Determine the profit function P(x), if the revenue function and cost functions are R(x)=70x-1000 and C(x)=30x+1000 respectively. Also find the break-even point.
P = R-C
break-even when R=C (P=0)
To determine the profit function P(x), we need to subtract the cost function C(x) from the revenue function R(x).
Profit (P) is calculated as Revenue (R) minus Cost (C):
P(x) = R(x) - C(x)
Given that R(x) = 70x - 1000 and C(x) = 30x + 1000, we can substitute these values into the profit function equation:
P(x) = (70x - 1000) - (30x + 1000)
P(x) = 70x - 1000 - 30x - 1000
P(x) = 40x - 2000
Therefore, the profit function P(x) is given by: P(x) = 40x - 2000
To find the break-even point, we need to set the profit function P(x) equal to zero, since the break-even point is the point where the profit is zero (i.e., neither making a profit nor incurring a loss).
So, we set P(x) = 0 and solve for x:
0 = 40x - 2000
Adding 2000 to both sides of the equation:
40x = 2000
Dividing both sides of the equation by 40:
x = 2000/40
x = 50
Therefore, the break-even point is at x = 50.