Honors Math 3
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More Optimization Problems (continued)
6. Point B is 100 km east of point A. At noon, a truck leaves A and travels north at 60 km/hr
while a car leaves B and travels west at 80 km/hr. When will the distance between the
vehicles be a minimum and what is the minimum distance?
7. Suppose that a 1-meter wire is cut into two pieces, then one piece is used to form a circle
while the other piece is used to form a square. What is the maximum possible total area of
the circle and the square?
8. A manufacturer produces cardboard boxes with square bases. The bottom of the box has a
double layer of cardboard for strength. If each box has volume 12 ft 3, what dimensions will
minimize the amount of cardboard used?
9. A box company produces a box with a square base and no top that has a volume of 8 ft3.
Material for the bottom costs $6 / ft2 and the material for the sides costs $3 / ft2. Find the box
dimensions that minimize the manufacturing cost.
10. The graphs of y = x 2 and y = x 3 meet twice. For what value of x between the intersection
points is the vertical distance between the graphs the greatest?
11. A net enclosure for golf practice is open at one end and on the bottom,
as illustrated in the diagram. The volume of the enclosure is 100
cubic meters. Find the dimensions that require the smallest amount of
netting.
12. A rectangle is formed so that one side is on the x-axis and the opposite
side has endpoints on the part of the parabola y = 4 - 4x 2 above the x-axis. Find the
maximum area of the rectangle.
13. An isosceles trapezoid is inscribed in the parabola y = 9 - x 2, with one base of the trapezoid
having its endpoints on the x-axis at the intersection of the function y = 9 - x 2 and the x-axis
and the other base having its endpoints on the function y = 9 - x 2. Find the vertices of the
trapezoid that maximize the trapezoid’s area.
Answers
1. x = 3.486 inches
9. 2 ft by 2 ft by 2 ft
2. (1.714, 0.999)
10. x =
3. 55 additional trees should be planted
11. 5.313 m by 5.313 m by 3.542 m
4.
100
p
meters by 50 meters
5. max area of 16 (2 by 8)
6. min distance of 60 km at 12:48 PM
7. max area of
1
4p
when only make a circle
8. 2 ft by 2 ft by 3 ft
2
3
12. max area of 3.079
13. (–3, 0) (3, 0) (1, 8) (–1, 8)