Problema Solution
A company plans to sell pens for $2 each. The company’s financial planner estimates that the cost, y, of manufacturing the pens is a quadratic function with a y-intercept of 120 and a vertex of (250, 370). What is the minimum number of pens the company must sell to make a profit?
Answer provided by our tutors
The equation for the cost can be written as:
y = ax^2 + bx + c
Since y-intercept is 120 this means for x = 0, y = 120 that is:
a*0^2 + b*0 + c = 120
c = 120
The vertex is given by (-b/2a, (c - b^2)/4a) this means:
-b/2a = 250
b = -500a
Since the line goes trough (250, 370), that is for x = 250, y = 370:
a*250^2 + (-500a)*250 + 120 = 370
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click here to see the equation solved for a
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a = -0.004
b = -500*(-0.004)
b = 2
The equation is:
y = -0.004x^2 + 2x + 120
The company makes profit when y > 2x that is:
-0.004x^2 + 2x + 120 > 2x
-0.004x^2 > - 120
x^2 < 120/0.004
x^2 < 30000
x < 173.2
Thus the minimum number of pens the company must sell is 1.