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Calc 1 4.5

 

<== Calculus 1
Number Question Answer
See the work
1. Find two numbers A and B (with A &le B) whose difference is 40 and whose product is minimized.

A =

B =

A= -20

B= 20
The work

Point Cost: 3
2. Find two positive numbers whose product is 225 and whose sum is a minimum.

Answer: ,
15, 15
The work

Point Cost: 3
3. Find the length L and width W (with W &le L) of the rectangle with perimeter 44 that has maximum area, and then find the maximum area.
L =

W =

Maximum area =
L = 11

W = 11

Maximum area = 121

The work

Point Cost: 3
4. A rancher wants to fence in an area of 2000000 square feet in a rectangular field and then divide it in half with a fence down the middle parallel to one side. What is the shortest length of fence that the rancher can use? 6928.2
The work

Point Cost: 3
5. A fence is to be built to enclose a rectangular area of 240 square feet. The fence along three sides is to be made of material that costs 4 dollars per foot, and the material for the fourth side costs 13 dollars per foot. Find the length L and width W (with W &le L) of the enclosure that is most economical to construct.

L =

W =
L = 22.58

W = 10.627
The work

Point Cost: 3
6. A hippogriff rancher wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle (see the figure below). He has 600 feet of fencing available to complete the job. What is the largest possible total area of the four pens?




Largest area = 	extrm{ft}^2
Largest area = 9000
The work

Point Cost: 3
7. A box is to be made out of a 6 cm by 20 cm piece of cardboard. Squares of side length x cm will be cut out of each corner, and then the ends and sides will be folded up to form a box with an open top.

(a) Express the volume V of the box as a function of x.

V = 	extrm{cm}^3

(b) Give the domain of V in interval notation. (Use the fact that length and volume must be positive.)

(c) Find the length L, width W, and height H of the resulting box that maximizes the volume.
(Assume that W &le L).

L = cm

W = cm

H = cm

(d) The maximum volume of the box is 	extrm{cm}^3.

V= (6 - 2x)(20 - 2x)x

Domain: (0,3)

L = 17.259

W = 3.259

H = 1.371

max volume of the box= 77.087
The work

Point Cost: 4
8. If 1300 square centimeters of material is available to make a box with a square base and an open top, What is the largest possible volume of the box that can be found?
Volume = cubic centimeters.
V = 4510.276
The work

Point Cost: 4
9. A rectangular storage container with an open top is to have a volume of 50 cubic meters. The length of its base is twice the width. Material for the base costs 50 dollars per square meter. Material for the sides costs 7 dollars per square meter. Find the cost of materials for the cheapest such container.
Minimum cost = .
Minimum cost = 906.207
The work

Point Cost: 4
10. Find the point on the line 2 x + 5 y  - 6 =0 which is closest to the point ( 0, 2 ).

( , )

(-0.2759, 1.31)
The work

Point Cost: 3
11. Find the minimum distance from the parabola

x - y^2 = 0
to the point (0,3).

Minimum distance =

Minimum distance = 2.2361
The work

Point Cost: NA
12. Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle with sides of length 10 if one side of the rectangle lies on the base of the triangle.

Width = Height =

W = 5

H = 4.33
The work

Point Cost: NA
13. A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y= 2-x^2. What are the dimensions of such a rectangle with the greatest possible area?

Width = Height =

W = 1.633

H = 1.333
The work

Point Cost: NA
14. A Norman window has the shape of a semicircle 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 20 feet? 28.005
The work

Point Cost: NA
15. A cylinder is inscribed in a right circular cone of height 3 and radius (at the base) equal to 3. What are the dimensions of such a cylinder which has maximum volume?

Radius = Height =

R = 2

H = 1
The work

Point Cost: NA
16. A fence 3 feet tall runs parallel to a tall building at a distance of 6 feet from the building. We want to find the the length of the shortest ladder that will reach from the ground over the fence to the wall of the building.

Here are some hints for finding a solution:
Use the angle that the ladder makes with the ground to define the position of the ladder and draw a picture of the ladder leaning against the wall of the building and just touching the top of the fence.

If the ladder makes an angle 0.85 radians with the ground, touches the top of the fence and just reaches the wall, calculate the distance along the ladder from the ground to the top of the fence.

The distance along the ladder from the top of the fence to the wall is

Using these hints write a function L(x) which gives the total length of a ladder which touches the ground at an angle x, touches the top of the fence and just reaches the wall.
L(x) = .
Use this function to find the length of the shortest ladder which will clear the fence.
The length of the shortest ladder is feet.

Ground -> fence = 3.993

Fence -> wall = 9.091



shortest ladder = 12.486
The work

Point Cost: NA
17. Centerville is the headquarters of Greedy Cablevision Inc. The cable company is about to expand service to two nearby towns, Springfield and Shelbyville. There needs to be cable connecting Centerville to both towns. The idea is to save on the cost of cable by arranging the cable in a Y-shaped configuation. Centerville is located at (9,0) in the xy-plane, Springfield is at (0,6), and Shelbyville is at (0,- 6). The cable runs from Centerville to some point (x,0) on the x-axis where it splits into two branches going to Springfield and Shelbyville. Find the location (x,0) that will minimize the amount of cable between the 3 towns and compute the amount of cable needed. Justify your answer.

To solve this problem we need to minimize the following function of x:
f(x)=
We find that f(x) has a critical number at x=
To verify that f(x) has a minimum at this critical number we compute the second derivative f''(x) and find that its value at the critical number is , a positive number.
Thus the minimum length of cable needed is



x = 3.464

Critical Number: 0.2165

Minimum Length: 19.392
The work

Point Cost: NA
18. A small town is situated on an island in the pacific ocean that lies exactly 6 miles from P, which is the nearest point to the island along a perfectly straight shoreline. 10 miles down the shoreline from P is the closest source of fresh water. If it costs 1.7 times as much money to lay pipe in the water as it does on land, how far down the shoreline from P should the pipe from the island reach land in order to minimize the total construction costs?

Distance from P =
P= 4.364
The work

Point Cost: NA