72 Part I: The Questions
509. A piece of wire that is 20 meters long is cut
into two pieces. One is shaped into a
square, and the other is shaped into an
equilateral triangle. How much wire should
you use for the square so that the total area
is at a maximum?
510. A piece of wire that is 20 meters long is cut
into two pieces. One is bent into a square,
and the other is bent into an equilateral triangle. How much wire should you use for
the square so that the total area is at a
minimum?
511. The illumination of a light source is directly
proportional to the strength of the light
source and inversely proportional to the
square of the distance from the source. Two
light sources, one five times as strong as the
other, are placed 20 feet apart, and an
object is placed on the line between them.
How far from the bright light source should
the object be placed so that the object
receives the least illumination?
512. Find the area of the largest rectangle that
can be inscribed in the ellipse x
y
2
2
4
9
1
+
= .
503. A box with an open top and a square base
must have a volume of 16,000 cubic centimeters. Find the dimensions of the box that
minimize the amount of material used.
504. Find the point(s) on the ellipse 8x
2
+ y
2
= 8
farthest from (1, 0).
505. Find the point on the line y = 4x + 6 that is
closest to the origin.
506. A rectangular poster is to have an area of
90 square inches with 1-inch margins at the
bottom and sides and a 3-inch margin at the
top. What dimensions give you the largest
printed area?
507. At which x values on the curve f
(x) = 2 +
20x
3
– 4x
5
does the tangent line have the
largest slope?
508. A rectangular storage container with an
open top is to have a volume of 20 cubic
meters. The length of the base is twice
the width. The material for the base costs
$20 per square meter. The material for the
sides costs $12 per square meter. Find the
cost of the materials for the cheapest
such container. Round your answer
to the nearest cent.
509. A piece of wire that is 20 meters long is cut
into two pieces. One is shaped into a
square, and the other is shaped into an
equilateral triangle. How much wire should
you use for the square so that the total area
is at a maximum?
510. A piece of wire that is 20 meters long is cut
into two pieces. One is bent into a square,
and the other is bent into an equilateral triangle. How much wire should you use for
the square so that the total area is at a
minimum?
511. The illumination of a light source is directly
proportional to the strength of the light
source and inversely proportional to the
square of the distance from the source. Two
light sources, one five times as strong as the
other, are placed 20 feet apart, and an
object is placed on the line between them.
How far from the bright light source should
the object be placed so that the object
receives the least illumination?
512. Find the area of the largest rectangle that
can be inscribed in the ellipse x
y
2
2
4
9
1
+
= .
503. A box with an open top and a square base
must have a volume of 16,000 cubic centimeters. Find the dimensions of the box that
minimize the amount of material used.
504. Find the point(s) on the ellipse 8x
2
+ y
2
= 8
farthest from (1, 0).
505. Find the point on the line y = 4x + 6 that is
closest to the origin.
506. A rectangular poster is to have an area of
90 square inches with 1-inch margins at the
bottom and sides and a 3-inch margin at the
top. What dimensions give you the largest
printed area?
507. At which x values on the curve f
(x) = 2 +
20x
3
– 4x
5
does the tangent line have the
largest slope?
508. A rectangular storage container with an
open top is to have a volume of 20 cubic
meters. The length of the base is twice
the width. The material for the base costs
$20 per square meter. The material for the
sides costs $12 per square meter. Find the
cost of the materials for the cheapest
such container. Round your answer
to the nearest cent.