🔒 Closed Answer

[Kulantoy]

18. Gravel is being dumped from a conveyor belt at a rate of 30 cubic feet per minute so that it forms a pile in the shape of a cone whose height and base are always equal. How fast is the height of the pile increasing when the pile is 10 ft high?
a. 0.238 ft/min
b. 0.382 ft/min
c. 0.493 ft/min
d. 0.271 ft/min
19. A particle moves according to the equations = t^4+ 6t^3-12t^2 + C, where t is in seconds and 8 in meters. What time t will the particle come to rest?
a. 1.016 sec
c. 1.076 sec
b. 1.035 sec d. 1.057 sec 20. In the previous question, what is the range of values of t when the particle's speed is increasing? a. t≥ 0.256 sec b. t≥ 0.673 sec c. t2 0.451 sec d. t 2 0,562 sec 21. Find the area of the largest rectangle that can be inscribed in a right triangle with legs 3 cm and 4 cm if two sides of the rectangle lie along the legs.
a. 3 cm^2
b. 4 cm^3 c. 2 cm^2 d. 5 cm^2 22. A Norman window has the shape of a rectangle surmounted by a semicircle with the diameter of the semicircle equal to the width of the rectangle. If the perimeter of the window is 30 ft, find the width of the window so that the greatest possible amount of light is admitted. a. 8.4 ft c. 9.5 ft d. 4.8 ft 23. A wall 10 ft high is 8 ft from a house. A ladder was placed in such a way that the other end will reach the house and the other end rests on the ground outside the wall. Find the length of the shortest ladder that will reach the house.
a. 36.5 m
b. 7.3 ft
b. 25.4 m
c. 14.3 m
d. 42.5 m 24. A room in an art gallery contains a picture you are interested in viewing. The picture is two meters high and is hanging so the bottom of the picture is one meter above your eye level. How far from the wall on which the picture is hanging should you stand so that the angle of vision occupied by the picture is a maximum?
a. √3 m
c. √5 m
d. √6 m b. √2 m 25. Two posts, one 12 feet high and the other 28 feet high, stand 30 feet apart. They are to be stayed by two wires, attached to a single stake, running from ground level to the top of each post. How far from the 12- foot post should the stake be placed to use the least amount of wire?
a. 7 ft
b. 9 ft
c. 11 ft
d. 13 ft
26. A farmer has 100 m of fencing to make a rectangular enclosure for sheep. He will use an existing wall for one side of the enclosure, and leave an opening of 2 m for a gate. Find the maximum possible area. d. 1256 m^2 c. 1300.5 m^2 b. 2360.5 m^2 a. 1520 m^2 27. Find the area of the largest rectangle that has sides parallel to the coordinate axes, one corner at the origin and the opposite corner on the line 3x+2y=12 in the first quadrant. a. 6
b. 3
c. 8
d. 7 28. Find the area of the largest rectangle that can be inscribed in an ellipse with the equation 9x^2 +4y^2- 36x-8y + 4 = 0.
a. 10
b. 12
c. 15
d. 13 29. A certain cylindrical container has a volume of 355 cm^3. If it is to have an open top, what height will minimize the cost of metal to construct the can?
c. 4.83 cm
a. 3.72 cm 30. Find the height of a right circular cone of smallest volume about a sphere of volume 288m m^3. c. 22 m a. 18 m 31. In how many ways can the word "MATHEMATICS" be rearranged?
b. 8.34 cm
b. 20 m
b. 5,864,674
c. 4,542,234
d. 5.94 cm
d. 24 m
d. 4,744,234
a. 4,989,600 32. Three boys and three girls line up in a row. In how many was can they sit if all the boys and all the girls have to be together.
a. 72
b. 82
c. 87
33. In the previous question, how many ways can they sit if only the girls have to be together? b. 144 c. 633 a. 366
34. In how many ways can you invite one or more of your five friends in a party? a. 31
b. 51
c. 56
d. 56
d. 435
d. 65
35. How many ways can one chairperson, one vice chairperson, and one college representative be chosen from 14 student council members?
b. 2184
c. 8963
d. 6924
a. 5298 36. How many 3 digit numbers can be formed from the digits 0, 1, 2, 3, 4, 5, and 6 if each digit can be used only once?
b. 250
a. 587 37. In the previous question, how many of those numbers are greater than 330?
a. 105
b. 125
c. 180
c. 145
d. 136
d. 130
38. If a multiple-choice test consists of 5 questions with 4 possible answers of which only one is correct, how many different ways can a student check off one answer to each question? a. 2015
b. 1024
c. 1069
d. 2454
39. In the previous question, how many ways can a student check off one answer to each question and get all the questions wrong?
a. 250
b. 260
c. 243
d. 261
40. In how many ways can you arrange 4 identical blue boxes, 1 black box, and 3 identical red boxes in a
row?
a. 360
b. 270
c. 260
d. 280
41. The lotto uses numbers 1 to 42. A winning ticket uses 6 different numbers in any order. What is the probability of winning the lotto with only one ticket?
a. 1.906*10-7
b. 6.091*10-7
c. 9.016*10-7
d. 1.609*10-7

 

[🤖]

95. We know that the center of the ellipse is at (0,0) since the equation is in standard form. Let's call the distance from the center to point A "d". Using the distance formula, we have:

d = sqrt((√6 cos 8 - 0)^2 + (√2 sin 8 - 0)^2)
d = sqrt(6cos^2 8 + 2sin^2 8)
d = sqrt(6(cos^2 8 + sin^2 8) + 2(sin^2 8))
d = sqrt(6 + 4sin^2 8)

Since we know d = 2, we can set up the following equation:

2 = sqrt(6 + 4sin^2 8)

Solving for sin^2 8, we get:

sin^2 8 = (2^2 - 6)/4
sin^2 8 = -1/2

This is not possible since the range of sin is [-1, 1]. Therefore, there must be an error in the problem statement.

96. We can rearrange the given equation to get:

2x^2 + 4x - 3y^2 + 6y = 1

Completing the square for the x-terms, we add (4/2)^2 = 4 to both sides:

2(x+1)^2 - 3y^2 + 6y = 5

Simplifying, we have:

2(x+1)^2 - 3(y-1)^2 = 8

Dividing both sides by 8, we get:

(x+1)^2/4 - (y-1)^2/(8/3) = 1

This is the equation of an ellipse, so the answer is (d) Ellipse.

97. The latus rectum of a hyperbola is defined as the line segment through a focus of the hyperbola perpendicular to the major axis, and its length is equal to 2b^2/a, where a and b are the lengths of the major and minor axes, respectively.

We are given that 2b^2/a = 18, and we know that c, the distance between the foci, is 12. Since the conjugate axis is parallel to the y-axis, we have the equation:

y^2/a^2 - x^2/b^2 = 1

The distance between the foci is related to a, b, and c by the equation:

c^2 = a^2 + b^2

Substituting c = 12 and solving for b^2 in terms of a^2, we get:

b^2 = c^2 - a^2
b^2 = 144 - a^2

Substituting this into the equation for the latus rectum, we get:

2(144 - a^2)/a = 18
288 - 2a^2 = 18a
2a^2 + 18a - 288 = 0
(a+12)(2a-24) = 0

Since a cannot be negative, we have a = 12. Substituting this into the equation for b^2, we get:

b^2 = 144 - 12^2 = 144 - 144/4 = 108

Therefore, the length of the transverse axis is 2a = 24, and the answer is (b) 3.

98. We can rearrange the given equation to get:

9x^2 - 36x + 4y^2 - 8y + 4 = 0

Completing the square for the x-terms, we add (36/2)^2 = 324 to both sides:

9(x-2)^2 + 4y^2 - 8y + 328 = 324

Simplifying, we have:

9(x-2)^2 + 4(y-1)^2 = 8

Dividing both sides by 8 and taking the square root, we get:

[(x-2)^2]/(8/9) + [(y-1)^2]/2 = 1

This is the equation of an ellipse, so it is not a hyperbola. Therefore, there must be an error in the problem statement.

99. We can use the formula for the height of a parabolic arch at a certain distance from its vertex:

h = (w/8d)(w - d)

where h is the height of the arch, w is the width of the span, and d is the distance from the vertex to the center of the span.

In this case, w = 40ft and d = 20ft, so we have:

h = (40/8(20))(40 - 20)
h = (1/4)(20) = 5

Therefore, the height of the smallest arch that can be used is 10ft (the required clearance) plus 5ft = 15ft. However, this answer is not one of the choices given, so there may be an error in the problem statement.

100. Since the directrix is the line x = 4, the focus is at (-4,0) (the same distance from the vertex as the directrix). The standard equation for a parabola with vertex at the origin is:

y^2 = 4px

where p is the distance from the vertex to the focus (and from the vertex to the directrix).

In this case, p = 4, so we have:

y^2 = 16x

Therefore, the answer is (b) y^2 = 16x.