To calculate: The break-even point for P(x)=−1100+120x−x2

Question

To calculate: The break-even point for  P(x)=−1100+120x−x2

Suhadolahbb · Accepted Answer

Step 1
Given Information:
Profit function,

P (x) = - 1100 + 120 x - x^{2}

Formula Used:
Break-even point is the point, when profit is zero or revenue is equal to the cost.

Profit = Total revenue - Total cost

Zero product property:
The General form of quadratic equation is

a x^{2} + b x + c = 0

Where a, b, c are constants and

a \neq 0

For real numbers a and b.

a b = 0

if and only if

a = 0

or

b = 0

or both.
Step 2
Calculation:
To find the break-even point, put revenue is equal to the cost.

P (x) = 0

- 1100 + 120 x - x^{2} = 0

x^{2} - 110 x - 10 x + 1100 = 0

Take the common terms outside the parenthesis

x (x - 110) - 10 (x - 110) = 0

(x - 10) (x - 110) = 0

Use the zero product property,

x - 10 = 0

or

x - 110 = 0

x = 10

or

x = 110

It is given that

x \leq 100

Hence, the break-even point is 10 units.

Esta Hurtado · Accepted Answer

Step 1 P(x)=−1100+120x−x2 Quadratic polynomial can be factored using the transformation ax2+bx+c=a(x−x1)(x−x2) where x1 and x2 are the solutions of the quadratic equation ax2+bx+c=0 −x2+120x−1100=0 All equations of the form ax2+bx+c=0 can be solved using the quadratic formula: −b±b2−4ac2a. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction. x=−120±1202−4(−1)(−1100)2(−1) Square 120 x=−120±14400−4(−1)(−1100)2(−1) Multiply -4 times -1 x=−120±14400+4(−1100)2(−1) Multiply 4 times -1100 x=−120±14400−44002(−1) Add 14400 to -4400 x=−120±100002(−1) Take the square root of 10000 x=−120±1002(−1) Multiply 2 times -1 x=−120±100−2 Now solve the equation x=−120±100−2 when ± is plus Add -120 to 100 x=−20−2 Divide -20 by -2 x=10 Now solve the equation x=−120±100−2 when ± is minus Subtract 100 from -120 x=−220−2 Divide -220 by -2 x=110 Factro the original expression using ax2+bx+c=a(x−x1)(x−x2). Substitute 10 for x1 and 110 for x2 −x2+120x−1100=−(x−10)(x−110)

To calculate: The break-even point for P(x)=-1100+120x-x^{2}

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