A rectangle is inscribed below a straight line, with its left edge
against the y-axis and its bottom edge against the x-axis.
The dimensions of the rectangle can be changed. This can
change the area. If the base of the rectangle is too narrow, you
will get a small area.
The dimensions of the rectangle can be changed. This can
change the area. If the base of the rectangle is too wide, you
will also get a small area.
Suppose x denotes the width of the rectangle and y denotes
the height of the rectangle. What dimensions will maximize
the area of the inscribed rectangle.
A = xy
b=2
m=
∆y
0−2
1
=
=−
∆x
4−0
2
1
y = mx + b = − x + 2
2
µ
¶
1
A = xy = x − x + 2
2
µ
¶
1
1
A = x − x + 2 = − x2 + 2x
2
2
dA
= −x + 2
dx
d2 A
= −1
dx2
dA
= −x + 2
dx
dA
= 0 when x = 2
dx
1
y = − x + 2 = −1 + 2 = 1
2
The largest box will have a base of 2 and a height of 1.
If a company manufactures 2,000 bicycles per year, it can sell
them for $ 80 per bicycle. However, if it manufactures 1,000
bicycles, the price is $ 100 per bicycle.
Let x be the number of bicycles manufactured in a year.
Let y be the price per bicycle.
Let R = xy. R is called the revenue.
Find the value of x that maximizes the revenue.