Calc HW
flcoman89
calcoctober31.pdf
Kyle Taitt CSU Webwork
MATH 160 WeBWorK assignment M160-801-FA-3.2 due 10/31/2016 at 11:59pm MDT
1. (1 point) A parcel delivery service will deliver a pack- age only if the length plus the girth (distance around, taken per- pendicular to the length) does not exceed 108 inches. Find the maximum volume of a rectangular box with square ends that satisfies the delivery company’s requirements.
Maximum Volume = in3. Answer(s) submitted:
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2. (1 point) A fence is to be built to enclose a rectangular area of 300 square feet. The fence along three sides is to be made of material that costs 6 dollars per foot, and the material for the fourth side costs 15 dollars per foot. Find the dimensions of the enclosure that is most economical to construct.
Dimensions: x Answer(s) submitted:
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3. (1 point) Find two numbers differing by 30 whose product is as small as possible.
Enter your two numbers as a comma separated list, e.g. 2, 3. The two numbers are . Answer(s) submitted:
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4. (1 point) A fence 4 feet tall runs parallel to a tall building at a distance of 4 feet from the building.
What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?
Length of ladder = feet. Answer(s) submitted:
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5. (1 point) An open box is to be made out of a 6-inch by 18- inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume.
Dimensions of the bottom of the box: x Height of the box:
Answer(s) submitted:
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6. (1 point) Find the point on the line �4x+5y+1 = 0 which is closest to the point (�4,5).
Answer is Answer(s) submitted:
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7. (1 point) A rectangular storage container with an open top is to have a volume of 22 cubic meters. The length of its base is twice the width. Material for the base costs 15 dollars per square meter. Material for the sides costs 8 dollars per square meter. Find the cost of materials for the cheapest such container.
Total cost = (Round to the nearest penny and include monetary units. For example, if your answer is 1.095, enter $1.10 including the dollar sign and second decimal place.)
Answer(s) submitted:
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8. (1 point) A rectangle is inscribed with its base on the x- axis and its upper corners on the parabola y = 2 � x2. What are the dimensions of such a rectangle with the greatest possible area?
Width = Height =
Answer(s) submitted:
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9. (1 point) A cylinder is inscribed in a right circular cone of height 5.5 and radius (at the base) equal to 7. What are the dimensions of such a cylinder which has maximum volume?
Radius = Height = Answer(s) submitted:
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10. (1 point) A Norman window has the shape of a semi- circle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 46 feet?
Answer(s) submitted:
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11. (1 point) A right circular cylinder is inscribed in a sphere of radius r.
Find the dimensions of such a cylinder with the largest possible volume (your answer may depend on r).
base radius = height = Answer(s) submitted:
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12. (1 point) A steel pipe is being carried down a hallway 9 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?
ft Answer(s) submitted:
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13. (1 point) The upper right-hand corner of a piece of paper, 12 in. by
8 in., as in the figure, is folded over to the bottom edge. How would you fold it so as to minimize the length of the fold? In other words, how would you choose x to minimize y?
x = in Answer(s) submitted:
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14. (1 point)
A cone-shaped drinking cup is made from a circular piece of paper of radius R by cutting out a sector and joining the edges CA and CB. Find the maximum capacity of such a cup (Your answer may depend on R).
Answer(s) submitted:
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15. (1 point) How far from A should the point P be chosen so as to maxi-
mize the angle q?
Answer(s) submitted:
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16. (1 point) The Nearsighted Cow Problem: A Calculus Classic.
A rectangular billboard 6 feet in height stands in a field so that its bottom is 6 feet above the ground. A nearsighted cow with eye level at 4 feet above the ground stands x feet from the billboard. Express q, the vertical angle subtended by the bill- board at her eye, in terms of x. Then find the distance x0 the cow must stand from the billboard to maximize q.
q(x) = q is maximized when x0 =
(Click on image for a larger view ) Answer(s) submitted:
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17. (1 point) A rain gutter is to be constructed from a metal sheet of width
30 cm by bending up one-third of the sheet on each side through an angle q. How should q be chosen so that the gutter will carry the maximum amount of water?
q = radians 2
Answer(s) submitted:
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18. (1 point) A woman at a point A on the shore of a circular lake with radius r = 3 wants to arrive at the point C diametrically opposite A on the other side of the lake in the shortest possible time. She can walk at the rate of 10mph and row a boat at 5mph. What is the shortest amount of time it would take her to reach point C?
Answer (in hours): t = Answer(s) submitted:
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19. (1 point) Ornithologists have determined that some species of birds
tend to avoid flights over large bodies of water during daylight hours. It is believed that more energy is required to fly over wa- ter than land because air generally rises over land and falls over water during the day. A bird with these tendencies is released from an island that is 5 km from the nearest point B on a straight shoreline, flies to a point C on the shoreline, and then flies along the shoreline to its nesting area D. Assume that the bird instinc- tively chooses a path that will minimize its energy expenditure. Points B and D are 13 km apart.
(a) In general, if it takes 1.4 times as much energy to fly over the water as land, how far should point C be from B in order to minimize the total energy expended in returning to its nesting area?
km (b) Let W and L denote the energy (in joules) per kilome-
ter flown over water and land respectively. Determine the ratio W/L corresponding to the minimum energy expenditure given that the bird flies to the shore at a point x kilometers from B. (Your answer may depend only on x.)
W/L = (c) If the ornithologists observe that birds of a certain species
reach the shore at a point 6 km from B, how many times more energy does it take this species to fly over water than land?
Answer(s) submitted:
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20. (1 point)
A humidifier uses a vertical rotating disk of radius r which is partially submerged in water. Evaporation is maximized if the exposed wetted area is maximized. The accompanying Figure shows the wetted area as light blue, the water as dark blue, and the dry area as light gray. If you have a black and white hard copy the light and dark blue may show as dark gray and black, respectively. You are looking along the axis of the disk, and it is rotating (clockwise or counterclockwise, it does not matter).
We want to know how to pick h, the distance from the center of the disk to the top of the water, so as to maximize the wetted area.
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The submerged part of the disk forms a segment of a circle. If t is the angle indicated in the Figure, i.e., half the angle defining the segment, then the area of the submerged part is
A = tr2 � h p
r
2 � h2.
In terms of t and r, the area W of the wetted part is the area of the disk minus the area of the dry part minus the area of the submerged part, i.e.,
W = pr2 � ph2 � tr2 + h p
r
2 � h2. We now need to eliminate h or t. Since
cos t = h
r
we have h = r cos t
and we can see similarly that p
r
2 � h2 = r sin t.
Substituting these expressions in W gives W = and
W
0 = . (Your answers will depend on t and r.)
The equation W 0(t) = 0 can be solved for t, giving t = radians which is degrees.
This means that h = ar
where a = .
Check the solution after the set closes, for a more detailed discussion of these calculations.
Answer(s) submitted:
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