Question : Find two numbers whose sum is 320 and whose product is as large as possible : 2151727
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) Find two numbers whose sum is 320 and whose product is as large as possible.
A) 10 and 310
B) 160 and 160
C) 159 and 161
D) 1 and 319
2) Find two numbers x and y such that their sum is 60 and x2y is maximized.
A) x = 15, y = 45
B) x = 20, y = 40
C) x = 45, y = 15
D) x = 40, y = 20
3) Of all numbers whose difference is 20, find the two that have the minimum product.
A) 1 and 21
B) 0 and 20
C) 40 and 20
D) 10 and -10
4) Maximize Q = xy2, where x and y are positive numbers, such that x + y2 = 5.
A) x = 1, y = 2
B) x = (5/2), y = √((5/2))
C) x = 0, y = √(5)
D) x = √((5/2)), y = (5/2)
5) If the price charged for a candy bar is p(x) cents, then x thousand candy bars will be sold in a certain city, where p(x) = 114 - (x/36). How many candy bars must be sold to maximize revenue?
A) 4104 thousand candy bars
B) 4104 candy bars
C) 2052 candy bars
D) 2052 thousand candy bars
6) A rectangular field is to be enclosed on four sides with a fence. Fencing costs $2 per foot for two opposite sides, and $3 per foot for the other two sides. Find the dimensions of the field of area 830 ft2 that would be the cheapest to enclose.
A) 23.5 ft @ $2 by 35.3 ft @ $3
B) 19.2 ft @ $2 by 43.2 ft @ $3
C) 43.2 ft @ $2 by 19.2 ft @ $3
D) 35.3 ft @ $2 by 23.5 ft @ $3
7) If the price charged for a bolt is p cents, then x thousand bolts will be sold in a certain hardware store, where p = 39 - (x/18). How many bolts must be sold to maximize revenue?
A) 702 thousand bolts
B) 702 bolts
C) 351 bolts
D) 351 thousand bolts
8) A hotel has 200 units. All rooms are occupied when the hotel charges $100 per day for a room. For every increase of x dollars in the daily room rate, there are x rooms vacant. Each occupied room costs $36 per day to service and maintain What should the hotel charge per day in order to maximize daily profit?
9) A baseball team is trying to determine what price to charge for tickets. At a price of $10 per ticket, it averages 50,000 people per game. For every increase of $1, it loses 5,000 people. Every person at the game spends an average of $5 on concessions. What price per ticket should be charged in order to maximize revenue?
10) The stadium vending company finds that sales of hot dogs average 37,000 hot dogs per game when the hot dogs sell for $2.50 each. For each 50 cent increase in the price, the sales per game drop by 5000 hot dogs. What price per hot dog should the vending company charge to realize the maximum revenue?
11) Supertankers off-load oil at a docking facility shore point 5 miles offshore. The nearest refinery is 10 miles east of the docking facility. A pipeline must be constructed connecting the docking facility with the refinery. The pipeline costs $300,000 per mile if constructed underwater and $200,000 per mile if over land.
Locate point B to minimize the cost of construction.
A) Point B is 4.28 miles from Point A.
B) Point B is 4.47 miles from Point A.
C) Point B is 6.91 miles from Point A.
D) Point B is 3.27 miles from Point A.
12) Suppose c(x) = x3 - 22x2 + 30,000x is the cost of manufacturing x items. Find a production level that will minimize the average cost per item of making x items.
A) 10 items
B) 12 items
C) 11 items
D) 13 items
13) Recent research has shown that the population f(S) of cod in the North Sea next year as a function of this year's population S (measured in thousands of tons) can be described by the Shepherd model,
f(S) = (aS/1 + (S/b)c)
where a, b, and c are constants. The values of a, b, and c are 3.036, 243.96, and 3.21, respectively. Find the approximate value of this year's population that maximizes next year's population using this model.
A) 156,000 tons
B) 4000 tons
C) 191 tons
D) 191,000 tons
14) Find the dimensions that produce the maximum floor area for a one-story house that is rectangular in shape and has a perimeter of 123 ft.
A) 61.5 ft × 61.5 ft
B) 30.75 ft × 123 ft
C) 30.75 ft × 30.75 ft
D) 10.25 ft × 30.75 ft
15) An architect needs to design a rectangular room with an area of 76 ft2. What dimensions should he use in order to minimize the perimeter?
A) 19 ft × 19 ft
B) 8.72 ft × 8.72 ft
C) 15.2 ft × 76 ft
D) 8.72 ft × 19 ft
16) A piece of molding 180 cm long is to be cut to form a rectangular picture frame. What dimensions will enclose the largest area?
A) 13.42 cm × 13.42 cm
B) 45 cm × 45 cm
C) 36 cm × 36 cm
D) 13.42 cm × 45 cm
17) A company wishes to manufacture a box with a volume of 32 cubic feet that is open on top and is twice as long as it is wide. Find the width of the box that can be produced using the minimum amount of material.
A) 6.8 ft
B) 3.4 ft
C) 5.8 ft
D) 2.9 ft
18) From a thin piece of cardboard 50 in by 50 in, square corners are cut out so that the sides can be folded up to make a box. What dimensions will yield a box of maximum volume? What is the maximum volume? Round to the nearest tenth, if necessary.
A) 33.3 in by 33.3 in by 16.7 in; 18,518.5 in3
B) 33.3 in by 33.3 in by 8.3 in; 9259.3 in3
C) 25 in by 25 in by 12.5 in; 7812.5 in3
D) 16.7 in by 16.7 in by 16.7 in; 4629.6 in3
19) A private shipping company will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 90 in What dimensions will give a box with a square end the largest possible volume?
A) 15 in × 15 in × 75 in
B) 15 in × 30 in × 30 in
C) 15 in × 15 in × 30 in
D) 30 in × 30 in × 30 in