13.9 Applications of Extrema 805
13.9 Applications of Extrema
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find the minimum distance from the point to the plane .
a.
b.
c.
d.
e.
____ 2. Find the minimum distance from the point to the surface
.
a.
429
b.
415
c.
55
d.
607
e.
37
____ 3. Find three positive numbers x, y, and z whose sum is 24 and product is a maximum.
a.
b.
c.
d.
e.
806 Chapter 13: Functions of Several Variables
____ 4. Find three positive numbers x, y, and z whose sum is 30 and the sum of the squares is
a minimum.
a.
b.
c.
d.
e.
____ 5. Suppose a home improvement contractor is painting the walls and ceiling of a
rectangular room. The volume of the room is 640 cubic feet. The cost of wall paint is $0.076 per
square foot and the cost of ceiling paint is $0.19 per square foot. Let x, y, and z be the length, width,
and height of a rectangular room respectively. Find the room dimensions that result in a minimum
cost for the paint. Round your answers to two decimal places.
a.
b.
c.
d.
e.
____ 6. Suppose a home improvement contractor is painting the walls and ceiling of a
rectangular room. The volume of the room is 360 cubic feet. The cost of wall paint is $0.054 per
square foot and the cost of ceiling paint is $0.18 per square foot. Let x, y, and z be the length, width,
and height of a rectangular room respectively. Identify the room dimensions that result in a minimum
cost for the paint and use these dimensions to find the minimum cost for the paint. Round your
answer to the nearest cent.
a. $54.00
b. $8.64
c. $7.00
d. $19.44
e. $21.60
____ 7. The material for constructing the base of an open box costs 1.5 times as much per
unit area as the material for constructing the sides. For a fixed amount of money $150.00, find the
dimensions of the box of largest volume that can be made.
a.
b.
c.
d.
e.
13.9 Applications of Extrema 807
____ 8. Suppose a trough with trapezoidal cross sections is formed by turning up the edges of
a 30-inch-wide sheet of aluminum (see figure). Find the cross section of maximum area.
a.
b.
c.
d.
e.
____ 9. A company manufactures two types of sneakers: running shoes and basketball shoes.
The total revenue from units of running shoes and units of basketball shoes is:
,
where and are in thousands of units. Find and so as to maximize the revenue.
a.
b.
c.
d.
e.
808 Chapter 13: Functions of Several Variables
____ 10. Suppose a corporation manufactures candles at two locations. The cost of producing
units at location 1 is and the cost of producing units at location 2 is
. The candles sell for $43 per unit. Find the quantity that should be produced
at each location to maximize the profit .
a.
b.
c.
d.
e.
____ 11. Find the least squares regression line for the points shown in the graph.
a.
b.
c.
d.
e.
13.9 Applications of Extrema 809
____ 12. The least squares regression line for the points shown in the graph is .
Calculate S, the sum of the squared errors.
a.
b.
c.
d.
e.
810 Chapter 13: Functions of Several Variables
____ 13. Find the least squares regression line for the points .
a.
b.
c.
d.
e.
____ 14. Find the least squares regression line for the points
. Round numerical values in your answer to two
decimal places.
a.
b.
c.
d.
e.
____ 15. Find three positive numbers x, y, and z whose sum is 48 and is a maximum.
a. x = 10, y = 7, z = 31
b. x = 7, y = 31, z = 10
c. x = 31, y = 10, z = 7
d. x = 12, y = 24, z = 12
e. x = 12, y = 12, z = 24
13.9 Applications of Extrema 811
13.9 Applications of Extrema
Answer Section
MULTIPLE CHOICE