Geometry math problem

Question

Which of the following is too big to fit in a box 2 feet by 2 feet by 3 feet

(1) a sphere of radius 1 foot

(2) a cube of side 2 feet

(3) a cylinder of radius 1 foot and height 3 feet.

(4)a sphere of radius 2 feet

(5) a cylinder of diameter 2 feet and height 2 feet

Explain how and why.

Answer 1

Note that NONE OF THE THREE DIMENSIONS may be exceeded.

However, the volume of the item may be less than the volume of the box, but it still might not fit.

(4) exceeds all three dimensions !!

Answer 2

Is this a multiple choice question. Regardless, there is only one correct option --> 4

Cannot Fit: Option 4

1. A sphere of radius 2 feet indicates that the diameter of the sphere is 4 feet. In not possible configuration can a sphere that 4 feet wide fit in to a box that has 3 feet as its longest side. The box would need to be, at a minimum, 4 ft x 4 ft x 4ft.

Can Fit: Options 1, 2, 3, and 5

Answer 3

Step 1: Find the volume of a box that is 2 feet by 2 feet by 3 feet.

A rectangular prism's volume is length times width times height.

So the volume will be 2*2*3 = 12 feet³

Step 2: Find the volume of a sphere of radius 1 foot

A sphere's volume is (4/3)(π)(r³).

Volume of this sphere will be (4/3)(π)(1³) = 4.19 feet³

So the sphere will fit inside the box.

Step 3: Find the volume of cube with side 2 feet.

The volume of a cube is a side cubed.

So the volume of this cube is (2)³ = 8 feet³

That means the cube will fit inside the box.

Step 4: Find the volume of a cylinder of radius 1 foot and height 3 feet.

The volume of a cylinder is π(r)²h.

The volume of this cylinder will be π(1)²(3) = 9.42 feet³

The cylinder will fit inside the box.

Step 5: Determine if a sphere with radius 2 will fit. For a sphere of radius 2 feet, we immediately know it won't fit inside the box because the diameter of the sphere is 2 times radius which is 2(2) = 4 feet. The diameter is larger than all of the dimmensions of the box which is 2 feet by 2 feet by 3 feet.

We can prove it by finding its volume.

A sphere's volume is (4/3)(π)(r³).

This sphere's volume will be (4/3)(π)(2³) = 33.51 feet³

The volume of the sphere is way more than the box's volume so it won't fit.

Step 6: Find the volume of a cylinder of diameter 2 feet and height 2 feet.

Radius is diameter/2 so 2/2 = 1 feet. The volume of a cylinder is π(r)²h.

The volume of this cylinder is π(1)²(2) = 6.28 feet³

This will fit inside the box.

Answer 4

I recommend sketching all geometry problems. So you should try this technique for this problem.

Meanwhile, let's look at it mathematically:

The box given has dimensions 2 x 2 x 3. Therefore, if any of the answer choices has any dimension larger than 3, it automatically means it won't fit in the box. (There are other possible considerations, too, but let's just start with the simplest one.

Looking at choice 4, we see that if the radius of a sphere is 2, then its diameter is 4. This is greater than the size of the box, so this is the one that will NOT fit.

Analyze the other choices in a similar way, and you'll see that they would all fit (though barely, in some cases).