A toy is in the form of a right circular cone mounted on a hemisphere. If the height of the cone is twice the radius of the cone and the total height of the toy is 15 centimeters, what is the volume of the hemisphere in cubic centimeters? (The volume of a sphere is , where r is the radius)
To Find: Volume of hemisphere
Approach
Working Out
Answer: C
The figure above shows a cube one of whose faces has been painted with a blue circle that touches each edge of the face. If all the other faces of the cube are also to be painted in a similar manner, how much surface area of the cube will be left unpainted?
(1) The surface area of the cube is 1944 square units
(2) The area of the blue circle is 81π square units
Solution
Steps 1 & 2: Understand Question and Draw Inferences
Given:
To find:
Area left unpainted in all the 6 faces of the cube = 6(a^{2}−πr^{2})
Step 3: Analyze Statement 1 independently
Step 4: Analyze Statement 2 independently
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in each of steps 3 and 4, this step is not required
Therefore, the correct answer is Option D.
A rectangular solid brick of iron is melted and shaped into a cube. If the areas of different sides of the brick were 24, 36 and 54 square units respectively, what is the surface area of the cube in square units?
Given:
To find:
Approach and Working:
By looking at a rectangular box, a carpenter estimates that the length of the box is between 2 to 2.1 meters, inclusive, the breadth is between 1 to 1.1 meters, inclusive and the height is between 2 to 2.1 centimeters, inclusive. If the actual length, breadth and height of the box do indeed fall within the respective ranges estimated by the carpenter, which of the following is the closest to the maximum possible magnitude of the percentage error that the carpenter can make in calculating the volume of the rectangular box?
Given:
To find: Approx. value of the maximum % error in estimating the volume
Approach:
Working Out:
So, greatest possible calculated volume = = 210*110*2.1 = =210∗11∗21 cm^{3}
Finding the maximum % error
Looking at the answer choices, we see that the answer choice which is closest to the value we obtained is Option E (22%) So, Option E is the correct answer
A right circular cylinder having the radius of its base as 2 centimeters is filled with water upto a height of 2 centimeters. This water is then poured into an empty rectangular container the dimensions of whose base are 2π by 3 centimeters. If the volume of water in the rectangular container is increased by 50 percent by adding extra water, what is the final height, in centimeters, of the water level in centimeters in the rectangular container?
Given
To Find:
Approach and Working:
What is the longest distance between two points in a rectangular solid if the area of its faces is 50, 75 and150 square units respectively?
Given
To Find: Length of the longest distance between two points inside the solid.
Approach
Length of the longest distance between 2 points inside the solid = (length of the space diagonal) = So, we need to find the dimensions of the rectangular solid
Let’s assume the length, breadth and height of the rectangular solid be l, b and h respectively.
So, we have
Working Out
2. Length of space diagonal =
Answer: A
Water is taken out of a cylindrical bucket by filling a cylindrical mug to the brim. How many mugs of water does the bucket contain?
Steps 1 & 2: Understand Question and Draw Inferences
Given:
To find: Number of mugs of water in the bucket
Step 3: Analyze Statement 1 independently
Statement 1 says that ‘The height of the bucket is three times the height of the mug and the radius of the bucket is twice the diameter of the mug’
Step 4: Analyze Statement 2 independently
Statement 2 says that ‘The bucket is filled to 70 percent of its capacity’
Step 5: Analyze Both Statements Together (if needed)
Answer: Option C
A cube of side length x centimeters is placed on a floor. A cube of side length x/2 centimeters is placed at the center of the top of the cube of side length x centimeters. What fraction of the total surface area of the cubes is visible?
Given
To Find:
Approach
^{}Working Out
Visible area of cube B
a. We know that area of one face of cube B = x^{2}
b. So, visible areas of Cube A =Area of 4 faces of cube B +partial area of face 2 =
If a cube of side length 24 units is cut into 512 smaller cubes of equal dimensions, then what is the ratio of the combined surface area of the smaller cubes and the surface area of the original cube?
Given:
2. We’re given the number of smaller cubes. As the smaller cubes have been formed from the larger cubes,
(Volume of bigger cube) = (Combined Volume of 512 smaller cubes). So, we can find a from the above equation.
Working Out:
Looking at the answer choices, we see that the correct answer is Option C
A hall is in the form of a rectangular solid with its each side length equal to 12 feet. If the floor and one wall of the hall are to be covered with colored tiles measuring 2 feet by 2 feet, how many such tiles are required?
Given
To Find: Number of tiles required?
Approach
Working Out
Answer: D
The volume of a sphere with radius r is and the surface area is 4πr^{2}. If a solid spherical ball of iron with radius r centimetres is cut into two equal hemispheres, what is the ratio, in terms of r, of the volume and the surface area of each hemisphere?
Approach:
Working Out:
So, the correct answer is Option A
In the rectangular solid above, if each dimension of the rectangular solid is an integer greater than 1 and the area of two sides of the solid is 14 and 18 respectively, what is the volume of the solid?
Given:
To find: Volume of the rectangular solid
Approach:
Working Out:
Looking at the answer choices, we see that the correct answer is Option B
The space diagonal of cube A is more than 2√6 units longer than the space diagonal of cube B and the surface area of cube A is 192 square units greater than the surface area of cube B. If the lengths of both cubes are integers, by how many cubic units is the volume of cube A greater than the volume of cube B?
Given:
To find: (Volume of Cube A) – (Volume of Cube B) = ?
Approach:
Working Out:
Looking at the answer choices, we see that the correct answer is Option B
A right circular cylinder of height 1 meter and radius 9√2 centimeters is to be used to store cubes of side length 3 cm each. If in each layer of cubes stored in the cylinder, the cubes are arranged such that the top view of the layer is a square, what is the maximum number of cubes that can be stored in the cylinder? (1 meter = 100 centimeters)
Given:
To find: The maximum number of cubes that can be stored in the cylinder
Approach:
Working Out:
A right circular cylinder is cut parallel to its base into two halves. By what percentage is the combined total surface area of the smaller cylinders greater than the total surface area of the original cylinder?
(1) The sum of the radius and the height of the original cylinder is 3 times the radius of the original cylinder.
(2) If the original cylinder is melted and a part of the molten material is used to form into a sphere with the same radius as the original cylinder, the volume of the sphere thus formed will be 33 percent less than the volume of the original cylinder. (The volume of a sphere is , where r is the radius)
Steps 1 & 2: Understand Question and Draw Inferences
Radius of each of the smaller cylinder = r and height of each of the smaller cylinder = h/2
To Find: Percentage increase in total surface area
Step 3: Analyze Statement 1 independently
(1) The sum of the radius and the height of the original cylinder is 3 times the radius of the original cylinder.
Sufficient to answer.
Step 4: Analyze Statement 2 independently
(2) If the original cylinder is melted and a part of the molten material is used to form a sphere with the same radius as the original cylinder, the volume of the sphere thus formed will be 33 percent less than the volume of the original cylinder. (The volume of a sphere is
, where r is the radius)
Sufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from steps 3 and 4, this step is not required.
Answer: D
In a rectangular box of dimensions 20 centimetres by 20 centimetres by 100 centimetres, identical spherical balls are arranged in layers such that each layer has exactly 4 balls. The top view of this arrangement is shown in the figure above. What is the approximate percentage of space left empty inside the box, if the box contains the maximum possible number of layers of such spherical balls? (The volume of a sphere is , where r is the radius of the sphere)
Given
To Find: Approximate percentage of empty space left inside the box?
Approach
2. Volume of box
a. As we know the dimensions of the box, we can find the volume of the box
3. Volume of total spheres in the box
Working Out
Answer: C
In the rectangular solid above, if each dimension of the solid is an integer, what is the volume of the solid?
(1) The area of two sides of the rectangular solid is 4 and 20 respectively
(2) The area of two sides of the rectangular solid is 4 and 5 respectively.
Solution
Steps 1 & 2: Understand Question and Draw Inferences
Let the length, breadth, and height of the rectangular solid be L, B and H respectively.
We are given that L, B, H are integers.
We need to find the volume of the solid, that is, the value of the product LBH.
Step 3: Analyze Statement 1 independently
Step 4: Analyze Statement 2 independently
Statement 2 states that: The area of two sides of the rectangular solid is 4 and 5 respectively.
Step 5: Analyze Both Statements Together (if needed)
Since we have a unique answer from step 4, this step is not required.
Hence the correct answer is Option B
A paint can is in the shape of a right circular cylinder. A rectangular paper label of area 12 square units is pasted on the can as shown in the figure.If the label can be wrapped around the can only once with no extra paper hanging out, what is the curved surface area of the paint can?
(1) The center of the paper label is at a distance of 5 units each from the top and the bottom of the can
(2) The height of the can is five times the height of the paper label.
Steps 1 & 2: Understand Question and Draw Inferences
Given:
To find: The curved surface area of the paint can
Thus, in order to find the required area, we need to find the ratio H/h
Step 3: Analyze Statement 1 independently
(1) The center of the paper label is at a distance of 5 units each from the top and the bottom of the can
So, statement 1 is not sufficient.
Step 4: Analyze Statement 2 independently
(2) The height of the can is five times the height of the paper label.
Since we now know the value of , we can find a unique value of A.
Sufficient.
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Step 4, this step is not required
Answer: Option B
If x units are added to the length of the radius of a circle, what is the number of units by which the circumference of the circle is increased?
Let the radius of the circle be 'r' units.
The circumference of the circle will therefore be 2πr units.
If the radius is increased by 'x' units, the new radius will be (r + x) units.
The new circumference will be
2π(r+x) = 2πr +2πx
Or the circumference increases by 2πx units.
Correct Answer (4)
ABCD has area equal to 28. BC is parallel to AD. BA is perpendicular to AD. If BC is 6 and AD is 8, then what is CD?
The given shape is a trapezium.
Area of a trapezium
Height = 4.
BA is perpendicular to BC and AD.
So, drop a line parallel to BA from C to meet AD at E.
CED is a right triangle with side CE measuring 4 and ED measuring 2 units.
Hence, CD, the measure of the hypotenuse =
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