A chain saw requires 2 hours of assembly and a wood chipper 5 hours. A maximum of 20 hours of assembly time is available. The profit is 150 on a chain saw and 230 on a chipper. How many of each should be assembled for maximum profit.
chain saw assembly profit per hr = $75 /h
chipper assembly profit per hr = $46 /h
Assemble only chain saws.
You will assemble 10 and earn $750.
I don't see how calculus is needed for this question.
To find the number of chain saws and wood chippers that should be assembled for maximum profit, we can set up a system of equations.
Let's assume x represents the number of chain saws and y represents the number of wood chippers.
The total assembly time for chain saws is 2 hours per chain saw, so the total assembly time for x chain saws is 2x hours.
The total assembly time for wood chippers is 5 hours per wood chipper, so the total assembly time for y wood chippers is 5y hours.
We know that the total assembly time cannot exceed 20 hours, so we can write the equation:
2x + 5y ≤ 20
The profit from each chain saw is $150, so the total profit from x chain saws is 150x.
The profit from each wood chipper is $230, so the total profit from y wood chippers is 230y.
We want to maximize the total profit, so our objective function is:
Profit = 150x + 230y
To summarize, we have the following equations:
Constraint: 2x + 5y ≤ 20
Objective function: Profit = 150x + 230y
To find the maximum profit, we need to find the values of x and y that satisfy the constraint and maximize the objective function. This is a linear programming problem.
To determine the number of chain saws and wood chippers that should be assembled for maximum profit, we can use a mathematical approach called linear programming.
Let's assume we assemble x chain saws and y wood chippers.
The assembly time constraint is given as follows:
2x (chain saw assembly time) + 5y (wood chipper assembly time) ≤ 20 (available assembly time)
The profit function can be expressed as:
Profit = 150x (profit from chain saws) + 230y (profit from wood chippers)
So, our goal is to maximize the profit function while satisfying the assembly time constraint mentioned above.
To solve this problem, we can follow these steps:
1. Rewrite the assembly time constraint in terms of x:
2x + 5y ≤ 20
2. Next, we graph the assembly time constraint on a coordinate plane. To do this, we solve the inequality for y:
y ≤ (20 - 2x) / 5
3. The feasible region on the graph represents all the possible combinations of chain saws and wood chippers that satisfy the assembly time constraint.
4. Calculate the profit at each corner point of the feasible region, which can be done by plugging in the x and y values into the profit function:
Profit = 150x + 230y
5. Compare the profits at each corner point to determine the maximum profit. The combination of x and y that results in the highest profit will give us the answer to the question.
By following these steps, you can determine how many chain saws and wood chippers should be assembled for maximum profit.