The profit function, y, for selling x shirts is given by the equation y=-0.1x^2+10x. What is the maximum value of the profit, y?

2 ways

1 calcluls

2. algebra

calculus, just take the deritivive and find wher it is 0

y=-0.2x+10

at x=50

sub 50 for x in original

y=250

algebra

for

y=a (x-h) ²+k

(h, k) is vertex

k is maximum value

complete the square

y = (-0.1x²+10x)

y=-0.1 (x²-100x)

take 1/2 of - 100 and square it and add negative and positive inside

-100/2=-50, (-50) ²=2500

y=-0.1 (x²-100x+2500-2500)

factor perfect square

y=-0.1 ((x-50) ²-2500)

exand

y=-0.1 (x-50) ²+250

vertex at (50,250)

producing 50 units yeilds a sale price of $250

max price is $250