Test: Rectangular Solids- 1 - GMAT MCQ

Step 1: Question statement and Inferences

A certain right circular cylinder is filled halfway with g gallons of water, so its volume is 2g. When placed on its side, the maximum depth of the water is c centimeters, so its radius is c.

Step 2: Finding required values

Set up the equation to solve for h. The volume of any right circular cylinder is V = πr²h. Knowing that V = 2g and that c = r, set up the equation.

Step 3: Calculating the final answer

Set up the equation like this:

2g = πc²h

To isolate h, divide both sides by πc²

h=2g/πc2

Answer: Option (B)

Step 1: Question statement and Inferences

For the volume of the rectangular solid, you need the length, width, and height. You have that the diagonal connecting A to E is twice the height, and that the length is √3.

Step 2: Finding required values

The dotted line creates a right triangle. If the hypotenuse is twice the height, call the height h and the hypotenuse 2h. With a base of √3 , plug these into the Pythagorean Theorem:

(√3)² + h² = (2h)²

3 + h² = 4h²

h² = 1

h = 1

Step 3: Calculating the final answer

Knowing the height EB is 1 and that BCDE is a square, the width BC is also 1. and the length AB is√3.

So, the volume of the solid = 1 × 1 × √3 = √3

Answer: Option (A)

Steps 1 & 2: Understand Question and Draw Inferences

For the volume of the cube, you need the edge length. You can find the edge length in different ways, such as from the area of a face or other dimensions of the cube. See if the statements provide something you can use to find the length of an edge.

Step 3: Analyze Statement 1

(1) Each face of the cube has an area of 5

From this, you can calculate that the edge length is √5, giving the cube a volume of 5√5

Statement (1) is sufficient.

Step 4: Analyze Statement 2

(2) The longest possible straight-line distance between any two points of the cube is √15

The longest possible straight-line distance is the body diagonal, represented by dotted line AD in the drawing below.

To find the edge, calculate the right triangles until you reach dotted line AD.

1. Call the edge e , which represents the lengths of edges BC, CD, DE, BE, and AB in the drawing.

2. With BCD as a right triangle, find the length of dotted line BD:

3. Now you have a new right triangle ABD, where AB = e, BD = e√2, and hypotenuse AD = √15 (from the statement). Set up the equation to solve for e:

And from this, you can calculate the cube volume of 5√5

Statement (2) is sufficient.

Step 5: Analyze Both Statements Together (if needed)

You get unique answers in steps 3 and 4, so this step is not required

Answer: Option (D)

Step 1: Question statement and Inferences

Let the radius, height and volume of bucket A be r, h and v. Let the radius, height and volume of bucket B be R, H and V.

Since three-fourths of bucket A raises the volume of bucket B rise from one-fourths to one-half of its volume,

ividing both sides of the equation by , we get

Step 2: Finding required values

We also know that r = 2R and H = 2. Substituting these values and, we have

Step 3: Calculating the final answer

Dividing both sides by R², we obtain

Correct Answer : B

Step 1: Question statement and Inferences

For the volume of the rectangular solid, you need the length, width, and height. You already have the length and the height, 4 and 2, respectively. Use the cylinders to calculate the width.

Step 2: Finding required values

Draw an equilateral triangle where each vertex is the center of one circle.

If the length of the rectangle is 4, the diameter of each circle is 2, and the radius is 1. The triangle is equilateral with a base of 2.

To find the height of the triangle, cut it in half as shown below, into two 30-60-90 triangles. With a hypotenuse of 2 and a base of 1 (half the side is the base), use the Pythagorean Theorem to find the third side, which is the height:

Add this height of the triangle to the circle radii above and below it for a rectangular solid width of (2 + √3)

Step 3: Calculating the final answer

By multiplying the length and height by the newly found width, we get:

The volume of the solid =  4*2*(2 + √3)

= 8*(2 + √3)

Answer: Option (D)

Steps 1 & 2: Understand Question and Draw Inferences

Let the radius and height of Cylinder A be R_A and H_A respectively, and

The radius and height of Cylinder B be R_B and H_B respectively.

Given:

R_A: H_A = R_B: H_B

This equation can also be written as:

R_A: R_B = H_A : H_B  . . . (1)

We know that the curved surface area of a right circular cylinder, CSA = 2πRH, where R and H are the radius and height of the cylinder respectively.

So, CSA_A : CSA_B = 2πR_A H_A : 2πR_B H_B  

CSA_A : CSA_B = R_A H_A : R_B H_B   . . . (2)

Substituting (1) in (2), we get:

CSA_A : CSA_B = R_A ²: R_B² = H_A ²: H_B²  . . . (3)

From Equation (3) it is clear that we will be able to determine the ratio of CSA_A : CSA_B if we know the ratio R_A  : R_B  or the ratio H_A  : H_B  

Step 3: Analyze Statement 1

(1)         The ratio of the radius of cylinder A to the radius of cylinder B is 2:1

Since the ratio R_A  : R_B  is given, we will be able to determine the ratio CSA_A : CSA_B

Statement 1 is sufficient.

Step 4: Analyze Statement 2

(2)         The ratio of the volume of cylinder A to the volume of cylinder B is 8:1

We know that volume of a right circular cylinder, V = πR² H, where R and H are the radius and height of the cylinder respectively.

So,

V_A : V_B = πR_A ²H_A : πR_B ²H_B  

V_A : V_B = R_A ²H_A : R_B ²H_B  . . . (4)

Substituting (1) in (4), we get:

V_A : V_B = R_A ³: R_B³   . . . (5)

We are given that V_A : V_B = 8:1

So, from Equation (5), we get:

R_A ³: R_B³   = 8:1

This means that,

R_A  : R_B  = 2:1

Since we have been able to determine the ratio R_A  : R_B  is given, we will be able to determine the ratio CSA_A : CSA_B

Statement 2 is sufficient.

Step 5: Analyze Both Statements Together (if needed)

You get a unique answers in steps 3 and 4, so this step is not required

Answer: Option (D)

Steps 1 & 2: Understand Question and Draw Inferences

Some general points to be noted:

(a) In a rectangular solid, each pair of adjacent faces has one common edge.

(b) The opposite faces have the same dimensions.

Some inference that can be drawn from the question:

(c) Since ‘exactly’ four faces have the same dimensions, the other two faces must have different dimensions.

(d) Two of the faces which have different dimensions compared the four faces which have the same dimensions must have the following properties:

(i) They must be opposite to each other and their areas must be equal.

(ii) They must both be of the shape of a square.

(iii) Each of these two faces has a common edge with each one of the other four faces.

(e) The lengths of edges are positive integers.

Step 3: Analyze Statement 1

The possible values of the lengths and breadths for the face whose area is 32 sq. units are: 32 and 1, OR 16 and 2 OR 8 and 4.

The possible values of the lengths and breadths for the face whose area is 16 sq. units are: 16 and 1 OR 8 and 2 OR 4 and 4.

Since one of the surfaces must be a square and the lengths of edges are integers, between the two areas given only the face with area of 16 sq. units makes a square. There will be only two such surfaces in this solid with edge lengths 4 units each.

The only possible lengths of the faces whose areas are 32 square units are 8 units and 4 units.

We now know all the dimensions of the rectangular solid (8 units, 4 units, 4 units) and hence its volume can be found out.

SUFFICIENT.

Step 4: Analyze Statement 2

Let’s say that the square face has edges of lengths ‘a’ units each. This means that the length of the other side of the rectangular faces is ‘2a’ or ‘a/2’. (Depending on which length is the bigger one.)

Since we do not have the values of the edges or any other numeric information, we cannot ascertain the volume of this rectangular solid.

INSUFFICIENT.

Step 5: Analyze Both Statements Together (if needed)

Since the answer has been obtained in Step 3, there is no need to combine the statements.

Correct Answer: A

Steps 1 & 2: Understand Question and Draw Inferences

For the longest straight-line distance between any two points, you need the length, width, and height of the rectangular solid.  You know that the height, the width, and the length are in ascending order of magnitude.

Step 3: Analyze Statement 1

(1) The length, width, and height of the rectangular solid are the squares of prime integers. 

This narrows the possibilities: L, W, and H could be 4, 9, 25, 49, and so forth, but there are still too many possibilities. 

Statement (1) is not sufficient.

Step 4: Analyze Statement 2

(2) The roots of the equation x² – 13x = -36 are the width and the height of the rectangular solid.

Factor the expression to (x – 9)(x – 4) for the roots. Thus the height and the width are 4 and 9, respectively (Since H < W).

However, knowing H < W < L, the length could be any number greater than 9.

Statement (2) is not sufficient.

Step 5: Analyze Both Statements Together (if needed)

You know from the question that H < W < L.

You know from Statement 1 that L, W, and H could be 4, 9, 25, 49, and so forth.

You know from Statement 2 that H and W are 4 and 9, respectively.

Taken together, L could be 25, 49, 121, or the square of any prime number greater than 3.

Hence both the statements together are not sufficient.

Answer: Option (E)

Volume of 4 cm cube = 64 cc. When it is cut into 1 cm cube, the volume of each of the cubes = 1cc
Hence, there will be 64 such cubes. Surface area of small cubes = 6 (1²) = 6 sqcm.
Therefore, the surface area of 64 such cubes = 64 * 6 = 384 sqcm.
The surface area of the large cube = 6(4²) = 6*16 = 96.

A regular hexagon comprises 6 equilateral triangles, each of them having one of their vertices at the center of the hexagon. The sides of the equilateral triangle are equal to the radius of the smallest circle inscribing the hexagon. Hence, each of the side of the hexagon is equal to the radius of the hexagon and the perimeter of the hexagon is 6r.