Differential Calculus

A balloon is rising vertically over a point A on the ground at the rate of 15 ft/sec. A point B on the ground level with and 30 ft from A. When the balloon is 40 ft. from A, at what rate is its distance from B changing? A. 13 ft/s B. 15 ft/s C. 12 ft/s D. 10 ft/s

C

The sum of the positive numbers is 50. What are the numbers if their product is to be the largest possible. A. 24 & 26 B. 28 & 22 C. 25 & 25 D. 20 & 30

C

The surface area of the sphere (initially zero) increases uniformly at the rate of 26 sq. cm. per second. Find the rate at which the radius is increasing after two seconds. A. 0.59 cm/sec B. 0.62 cm/sec C. 0.509 cm/sec D. 0.52 cm/sec

C

The volume of the closed cylindrical tank is 11.3 cubic meter. If the total surface area is a minimum, what is its base radius, in m? A. 1.44 B. 1.88 C. 1.22 D. 1.66

C

The distance a body travels is a function of time and is given by x(t) = 16t + 8t^2. Find its velocity at t = 3. A. 64 B. 56 C. 54 D. 44

A

As x increases uniformly at the rate of 0.002 feet per second, at what rate is the expression (1 + x) to the 3rd power increasing when x becomes 8 feet? A. 430 cfs B. 0.300 cfs C. 0.486 cfs D. 0.346 cfs

C

Find the height of the largest right pyramid with a square base that can be inscribed in a sphere of radius 10 cm. A. 12.2 cm B. 14.4 cm C. 13.3 cm D. 15.5 cm

C

Find the point of inflection of the curve y = x3 - 3x2 + 6. a. (0, 2) b. (1,3) c. (1, 4) d. (2, 1)

C

The shortest distance from the point (5, 10) to the curve x2 = 12y is: A. 4.331 B. 3.474 C. 5.127 D. 6.445

A

A rectangular lot has an area of 1600 sq. m. find the least amount of fence that could be used to enclose the area. a. 100 m. b. 160 m. c. 200 m. d. 300 m.

B

An open top rectangular tank with square bases is to have a volume of 10 cu. m. The materials for its bottom are to cost P15 per square meter and that for the sides, P6 per square meter. Find the most economical dimensions for the tank. A. 1.5m x 1.5m x 4.4m B. 2m x 2m x 2.5m C. 4m x 4m x 0.6m D. 3m x 3m x 1.1m

B

An open top rectangular tank with square bases is to have a volume of 10 cubic meters. The material for its bottom cost P 150.00 per square meter, and that for the sides is P 60.00 per square meter. The most economical height is: A. 2 meters B. 2.5 meters C. 3 meters D. 3.5 meters

B

The acceleration of the particle is given by a = 2 + 12t in m/s2 where t is the time in minutes. If the velocity of this particle is 11 m/s after 1 min, find the velocity after 2 mins. a. 26 m/sec b. 31 m/sec c. 37 m/sec d. 45 m/sec

B

A car starting at 12:00 noon travels west at a speed of 30 kph. Another car starting from the same point at 2:00 pm travels north at 45 kph. Find how fast the two are separating at 4:00 pm? A. 55 B. 57 C. 51 D. 53

C

A machine is rolling a metal cylinder under pressure. The radius of the cylinder is decreasing at the rate of 0.05 cm per second and the volume V is 128π cu.cm. At what rate is the length "h" changing when the radius is 2.5 cm. a. 0.8192 cm/sec b. 0.7652 cm/sec c. 0.6178 cm/sec d. 0.5214 cm/sec

A

A man 6 ft. tall is walking toward a building at the rate of 5 ft/sec. If there is a light on the ground 50 ft. from the building, how fast is the man/s shadow on the building growing shorter when he is 30 ft. from the building? a. -3.75 fps b. -7.35 fps c. -5.37 fps d. -4.86 fps

A

A norman window is in the shape of a rectangle surmounted by a semi-circle. What is the ration of the width of the rectangle to the total height so that it will yield a window admitting the most light for a given perimeter? A. 1 B. 1/2 C. 2 D. 2/3

A

A rectangular box with square base and open at the top is to have a capacity of 16823 cu.cm. Find the height of the box that requires minimum amount of material required. a. 16.14 cm b. 14.12 cm c. 12.14 cm d. 10.36 cm

A

A rectangular window surmounted by a right isosceles triangle has a perimeter equal to 54.14 m. Find the height of the rectangular window so that the window will admit the most light. a. 10 b. 22 c. 12 d. 27

A

A student club on a college campus charges annual membership dues of P10, less 5 centavos for each member over 60. How many members would give the club the most revenue from annual dues? a. 130 members b. 420 members c. 240 members d. 650 members

A

A triangle has variable sides x,y,z subject to the constraint such that the perimeter is fixed to 18cm. What is the maximum possible area for the triangle? A. 15.59 cm^2 B. 18.71 cm^2 C. 17.15 cm^2 D. 14.03 cm^2

A

A wall 2.245 m high, is "x" meters away from a building. The shortest ladder that can reach the building with one end resting on the ground outside the wall is 6 m. What is the value of x? a. 2 m b. 4 m c. 6 m d. 8 m

A

ABC company manufactures computer spare parts. With its present machines, it has an output of 500 units annually. With the addition of the new machines, the company could boost its yearly production to 750 units. If it produces "x: parts it can set a price of P = 200 - 0.15x pesos per unit and will have a total yearly cost of C = 6000 + 6x + 0.003x2 in pesos. What production level maximizes total yearly profit? a. 660 units b. 237 units c. 560 units d. 243 units

A

An open top rectangular tank with square bases is to have a volume of 10 cubic meters. The material for its bottom is to cost 15 cents per square meter and that for the sides 6 cents per square meter. Find the most economical dimension for the tank. a. 2 x 2 x 2.5 b. 2 x 5 x 2.5 c. 2 x 3 x 2.5 d. 2 x 4 x 2.5

A

Divide the number 60 into two parts so that the product P of one part and the square of the other is the maximum. Find the smallest part. a. 20 b. 10 c. 22 d. 27

A

Find the minimum amount of tin sheet that can be made into a closed cylinder having a volume of 108 cu. inches in square inches. A. 125.50 B. 127.50 C. 129.50 D. 123.50

A

Find the radius of a right circular cone having a lateral area of 544.12 sq. m. to have a maximum value. a. 10 b. 20 c. 17 d. 19

A

Find the radius of curvature of the curve x = y3 at point (1, 1). a. -1.76 b. -1.24 c. 2.19 d. 2.89

A

If the sum of two numbers is C, find the minimum value of the sum of their squares. A. C^2/2 B. C^2/4 C. C^2/6 D. C^2/8

A

If x units of a certain item are manufactured, each unit can be sold for 200 - 0.01x pesos. How many units can be manufactured for maximum revenue? What is the corresponding unit price? a. 10000, P100 b. 10500, P300 c. 20000, P200 d. 15000, P400

A

In manufacturing and selling "x" units of a certain commodity, the selling price per unit is P = 5 - 0.002x and the production cost in pesos is C = 3 + 1.10x. Determine the production level that will produce the maximum profit and what would this profit be? a. 975, P1898.25 b. 800, P1750.75 c. 865, P1670.50 d. 785, P1920.60

A

The hypotenuse of a right triangle is 20 cm. What is the maximum possible area of the triangle in square centimeters? a. 100 b. 170 c. 120 d. 160

A

Two sides of a triangle are 30 cm and 40 cm respectively. How fast is the area of the triangle increasing if the angle between the given sides is 60° and is increasing at the rate of 4°/sec. a. 20.94 m2 /sec b. 29.34 m2 /sec c. 14.68 m2 /sec d. 24.58 m2 /sec

A

Water is flowing into a conical cistern at the rate of 8 m^3/min. If the height of the inverted cone is 12m and the radius of its circular opening is 6m. How fast is the water level rising when the water is 4m deep? A. 0.64 m/min B. 0.56 m/min C. 0.75 m/min D. 0.45 m/min

A

Water is pouring into a swimming pool. After T hours, there are t +√t gallons in the pool. At what rate is the water pouring into the pool when t = 9hours? A.7/6 gph B. 8/7 gph C. 6/5 gph D. 5/4 gph

A

What is the maximum length of the perimeter if the hypotenuse of a right triangle is 5 m long? a. 12.08 m b. 15.09 m c. 20.09 m d. 8.99 m

A

Y = x3 - 3x. Find the maximum value of y. a. 2 b. 1 c. 0 d. 3

A

A boatman is at A, which is 4.5 km from the nearest point B on a straight shore BM. He wishes to reach, in minimum time, a point C situated on the shore 9 km from B. How far from C should he land if he can row at the rate of 6 Kph and walk at the rate of 7.5 Kph? A. 1 km B. 3 km C. 5 km D. 8 km

B

A box is to be constructed from a piece of zinc 20 sq.in by cutting equal squares from each corner and turning up the zinc to form the side. What is the volume of the largest box that can be so constructed? A. 599.95 cu in. B. 592.59 cu in. C. 579.50 cu in. D. 622.49 cu in.

B

A car standing at 12:00 noon travels west at a speed of 30 kph. Another car starting from the same point at 2:00 P.M. travels north at 45 kph. Find how (in kph) fast the two are separating at 4:00 P.M.? A. 49 B. 51 C. 53 D. 55

B

A fencing is limited to 20 ft. length. What is the maximum rectangular area that can be fenced in using two perpendicular corner sides of an existing wall? A. 120 B. 100 C. 140 D. 190

B

A school sponsored trip will cost each student 15 pesos if not more than 150 students make the trip. However, the cost will be reduced by 5 centavos for each student in excess of 150. How many students should make the trip in order for the school to receive the largest group income? a. 250 b. 225 c. 200 d. 195

B

A vehicle moves along a trajectory having coordinates given as x = t3 and y = 1 - t2 . The acceleration of the vehicle at any point of the trajectory is a vector, having magnitude and direction. Find the acceleration when t = 2. a. 13.20 b. 12.17 c. 15.32 d. 12.45

B

A wall "h" meters high is 2m away from the building. The shortest ladder that can react the building with one end resting on the ground outside the wall is 6m. How high is the wall in meters? A. 2.34 B. 2.24 C. 2.44 D. 2.14

B

The number of newspaper copies distributed is given by C = 50t2 - 200t + 10000, where t is in years. Find the minimum number of copies distributed from 1995 to 2002. A. 9850 B. 9800 C. 10200 D. 7500

B

With only 381.7 square meter of materials, a closed cylindrical tank of maximum volume. What is to be the height of the tank in m? a. 7 m b. 9 m c. 11 m d. 13 m

B

A certain spare parts has a selling price of P150 if they would sell 8000 units per month. If for every P1.00 increase in selling price, 80 units less will be sold out per month. If the production cost is P100 per unit, find the price per unit for maximum profit per month. a. P150 b. P250 c. P175 d. P225

C

A helicopter is rising vertically from the ground at a constant rate of 4.5 meters per second. When it is 75m off the ground, a jeep passed beneath the helicopter travelling in a straight line at a constant rate of 80kph. Determine how fast the distance between them changing after 1 second. A. 12.34 m/s B. 11.10 m/s C. 10.32 m/s D. 9.85 m/s

C

A poster is to contain 300 (cm square) of printed matter with margins of 10cm at the top and bottom and 5cm at each side. Find the overall dimensions if the total area of the poster is minimum. A. 27.76 cm, 47.8 cm B. 20.45 cm, 35.6 cm C. 22.24 cm, 44.5 cm D. 25.55 cm, 46.7 cm

C

A standard cell has an emf "E" of 1.2 volts. If the resistance "R" of the circuit is increasing at the rate of 0.03 ohm/sec, at what rate is the current "I" changing at the instant when the resistance is 6 ohms? Assume Ohm's law E = IR. A. -0.002 amp/sec B. 0.004 amp/sec C. -0.001 amp/sec D. 0.003 amp/sec

C

A statue 3 m high is standing on a base of 4 m high. If an observer's eye is 1.5 m above the ground, how far should he stand from the base in order that the angle subtended by the statue is a maximum? A. 3.41 m B. 3.51 m C. 3.71 m D. 4.41 m

C

A statue 3m high is standing on a base of 4m high. If an observer's eye is 1.5m above the ground, how far should he stand from the base in order that the angle subtended by the statue is a maximum. A. 3.41m B. 3.51m C. 3.71m D. 4.41m

C

A trapezoidal gutter is to be made from a strip of metal 22 m wide by bending up the sides. If the base is 14 m, what width across the top gives the greatest carrying capacity? a. 10 b. 22 c. 16 d. 27

C

Find two numbers whose sum is 20, if the product of one by the cube of another is to be the maximum. a. 4 and 16 b. 10 and 10 c. 5 and 15 d. 8 and 12

C

If the sum of the two numbers is 4, find the minimum value of the sum of their cubes. a. 10 b. 18 c. 16 d. 32

C

What is the allowable error in measuring the edge of the cube that is intended to hold 8 cu. m, if the error of the computed volume is not to exceed 0.03 cu. m? A. 0.002 B. 0.003 C. 0.0025 D. 0.001

C

A company estimates that it can sell 1000 units per week if it sets the unit price at P3.00, but it's weekly sales will rise by 100 units for each P0.10 decrease in price. Find the number of units sold each week and its unit price per maximum revenue. a. 1500 ; P1.50 b. 1000 ; P3.00 c. 2500 ; P2.50 d. 2000 ; P2.00

D

A function is given below, what x value maximizes y? y2 + y + x2 - 2x = 5 A. 2.23 B. -1 C. 5 D. 1

D

A particle moves in a plane according to the parametric equations of motions: x = t2, y = t3. Find the magnitude of the acceleration when the t = 0.6667. a. 6.12 b. 5.10 c. 4.90 d. 4.47

D

A piece of plywood for a billboard has an area of 24 sq. feet. The margins at the top and bottom are 9 inches and at the sides are 6 in. Determine the size of the plywood for maximum dimensions of the painted area. a. 3 x 8 b. 3 x 4 c. 4 x 8 d. 4 x 6

D

Car A moves due East at 30 kph at the same instant car B is moving S 30 deg. E, with a speed of 60 kph. The distance from A to B is 30 km. Find how fast is the distance between them separating after one hour. A. 36 kph B. 38 kph C. 40 kph D. 45 kph

D

Divide 120 into two parts so that product of one and the square of the other is maximum. Find the numbers. A. 60 & 60 B. 100 & 20 C. 70 & 50 D. 80 & 40

D

Find the radius of curvature of the curve y = 2x3 + 3x2 at (1,5). a. 90 b. 84 c. 95 d. 97

D

If y = x to the 3rd power - 3x. find the maximum value of y. A. 0 B. -1 C. 1 D. 2

D

Two cities A and B are 8 km and 12 km, respectively, north of a river which runs due east. City B being 15 km east of A. a pumping station is to be constructed (along the river) to supply water for the two cities. Where should the station be located so that the amount of pipe is a minimum? A. 3 km east of A B. 4 km east of A C. 9 km east of A D. 6 km east of A

D

Two sides of a triangle are 15 cm and 20 cm long respectively. How fast is the third side increasing if the angle between the given sides is 60° and is increasing at the rate of 2°/sec. a. 3.60 cm/sec b. 2.70 cm/sec c. 1.20 cm/sec d. 0.05 cm/sec

D

Two vertices of a rectangle are on the x axis. The other two vertices are on the lines whose equations are y = 2x and 3x + y = 30. 57. If the area of the rectangle is maximum, find the value of y. a. 8 b. 7 c. 9 d. 6

D

A Norman window is in the shape of a rectangle surmounted by a semi-circle. What is the ratio of the width of the rectangle to the total height so that it will yield a window admitting the most light for a given perimeter? A. 1 B. 2/3 C. 1/3 D. ½

A

A farmer has enough money to build only 100 meters of fence. What are the dimensions of the field he can enclose the maximum area? A. 25m x 25m B. 15m x 35m C. 20m x 30m D. 22.5m x 27.5m

A

A gutter with trapezoidal cross section is to be made from a long sheet of tin that is 15 cm. wide by turning up one third of its width on each side. What is the width across the top that will give a maximum capacity? a. 10 b. 20 c. 15 d. 13

A

A hemispherical dome has a diameter of 100m. A searchlight was placed at a point A located at the circumstance at the base. At the middle of the dome at B, a balloon was released vertically at a velocity of 4 m/s. How fast is the shadow of the balloon move along the roof when the balloon is 25 m high? A. 6.4 m/s B. 4.6 m/s C. 7.2 m/s D. 4 m/s

A

What is the least amount of tin sheet, in sq. inches, that can be made into a closed cylindrical can having a volume of 108 cu. inches? A. 125 square meter B. 137 square meter C. 150 square meter D. 120 square meter

A

The coordinates (x,y) in feet of a moving particle P are given by x=cost - 1 and y = 2sint + 1, Where t is the time in seconds. At what extreme rates in fps is P moving along the curve? A. 3 and 2 B. 3 and 1 C. 2 and 0.5 D. 2 and 1

D

A monthly overhead of a manufacturer of a certain commodity is P6000 and the cost of the material is P1.0 per unit. If not more than 4500 units are manufactured per month, labor cost is P0.40 per unit, but for each unit over 4500, the manufacturer must pay P0.60 for labor per unit. The manufacturer can sell 4000 units per month at P7.0 per unit and estimates that monthly sales will rise by 100 for each P0.10 reduction in price. Find the number of units that should be produced each month for maximum profit. A. 4700 units B. 2600 units C. 6800 units D. 9900 units

A

A printed page must contain 60 sq.m. of printed material. There are to be margins of 5 cm. on either side and the margins of 3 cm. on top and bottom. How long should the printed lines be in order to minimize the amount of paper used? a. 10 b. 18 c. 12 d. 15

A

An LTR train 6m above the ground crosses a street at 9 m/s at the instant that a car approaching at a speed of 4 m/s is 12m up the street. Find the rate of the LRT train and the car separating one second later. A. 3.64 m/s B. 3.94 m/s C. 4.24 m/s D. 4.46 m/s

A

Compute the maximum area of the rectangle. a. 30 sq. units b. 70 sq. units c. 90 sq. units d. 40 sq. units

A

Determine the shortest distance from point (4,2) to the parabola y^2 = 8x. A. 2.83 B. 3.54 C. 2.41 D. 6.32

A

Given a cone of diameter x and altitude of h. What percent is the volume of the largest cylinder which can be incribed in the cone to the volume of the cone? A. 44% B. 46% C. 56% D. 65%

A

If the hypotenuse of a right triangle is known, what is the ratio of the base and the altitude of the right triangle when its area is maximum? a. 1:1 b. 1:2 c. 1:3 d. 1:4

A

The altitude of a cylinder of maximum volume that can be inscribed in a right circular cone of radius r and height h is: A. h/3 B. 2h/3 C. 3h/2 D. h/4

A

The distance traveled by a train is given by the equation x = 3t^2 + 2t + 4 where x is the distance in kilometers and t is the time in hours. Determine the velocity of the train after travelling 10 kilometers. A. 8.72 kph B. 18.72 kph C. 23.5 kph D. 75.3 kph

A

The edges of a rectangular box are to be reinforced with narrow metal strips. If the box will have a volume of 8 cubic meters, what would its dimension be to require the least total length of strips? a. 2 x 2 x 2 b. 4 x 4 x 4 c. 3 x 3 x 3 d. 2 x 2 x 4

A

The highway department is planning to build a picnic area for motorist along a major highway. It is to be rectangular with an area of 5000 sq. m. is to be fenced off on the three sides not adjacent to the highway. What is the least amount of fencing that will be needed to complete the job? a. 200 m. b. 300 m. c. 400 m. d. 500 m.

A

The motion of a particle is defined by the parametric equation x =t^3 and y =2t^2. Determine the velocity when t = 2. A. 14.42 B. 16.25 C. 12.74 D. 18.63

A

The sum of two numbers is K. The product of one by the cube of the other is to be a maximum. Determine one of the numbers. A. 3K/4 B. 3K/8 C. 3K/2 D. 3K/7

A

The volume of the sphere is increasing at the rate of 6cm3/hr. At what rate is its surface area increasing when the radius is 50 cm (in cm3/hr) a. 20.94 m2/sec b. 29.34 m2/sec c. 14.68 m2/sec d. 24.58 m2/sec

A

Three sides of a trapezoid are each 8 cm long. How long is the 4th side, when the area of the trapezoid has the greatest value? A. 16 cm B. 15 cm C. 12 cm D. 10 cm

A

Two railroad tracks are perpendicular to each other. At 12:00 P.M. there is a train at each track approaching the crossing at 50 kph, one being 100 km and the other 150 km away from the crossing. How fast on kph is the distance between the two trains changing at 4:00 P.M.? A. 67.08 B. 68.08 C. 69.08 D. 70.08

A

What is the curvature of the curve y^2 = 16x at the point (4,8)? A. -0.044 B. -0.088 C. -0.066 D. -0.033

A

A balloon is rising vertically over a point A on the ground at the rate of 15 ft/sec. A point B on the ground is level with and 30 ft. from A. When the balloon is 40 ft. from A, at what rate is its distance from B changing? A. 10 ft/s B. 12 ft/s C. 13 ft/s D. 15 ft/s

B

A boatman is at A which is 4.5 km from the nearest point B on a straight shore BM. He wishes to reach in minimum time a point C situated on the shore 9km from B. How far from C should he land if he can row at the rate of 6 kph and can walk at the rate of 7.5 kph? A. 4.15 km B. 3.0 km C. 3.25 km D. 4.0 km

B

A normal widow is in the shape of a rectangle surrounded by a semi-circle. If the perimeter of the window is 71.416, what is the radius and the height of the rectangular portion so that it will yield a window admitting the most light. a. 12 b. 20 c. 22 d. 27

B

A particle moves according to the parametric equations: y = 2t^2 x = t^3 where x and y are displacements (in meters) in x and y direction, respectively and t is time in seconds. Determine the acceleration of the body after t=3 seconds. A. 12.85 m/s2 B. 18.44 m/s2 C. 21.47 m/s2 D. 5.21 m/s2

B

A particle moves along a path whose parametric equations are x = t3 and y = 2t2 . What is the acceleration when t = 3sec? a. 15.93 m/sec2 b. 18.44 m/sec2 c. 23.36m/sec2 d. 10.59 m/sec2

B

A rectangular box having a square base and open at the top is to have a capacity of 16823 cc. Find the height of the box to use the least amount of material. A. 16.14 cm B. 32.28 cm C. 18.41 cm D. 28.74 cm

B

A rectangular field is to be fenced into four equal parts. What is the size of the largest field that can be fenced this way with a fencing length of 1500 feet if the division is to be parallel to one side? A. 65,200 B. 62,500 C. 64,500 D. 63,500

B

A van is 5km due north of a bus at 2:00 p.m. If the van is travelling northward at the rate of 60kph and the bus is travelling westwards at the rate of 75kph, how fast will the two be separating at 5:00 p.m.? A. 85.36 kph B. 96.04 kph C. 98.65 kph D. 102.53 kph

B

Compute the radius of curvature of the curve x = 2y3 - 3y2 at (4, 2). a. -99.38 b. - 97.15 c. -95.11 d. -84.62

B

Find the minimum distance from the point (4,2) to the parabola y^2 = 8x A. 4√3 B. 2√2 C. √3 D. 2√3

B

Find the radius of curvature of a parabola y2 - 4x = 0 at point (4, 4). a. 25.78 b. 22.36 c. 20.33 d. 15.42

B

Given the following profit-versus-production function for a certain commodity: P = 200000 - x - [1.1 / (1 + x)]8 . Where P is the profit and x is unit of production. Determine the maximum profit. A. 190000 B. 200000 C. 250000 D. 550000

B

If the distance y from the origin at time t is given by y = -16t^2 + 3000t + 50000, find the initial velocity when t = 0. A. 0 B. 3000 C. 5300 D. 50000

B

Sand is falling off a conveyor onto a conical pile at the rate of 15 cm3 /min. The base of the cone is approximately twice the altitude. Find the height of the pile if the height of the pile is changing at the rate 0.047746 cm/min. a. 12 cm b. 10 cm c. 8 cm d. 6 cm

B

The cost C of a product is a function of the quantity x of the product is given by the relation: C(x) = x2 - 4000x + 50. Find the quantity for which the cost is a minimum. A. 3000 B. 2000 C. 1000 D. 1500

B

The speed of the traffic flowing past a certain downtown exit between the hours of 1:00 P.M. and 6:00 P.M. is approximately V = t^3 - 10.5t^2 + 30t + 20 miles per hour, where t = number of hours past noon. What is the fastest speed of the traffic between 1:00 P.M. and 6:00 P.M. in mph? A. 50 B. 46 C. 32.5 D. 52

B

The sum of two numbers is 12. Find the minimum value of the sum of their cubes. a. 644 b. 432 c. 346 d. 244

B

Two railroad tracks are perpendicular with each other. At 12:00 P.M. there is a train on each track approaching the crossing at 50 kph, one being 100 km., the other 150 km. away from the crossing. How fast in kph is the distance between the two trains changing at 4:00 PM.? A. 68.08 B. 67.08 C. 69.08 D. 70.08

B

Water flows into a tank having the form of a frustum

B

What is the minimum profit when the profit-versus-production function is as given below? P is profit and x is unit of production. P = 200,000 - x - (1.1/x+1)^8 A. 285,000 B. 200,000 C. 250,000 D. 305,000

B

A 3-meter long steel pipe has its upper end leaning against a vertical wall and lower end on a level ground. The lower end moves away at a constant rate of 2 cm/s. How fast is the upper end moving down, in cm/s, when the lower end is 2m from the wall? A. 1.81 B. 1.66 C. 1.79 D. 1.98

C

A box is to be constructed from a piece of zinc 20 inches square by cutting equal squares from each corner and turning up the zinc to form the side. What is the volume of the largest box that can be so constructed? A. 599.95 cubic inches B. 579.50 cubic inches C. 592.59 cubic inches D. 622.49 cubic inches

C

A certain travel agency offered a tour that will cost each person P 1500.00 if not more than 150 persons will join, however the cost per person will be reduced by P 5.00 per person in excess of 150. How many persons will make the profit a maximum? A. 75 B. 150 C. 225 D. 250

C

A cylindrical steam boiler is to constructed having a capacity of 1000 cu. m. The material for the sides cost P2000.00 per square meter and for the ends P3000.00 per square meter. Find the radius so that the cost is least. A. 3.52 m B. 4.12 m C. 4.73 m D. 5.25 m

C

A man walks across a bridge at the rate of 5 fps as a boat passes directly beneath him at 10 fps. If the bridge is 10 feet above the boat, how fast are the man and the boat separating 1 second later? A. 8 fps B. 8.25 fps C. 8.33 fps D. 8.67 fps

C

A manufacturer estimates that the cost of production of "x" units of a certain item is C = 40x - 0.02x2 - 600. How many units should be produced for minimum cost? a. 10 units b. 100 units c. 1000 units d. 10000 units

C

At any distance x from the source of light, the intensity of illumination varies directly as the intensity of the source and inversely as the square of x. Suppose that there is a light at A, and another at B, the one at B having an intensity 8 times that of A. The distance AB is 4m. At what point from A on the line AB will be intensity of illumination be least? A. 2.15m B. 1.33m C. 1.50m D. 1.92m

C

At what point from the intersection of the x and y axes will the farthest vertex of the rectangle be located along the x axis so that its area is maximum. a. 9 units b. 7 units c. 8 units d. 6 units

C

Determine the diameter of a closed cylindrical tank having a volume of 11.3 cu. m to obtain minimum surface area. A 1.22 B. 1.64 C. 2.44 D. 2.68

C

The cost of C of a product is a function of the quantity x of the product: C(x) = x2 - 400x + 50. Find the quantity for which the cost is minimum. A. 1000 B. 1500 C. 2000 D. 3000

C

The cost of fuel in running a locomotive is proportional to the square of the speed and in $25 per hour for a speed of 25 miles per hour. Other costs amount to $100 per hour, regardless of the speed. What is the speed which will make the cost per mile a minimum? A. 40 B. 55 C. 50 D. 45

C

Water is pouring into a conical vessel 15cm deep and having a radius of 3.75cm across the top. If the rate at which the water rises is 2 cm/sec, how fast is the water flowing into the conical vessel when the water is 4cm deep? A. 2.37 m^3/sec B. 5.73 m^3/sec C. 6.28 m^3/sec D. 4.57 m^3/sec

C

Water is running into a hemispherical bowl having a radius of 10cm at a constant rate of 3cm^3/min. When the water is x cm. deep, the water level is rising at the rate of 0.0149 cm/min. What is the value of x? A. 3 B. 2 C. 4 D. 5

C

A balloon is released from the ground 100 meters from an observer. The balloon rises directly upward at the rate of 4 meters per second. How fast is the balloon receding from the observer 10 seconds later? A. 1.68 m/s B. 1.36 m/s C. 1.55 m/s D. 1.49 m/s

D

A car drives east from point A at 30 kph. Another car starting from B at the same time, drives S 30 deg. W toward A at 60 kph. B is 30 km away from A. How fast in kph is the distance between the two cars changing after one hour? A. 76.94 kph B. 78.94 kph C. 75.94 kph D. 77.94 kph

D

A closed cylindrical tank has a capacity of 576.56 cubic meters. Find the minimum surface area of the tank. a. 218.60 cubic meters b. 412.60 cubic meters c. 516.32 cubic meters d. 383.40 cubic meters

D

A load of 40 kN is to be raised by means of a lever weighing 250 N/m, which is supported at one end. If the load is placed 1m from the support, how long should the lever be so that the force required be a minimum? A. 13.43 m B. 20.19 m C. 18.56 m D. 17.89 m

D

An airplane is flying horizontally at an altitude of 9000 m. An observer on the ground noticed that when the angle of elevation of the plane is 60 deg. , the angle decreases at the rate of 0.15 radians/second. What is the velocity of the airplane? A. 600 m/s B. 900 m/s C. 1400 m/s D. 1800 m/s

D

An iron bar 20 m long is bent to form a closed plane area. What is the largest area possible? A. 21.56 square meter B. 25.68 square meter C. 28.56 square meter D. 31.83 square meter

D

An object moves along a straight line such that, after t minutes, its distance from its starting point is D = 20t + 5/ (t + 1) meters. At what speed, in m/minute will it be moving at the end of 4 minutes? A. 39.8 B. 49.8 C. 29.8 D. 19.8

D

Sand is pouring from a spout at the rate of 25 cc/sec. It forms a cone whose height is always 1/3 the radius of its base. At what rate in cm/sec is the height increasing when the cone is 50 cm high? A. 0.000785 B. 0.000214 C. 0.000447 D. 0.000354

D

Suppose that x years after founding in 1975 a certain employee association had a membership of f(x) = 100(2x^3 - 45x^2 + 264x), at what time between 1975 and 1989 was the membership smallest? A. 1983 B. 1985 C. 1984 D. 1986

D

The cost per hour of running a motor boat is proportional to the cube of the speed. At what speed will the boat run against a current of 8km/hr on order to go a given distance most economically? A. 10 kph B. 13 kph C. 11 kph D. 12 kph

D

The height of a right circular cylinder is 50 inches and decreases at the rate of 4 inches per second, while the radius of the base is 20 inches and increases at the rate of one inch per second. At what rate is the volume changing? A. 11310 cu. in./sec B. 1275 cu. in./sec C. 11130 cu. in./sec D. 1257 cu. in./sec

D

There is a constant inflow of a liquid into a conical vessel 15 feet deep and 7.5 feet in diameter at the top. Water is rising at the rate of 2 feet per minute when the water is 4 feet deep. What is the rate if inflow in cu. ft. per minute? A. 8.14 B. 7.46 C. 9.33 D. 6.28

D