Stewart_Calc_7ET ch04sec07
MULTIPLE CHOICE
1. What is the shortest possible length of the line segment that is cut off by the first quadrant
and is tangent to the curve at some point?
a.
b.
c.
d.
None of these
e.
2. Find two positive numbers whose product is and whose sum is a minimum.
a.
b.
3, 48
c.
6, 24
d.
e.
2, 72
3. A rectangular storage container with an open top is to have a volume of 10 . The length
of its base is twice the width. Material for the base costs $12 per square meter. Material for
the sides costs $5 per square meter. Find the cost of materials for the cheapest such
container.
a.
$153.92
b.
$152.9
c.
$152.4
d.
$153.9
e.
$158.1
f.
$151.6
2 10
4. Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral
triangle of side L = 9 cm if one side of the rectangle lies on the base of the triangle.
Round your answer to the nearest tenth.
a.
9.5 cm, 3.9 cm
b.
7.5 cm, 2.9 cm
c.
5.5 cm, 4.4 cm
d.
4.5 cm, 4 cm
e.
4 cm, 3.91 cm
f.
4.5 cm, 3.9 cm
5. A woman at a point A on the shore of a circular lake with radius wants to arrive at the
point C diametrically opposite on the other side of the lake in the shortest possible time. She
can walk at the rate of and row a boat at . How should she proceed? (Find ).
Round the result, if necessary, to the nearest hundredth.
a.
radians
b.
radians
c.
radians
d.
She should walk around the lake from point A to point C.
e.
She should row from point A to point C radians
6. Find the point on the line that is closest to the origin.
a.
b.
c.
d.
e.
7. A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the
other is bent into an equilateral triangle. How should the wire be cut for the square so that
the total area enclosed is a minimum?
Round your answer to the nearest hundredth.
a.
3.25 m
b.
5.35 m
c.
0 m
d.
4.35 m
e.
4.4 m
8. A rectangular beam will be cut from a cylindrical log of radius inches. Suppose that
the strength of a rectangular beam is proportional to the product of its width and the square
of its depth. Find the dimensions of the strongest beam that can be cut from the cylindrical
log.
a.
in, in
b.
in, in
c.
in, in
d.
in, in
e.
in, in
9. Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of
radius .
a.
b.
c.
d.
e.
10. Find the dimensions of a rectangle of area 64 that has the smallest possible perimeter.
a.
4 ft by 16 ft
b.
2 ft by 32 ft
c.
1 ft by 64 ft
d.
8 ft by 8 ft
11. A production editor decided that a promotional flyer should have a 1-in. margin at the top
and the bottom, and a -in. margin on each side. The editor further stipulated that the flyer
should have an area of 392 . Determine the dimensions of the flyer that will result in
the maximum printed area on the flyer.
a.
7 in. 56 in.
b.
in. in.
c.
14 in. 28 in.
d.
in. in.
12. Find the dimensions of the rectangle enclosed in the semicircle with the
largest possible area.
a.
5 in. 7 in.
b.
in. in.
c.
in. in.
d.
in. in.
13. An apple orchard has an average yield of 32 bushels of apples per tree if tree density is 30
trees per acre. For each unit increase in tree density, the yield decreases by 2 bushels per
tree. How many trees per acre should be planted to maximize yield?
a.
27 trees/acre
b.
30 trees/acre
c.
37 trees/acre
d.
23 trees/acre
14. A manufacturer has been selling 1,200 television sets a week at $400 each. A market survey
indicates that for each $30 rebate offered to the buyer, the number of sets sold will increase
by 60 per week. Find the demand function.
a.
b.
c.
d.
e.
NUMERIC RESPONSE
1. A farmer with 710 ft of fencing wants to enclose a rectangular area and then divide it into
four pens with fencing parallel to one side of the rectangle. What is the largest possible total
area of the four pens?
2. What is the minimum vertical distance between the parabolas and ?
3. The sum of two positive numbers is . What is the smallest possible value of the sum of
their squares?
4. The average cost of producing x units of a commodity is given by the equation
.
Find the marginal cost at a production level of 1,255 units.
5. Find an equation of the line through the point that cuts off the least area from the
first quadrant.
6. A steel pipe is being carried down a hallway 14 ft wide. At the end of the hall there is a
right-angled turn into a narrower hallway 6 ft wide. What is the length of the longest pipe
that can be carried horizontally around the corner?
7. Find the maximum area of a rectangle that can be circumscribed about a given rectangle
with length L = 8 and width W = 3.
SHORT ANSWER
1. Find two numbers whose difference is 170 and whose product is a minimum.
2. The owner of a ranch has 4000 yd of fencing with which to enclose a rectangular piece of
grazing land situated along a straight portion of a river. If fencing is not required along the
river, what are the dimensions of the largest area he can enclose? What is the area?
3. If an open box is made from a metal sheet 9 in. square by cutting out identical squares from
each corner an bending up the resulting flaps, determine the dimensions of the box with the
largest volume that can be made.
4. A rectangular box having a top and a square base is to be constructed at a cost of $1. If the
material for the bottom costs $0.35 per square foot, the material for the top costs $0.15 per
square foot, and the material for the sides costs $0.20 per square foot, find the dimensions
and volume of the box of maximum volume that can be constructed.
5. A poster of height 33 in. is mounted on a wall so that its lower edge is 15 in. above the eye
level of an observer. How far from the wall should the observer stand so that the viewing
angle subtended at his eye by the poster is as large as possible (see figure – not drawn to
scale)?
Round your answer to the nearest integer.
33
15