Find the breakeven point for the firms with total revenue R(x)=〖-4x〗^2+72x and a cost function of C(x)=16x+180. Sketch the graph to show the breakeven points for the the firms.
Question
Find the breakeven point for the firms with total revenue R(x)=〖-4x〗^2+72x and a cost function of C(x)=16x+180. Sketch the graph to show the breakeven points for the the firms.
Solution 1
The breakeven point is the point where total revenue equals total cost. So, we need to set R(x) equal to C(x) and solve for x.
R(x) = C(x) -4x^2 + 72x = 16x + 180
Rearrange the equation to one side: -4x^2 + 72x - 16x - 180 = 0 -4x^2 + 56x - 180 = 0
Divide the entire equation by -4 to simplify: x^2 - 14x + 45 = 0
This is a quadratic equation in the form of ax^2 + bx + c = 0. We can solve for x using the quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / 2a:
x = [14 ± sqrt((-14)^2 - 4145)] / 2*1 x = [14 ± sqrt(196 - 180)] / 2 x = [14 ± sqrt(16)] / 2 x = [14 ± 4] / 2 x = 10 or x = 5
So, the breakeven points are x = 5 and x = 10.
To sketch the graph, plot the total revenue and total cost functions on the same graph. The x-coordinates where the two functions intersect are the breakeven points. The total revenue function is a downward-opening parabola, and the total cost function is a straight line. The breakeven points are where the parabola intersects the line at x = 5 and x = 10.
Solution 2
The breakeven point is the point where total revenue equals total cost. So, we need to set R(x) equal to C(x) and solve for x.
R(x) = C(x) -4x^2 + 72x = 16x + 180
Rearrange the equation to one side: -4x^2 + 72x - 16x - 180 = 0 -4x^2 + 56x - 180 = 0
Divide the equation by -4 to simplify: x^2 - 14x + 45 = 0
This is a quadratic equation in the form ax^2 + bx + c = 0. We can solve for x using the quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / 2a:
x = [14 ± sqrt((14)^2 - 4145)] / 2*1 x = [14 ± sqrt(196 - 180)] / 2 x = [14 ± sqrt(16)] / 2 x = [14 ± 4] / 2 x = 10, 5
So, the breakeven points are x = 5 and x = 10.
To sketch the graph, plot the total revenue and total cost functions on the same graph. The x-coordinates where the two functions intersect are the breakeven points. The total revenue function is a downward-opening parabola, and the total cost function is a straight line. The breakeven points are where the parabola intersects the line at x = 5 and x = 10.
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The intersection point of the total cost and revenue lines represent in breakeven analysis shows Maximum profit Breakeven point Fixed cost Variable cost
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