ARC Refresher

Question 1

A laboratory technician needs to prepare 500 mL of a 25% acid solution by mixing a 10% acid solution with a 40% acid solution. How much of each solution should be mixed to create the desired concentration?

A. 250 mL of the 10% acid solution and 250 mL of the 40% acid solution
B. 150 mL of the 10% acid solution and 350 mL of the 40% acid solution
C. 200 mL of the 10% acid solution and 300 mL of the 40% acid solution
D. 100 mL of the 10% acid solution and 400 mL of the 40% acid solution

Answer 1

A. 250 mL of the 10% acid solution and 250 mL of the 40% acid solution

Question 2

A jeweler needs to create 100 grams of a gold alloy that is 70% gold. They have two types of alloys available: one that is 60% gold and another that is 90% gold. How many grams of each alloy should the jeweler mix to create the desired 70% gold alloy?

A. 33.33 grams of the 60% gold alloy and 66.67 grams of the 90% gold alloy
B. 60 grams of the 60% gold alloy and 40 grams of the 90% gold alloy
C. 66.67 grams of the 60% gold alloy and 33.33 grams of the 90% gold alloy
D. 40 grams of the 60% gold alloy and 60 grams of the 90% gold alloy

Answer 2

C. 66.67 grams of the 60% gold alloy and 33.33 grams of the 90% gold alloy

Question 3

A chemist has a 40% salt solution and needs to dilute it to create 200 mL of a 15% salt solution. How much water should be added to achieve the desired concentration?

A. 100mL
B. 125mL
C. 150mL
D. 175mL

Answer 3

B. 125mL

Question 4

Two lines are given by the equations:
Line 1: y = 3x + 2y
Line 2: y = (−1/3)x + 5
Find the angle between these two lines in degrees.

A. 30deg
B. 45deg
C. 60deg
D. 90deg

Answer 4

D. 90deg

Question 5

Find the distance from the point (4, −3) to the line given by the equation 3x − 4y + 5 = 0.

A. 5.2
B. 5.4
C. 5.6
D. 5.8

Answer 5

D. 5.8

Question 6

Find the distance between the parallel lines:
Line 1: 3x − 4y + 12 = 0
Line 2: 3x − 4y − 8 = 0

A. 1
B. 2
C. 3
D. 4

Answer 6

Question 7

A stone is thrown so that it will hit a bird at the top of a pole. However, at the instant the stone is thrown, the bird flies away in a horizontal straight line at a speed of 10m/s. The stone reaches a max height that is twice the height of the pole and on its descent it touches the bird. Find horizontal component of velocity of the stone.

A. 11.1 m/s
B. 12.1 m/s
C. 13.1 m/s
D. 14.1 m/s

Answer 7

B. 12.1 m/s

Question 8

An ellipse is given by the equation:

If the dimensions of the possible rectangle is to be maximized that can be inscribed in this ellipse, where the sides of the rectangle are parallel to the coordinate axes, find the area of the largest rectangle that can be inscribed in the given ellipse.

A. 6
B. 9
C. 12
D. 15

Answer 8

C. 12

Question 9

A long, straight pipe needs to be carried horizontally through a hallway that makes a 90-degree turn. The first hallway is 3 meters wide, and the second hallway is 4 meters wide. What is the maximum length of the pipe that can be carried around the corner without tilting it vertically?

A. 8.78m²
B. 9.87m²
C. 10.03m²
D. 11.18m²

Answer 9

B. 9.87m²

Question 10

Two vertical posts, one 20 meters high and the other 30 meters high, are 50 meters apart. A cable is to be strung from the top of the first post to a point on the ground between the posts, and then from that point to the top of the second post. Where should the point on the ground be located to minimize the length of the cable?

A. 10m from the shorter post
B. 20m from the shorter post
C. 30m from the shorter post
D. 40m from the shorter post

Question 11

What is the minimum length of the cable of the previous problem?

A. 50.92m
B. 64m
C. 70.71m
D. 81m

Question 12

You have n positive numbers whose sum is 100. Determine the values of n that will maximize their product.

A. 34
B. 35
C. 36
D. 37

Question 13

The total sum of x and y is 100. Determine the values of x and y that will maximize the product:
P = x³y²

A. 40, 60
B. 50, 50
C. 60, 40
D. 20, 80

Question 14

What is the maximum possible product of the previous problem?

A. 345,600,000
B. 350,000,000
C. 356,400,000
D. 360,000,000

Question 15

A Norman window has the shape of a rectangle surmounted by a semicircle. Suppose the total perimeter of the window is 30 meters. Determine the dimensions of the window that will allow the maximum amount of light to be admitted, assuming that maximizing the area of the window maximizes the light admitted. What is the maximum area of the window?

A. 63.01m²
B. 64m²
C. 65.01m²
D. 66m²

Question 16

A window is in the shape of a rectangle topped with a right isosceles triangle. Suppose the total perimeter of the window is 20 meters. The dimensions of the window (width of the rectangle, height of the rectangle, and the height of the triangular part) will maximize the area of the window. What is that maximum area?

A. 23.24m²
B. 24.36m²
C. 25m²
D. 26.12m²

Question 17

A rectangular garden is to be constructed along an existing wall, which will serve as one of the sides of the garden. The total area of the garden must be 200 m² . Only three sides of the garden need to be fenced, as the existing wall will serve as the fourth side. The dimensions of the garden will minimize the amount of fencing required. What is the minimum fencing perimeter?

A. 40m
B. 50m
C. 60m
D. 80m

Question 18

Consider a rectangle with a fixed perimeter of 40 units. Let the dimensions of the rectangle be x and y, representing the length and the width, respectively. The objective is to maximize the area of the rectangle. Evaluate the maximum area.

A. 100
B. 200
C. 300
D. 400

Question 19

Consider a rectangle with a fixed area of 120 square units. Let the dimensions of the rectangle be x and y, representing the length and the width, respectively. The objective is to minimize the perimeter of the rectangle. Evaluate the minimum perimeter.

A. 600
B. 700
C. 800
D. 900

Question 20

Consider a right triangle with a base of length b = 12 units and a height of length h = 8 units. We want to find the largest rectangle that can be inscribed inside this triangle, where the rectangle shares one side with the base of the triangle. What is the area of the needed rectangle?

A. 12
B. 24
C. 36
D. 48

Question 21

Consider a semicircle with a radius of R = 10 units. We want to find the dimensions of the largest rectangle that can be inscribed inside this semicircle, with the rectangle’s base lying along the diameter of the semicircle. What is the area of the rectangle?

A. 100
B. 200
C. 300
D. 400

Question 22

Consider a closed cylindrical vessel with a fixed surface area of S = 150 square units. The objective is to maximize the volume of the cylinder. What is the volume of the cylinder?

A. 100
B. 124.62
C. 141.05
D. 250

Question 23

Consider a closed cylindrical vessel with a fixed volume of V = 500 cubic units. The objective is to minimize the surface area of the cylinder. Let r be the radius of the base, and h be the height of the cylinder. What is the surface area of the smallest cylinder?

A. 213.46
B. 348.73
C. 447.28
D. 503.78

Question 24

Consider an open-top cylindrical vessel with a fixed surface area of S = 600 square units. The objective is to maximize the volume of the cylinder. What is the volume of the cylinder?

A. 729.28
B. 802.97
C. 992.37
D. 1036.48

Question 25

Consider an open-top cylindrical vessel with a fixed volume of V = 500 cubic units. The objective is to minimize the surface area of the cylinder. What is the surface area of the cylinder?

A. 369.05
B. 401.89
C. 512.34
D. 607.98

Question 26

Consider an inverted conical vessel with a height of 12 meters and a base radius of 4 meters. Water is being pumped into the vessel at a rate of 2 cubic meters per minute. We want to determine the rate at which the water level h is rising when the water is 6 meters deep.

A. 1/pi m/min
B. 1/(2pi) m/min
C. 1/(3pi) m/min
D. 1/(4pi) m/min

Question 27

Consider an inverted conical vessel with a height of 15 meters and a base radius of 5 meters. Water is being pumped into the vessel at a rate of 3 cubic meters per minute, and the rate at which the water level is rising is 0.1 meters per minute. We want to determine the height of the water in the vessel at that instant.

A. 5.17m
B. 7.71m
C. 9.27m
D. 11.87m

Question 28

Consider a hemispherical vessel with a radius of 10 meters. Water is being pumped into the vessel at a rate of 5 cubic meters per minute. We want to determine the rate at which the water level h is rising when the depth of the water is 4 meters.

A. 0.0124
B. 0.0248
C. 0.0497
D. 0.0746

Question 29

Consider a hemispherical vessel with a radius of 8 meters. Water is being pumped into the vessel at a rate of 6 cubic meters per minute, and the rate at which the water level is rising is 0.2 meters per minute. We want to determine the height of the water level h at this instant.

A. 0.31m
B. 0.62m
C. 1.24m
D. 2.48m

Question 30

Consider a car that is accelerating at a constant rate of 2 meters per second squared. At time t = 0 seconds, the car is at rest, and its position (displacement) s(t) is measured from a fixed starting point. We want to determine how fast the car is moving (velocity) after 5 seconds and how far it has traveled by then.

A. v = 5m/s; s = 11m
B. v = 6m/s; s = 12m
C. v = 8m/s; s = 16m
D. v = 10m/s; s = 25m