A Cube with an Edge Length of 6 Contains the Largest Possible Sphere GMAT Problem Solving

Question: A cube with an edge length of 6 contains the largest possible sphere that completely fits inside of it. If the volume V of a sphere with radius r is given by the formula
V = \(\frac{4}{3}\pi r^3\), what is the volume of the empty space inside the cube?

1. \(36(\pi^3 - 6)\)
2. \(36\pi\)
3. \(36=(6-\pi)\)
4. \(36(\pi-1)\)
5. \(36(4-\pi)\)

Correct Answer: C
Solution and Explanation:

Approach Solution 1:

The largest possible sphere that fits inside the cube has its diameter equal to the length of the edge of the cube. Since the edge of the cube has a length of 6, the diameter of the sphere also has a length of 6, and thus its radius has a length of 3.

Since the volume of the cube is \(6^3=216\) and that of the sphere is \(\frac{4}{3}*\pi*3^3=36\pi\)

The volume of the empty space inside the cube is \(216 - 36\pi= 36(6 -\pi)\)

Approach Solution 2:

Since the largest sphere is fitted inside the cube of 6 cm; then its diameter = 6 cm or radius = 3 cm

Area of the sphere = \(\frac{4}{3}*27*\pi=36\pi\)

And the area of cube = \(6^3 = 216\)

So the empty space = \(216-36\pi\)

Required answer = \(36(6 - \pi)\)

Approach Solution 3:

The problem statement states that:

Given:

• A cube with an edge length of 6 contains the largest possible sphere that completely fits inside of it.
• The volume V of a sphere with radius r is given by the formula V = \(\frac{4}{3}\pi r^3\)

Find Out:

• The volume of the empty space inside the cube.

Let the largest possible sphere of radius "r" completely fits inside of the cube of side "a".
Therefore, we can say as per the mathematical formula:
2r =a =6
=> r=3.

Empty space inside cube = cube volume - sphere volume
Therefore, Empty space inside cube = \(a^3− \frac{4}{3}π r^3= 6^3 − \frac{4}{3}π 3^3 = 6^3 − 6^2π = 36(6−π)\)

Hence, Empty space inside cube = \(36(6−π)\)

“A cube with an edge length of 6 contains the largest possible sphere”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The candidates can go through GMAT Quant practice papers to strenthen their mathematical understanding.

Suggested GMAT Problem Solving Samples

• If 10 Millimeters Equal 1 Centimeter, How many Square Centimeters does 1 Square Millimeter Equal? GMAT Problem Solving
• Assume that all 7-Digit Numbers That do not Begin with 0 or 1 are Valid Phone Numbers. GMAT Problem Solving
• A Contractor Estimated that his 10-Man Crew Could Complete the Construction in 110 Days if There Was no Rain. GMAT Problem Solving
• GMAT Problem Solving - How Much Pure Alcohol Should Be Added To 400 ml of a 15% Solution To Make The Strength Of The Solution 32%?
• Find The Altitude Of An Equilateral Triangle Whose Side is 20 GMAT Problem Solving
• Which of the following is the value of √3√0.000064? GMAT Problem Solving
• GMAT Problem Solving – If @ x=x^2/2x^2-2 , What is the Units Digit of @ (@4)?
• GMAT Problem Solving – What is the product of all possible solutions of the equation |x+2|- 5|x+2| = -6?
• A Welder Received an Order to Make a 1 Million Litre Cube-Shaped Tank. GMAT Problem Solving
• A Right Angled Triangle has its Sides in Arithmetic Progression and Being Integers GMAT Problem Solving
• What is the Largest Power of 3 Contained in 200! GMAT Problem Solving
• If g is an integer what is the value of\((-1)^{g^{4-1}}\)?GMAT Problem Solving
• If @\(x=\frac{x^2}{2x^2}-2\), What is the Units Digit of @(@4)? GMAT Problem Solving
• GMAT Problem Solving - Iqbal Dealt Some Cards to Mushtaq and himself from a Full Pack of Playing Cards
• Running at the Same Constant Rate, 6 Identical Machines GMAT Problem Solving
• A conference room is equipped with a total of 45 metal or wooden chairs GMAT Problem Solving
• The Greatest 6-Digit Number When Divided by 6, 7 ,8 , 9, and 10 Leaves a Remainder of 4, 5, 6, 7, and 8 Respectively GMAT Problem Solving
• The square of 5√2 =?GMAT Problem Solving
• If \(x^2\)=\(2^x\), What is the Value of x ?GMAT Problem Solving
• A Shop Stores x kg of Rice. The First Customer Buys half this Amount Plus half a kg of Rice GMAT Problem Solving
• In a Class of 120 Students Numbered 1 to 120, All Even Numbered Students Opt for Physics GMAT Problem Solving
• Machine A Produces bolts at a Uniform Rate of 120 Every 40 seconds GMAT Problem Solving
• Out of 7 Consonants and 4 Vowels, How Many Words of 3 Consonants and 2 Vowels Can be Formed? GMAT Problem Solving
• 4 Bells Toll Together at 9:00 A.M. They Toll After 7, 8, 11 and 12 Seconds GMAT Problem Solving
• 12 Marbles are Selected at Random from a Large Collection of White, Red, Green and Yellow Marbles GMAT Problem Solving
• A welder received an order to make a 1 million litre cube-shaped tank GMAT Problem Solving
• After 6 games, Team B had an average of 61.5 points per game GMAT Problem Solving
• A Man has 1044 Candles. After Burning, He can Make a New Candle GMAT Problem Solving
• If k is an Integer and 2 < k < 7, for How Many Different Values of k is There a Triangle With Sides of Lengths 2, 7, and k? GMAT Problem Solving
• Excluding Stoppages, The Speed of a Bus is 54km/hr and Including Stoppage, It is 45km/hr GMAT Problem Solving
• At a Dog Competition, a Dog is Awarded 10 Points if it Runs Through 4 Pipes, Makes 10 Jumps, and Walks on 2 Beams. GMAT Problem Solving
• If m Lies Between The Integers p and s On the Number Line Shown GMAT Problem Solving
• A Cube Of Side 7 cm Is Coloured On Pair of Opposite Faces By Red, Green and Yellow Shades GMAT Problem Solving
• If x^2 − 5x − 6 = 0, which of the following could be x ? GMAT Problem Solving
• How Many Numbers Between 1 and 1000, Inclusive Have an Odd Number of Factors? GMAT Problem Solving
• How Many Factors of 80 are Greater Than\(\sqrt80\)? GMAT Problem Solving
• If \(2^{98}=256L+N\) , Where L and N are Integers and \(0\leq{N}\leq{4}\) , What is the Value of N? GMAT Problem Solving
• Two Dice are Thrown Simultaneously. What is the Probability of Getting Two Numbers Whose Product is Even? GMAT Problem Solving
• The Smallest 3-Digit Positive Integer Obtained By Adding Two Positive Two-Digit Numbers GMAT Problem Solving
• Train A Leaves New York at 9am Eastern Time on Monday, Headed for Los Angeles at a Constant Rate GMAT Problem Solving
• An Express Train Traveled At An Average Speed of 100 Kilometers Per Hour GMAT Problem Solving
• The Hexagon ABCDEF is Regular GMAT Problem Solving
• If x is not equal to zero, and x + 1/x = 3, then what is the value of x^4 + 1/x_4? GMAT Problem Solving
• If mn ≠ 0 and 25 Percent of n Equals 37 1/2 Percent of m, What is the Value of 12n/m? GMAT Problem Solving
• How many multiples of 7 are there between 21 and 343, exclusive? GMAT Problem Solving
• How many odd factors does 210 have? GMAT Problem Solving
• How many three-digit integers are not divisible by 3? GMAT Problem Solving
• How many three-digit numbers are there such that all three digits are different and the first digit is not zero? GMAT Problem Solving
• After 6 Games, Team B Had an Average of 61.5 Points Per Game. GMAT Problem Solving