, we get y = 80, so one point is (0, 80).
By substituting y = 0, we get x = 80, so another point is (80, 0).

x + 2y = 100:
By substituting x = 0, we get y = 50, so one point is (0, 50).
By substituting y = 0, we get x = 100, so another point is (100, 0).

Now, plot these corner points on the graph and shade the area inside the feasible region.

To maximize the objective function P = 50x + 70y, evaluate the objective function at each corner point within the shaded region and identify the point that gives the maximum value. This point will indicate the optimal number of bicycles and tricycles that maximize profits per month.

After identifying the optimal point, substitute the values of x and y into the objective function to find the maximum profit (P).

I hope this explanation helps you in solving the problem. Good luck!