Could somebody help me with this problem pls.

Question

Marc has a summer-tree trimming business.

Marc knows that P=-4x²+100-600 models his profit in dollars, where x is the amount he charges per tree.

How much does he need to charge if he wants to break even? Explain.

and how much does he need to charge if he wants to have a profit of $25.

Answer 1

-4x^2 +100x -600 = P =0 to break even

break even means zero profit

divide by-4

x^2 -25x +150 = 0

factor

(x-15)(x-10)

set each factor =0

x =$15 or $10 per tree to break even

set -4x^2-100x +600 = 25 for a $25 profit

4x^2 +100x -625 = 0

divide by 4

complete the square

x^2 +25x + 156.25/4 = -625/4 + 156.25

(x-12.5)^2 = 0

x-12.5 = 0

x = $12.50 per tree for a $25 profit (which also happens to be the maximum profit possible)

or use the quadratic formula

x = -100/-8 + or - (1/8)sqr(100^2+16(625))

x = (100 +0/8 = $12.50 per tree for a $25 profit

max profit is when the derivative =0

P' = -4x +100 =0

x = 100/4 = $12.50

P(12.50) = -4(12.5)^2 + 100(12.5) - 600

-625 + 1250 -600 = $25 profit

P(10) =-4(10^2) +100(10) -600

= -400 + 1000-600

=0 profit

P(15) = -4(15^2) +100(15) -600

= -900 +1500-600

=0 profit

The Profit function P is a downward opening parabola with vertex = maximum point = (12.5, 25) if you graph x on the x axis and P on the y axis

it's symmetric with line of symmetry x=12.5

15 and 10 are equistant from 12.5

they are the two zeros, roots, solutions or x intercepts