SOLUTION: at the movies a container that holds popcorn has a square bottom base with side length of 4in and a square top with sides of 6in. The height of the container is 10 in. (front view)

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Question 1084033: at the movies a container that holds popcorn has a square bottom base with side length of 4in and a square top with sides of 6in. The height of the container is 10 in. (front view) Select two choices that are greater than the volume of this container
A.a cylinder with a diameter of 5.1in and a height of 10
B.A square prism with lengths of the bases sides of 6.5 inches and a height of 10inches
C.a square pyramid with a base of 36 inches and height of 10
D. a sphere with a radius of 4.5 inches.
Found 2 solutions by KMST, jim_thompson5910:
Answer by KMST(5328) (Show Source):
You can put this solution on YOUR website!
If option C is a square base pyramid with base sides measuring 36 inches,
that is definitely larger.

If option C is a square base pyramid with base area measuring 36 square inches,
that is smaller than all the other options.

The sphere (D) and the square base prism (B) are larger than the popcorn container.

Here is a side view of the three items:

It is obvious that the popcorn container fits inside the prism.

The comparison with the sphere is not so obvious.

The shape of the popcorn container is called a frustum of a pyramid.
That is the "stump" left when we cut off a pyramid parallel to its base.
Imagine a pyramid square bottom base with side length of 6 in and a height of 30 inches.
Its volume is .
Now cut it at a height of 10 inches.
The part above the cut is a smaller pyramid scale replica of the original pyramid.
Its volume is times the volume of the original pyramid.
The bottom part is your frustum.
The volume of the frustum is times the volume of the original pyramid.
It is .

The volume of the sphere is
.

Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!

Answer: choice B and choice D
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Explanation:

It might help to set up a drawing of how this container looks.

Imagine you have a pyramid that is upside down. The apex of the pyramid is pointing down with the base at the top. What I'm decribing is shown in figure 1a (see image below).

Now imagine cutting the pyramid at a given red line. The red line would be parallel to the base. The cut forms a smaller square. See figure 2a. This figure shows a smaller pyramid in red.

If we remove the red pyramid you see in figure 2a, we'll end up with the 3D solid shown in figure 3a. In mathematical terms, is called a frustum. Specifically it's a square frustum. The initial popcorn container will look something like figure 3a.

Here's how the figure will be look visually. This is a 3D view

Note: Figures not to scale

If you wish, you can think of it in a 2D fashion. Imagine looking straight on from one of the sides

Note: Figures not to scale

As you can see in figure 1b, we start with a simple triangle.
In figure 2b, we highlight the portion (in red) we want to get rid of.
Then in figure 3b, we actually erase that red portion ending us up with a trapezoid.

If you shine a flashlight at the 3D solid in figure 3a, you'll end up with a shadow with the shape of figure 3b. The flashlight must be aimed such that the flashlight stick itself is parallel to the flat ground.

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We will use the volume formula found on this page to find the volume of the frustum.

That volume formula is

First we need to calculate and , which are the areas of the parallel bases
= area of smaller square base (with side length 4)
= area of larger square base (with side length 6)

If you refer back to figure 3a, is the red square and is the green square.

Area of the smaller square

Area of the larger square

Now we can use the volume formula to get...

Plug in the areas (found above) and the height h = 10.
Use a calculator

The volume of the popcorn container is roughly 253.333333 cubic inches (this is approximate accurate to 6 decimal places)
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We're looking for two volumes (from the answer choices) that will be larger than 253.333333 cubic inches.

Let's go through all of the answer choices to see which result will be larger than 253.333333.
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Part A)

radius = diameter/2
radius = 5.1/2
radius = 2.55 inches

Volume of Cylinder = pi*(radius)^2*(height)
Volume of Cylinder = pi*(r)^2*(h)
Volume of Cylinder = pi*(2.55)^2*(10)
Volume of Cylinder = pi*6.5025*(10)
Volume of Cylinder = pi*65.025
Volume of Cylinder = 65.025*pi
Volume of Cylinder = 204.282062299677

This value is NOT larger than 253.333333, so choice A can be eliminated
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Part B)

Volume of square prism = (area of base)*(height)
Volume of square prism = (6.5*6.5)*(10)
Volume of square prism = (42.25)*(10)
Volume of square prism = 422.5

This value is larger than 253.333333. Therefore choice B is one of the answers. We just need one more.
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Part C)

Note: I'm going to assume for this part it meant to say "square pyramid with base area 36 square inches". With this correction, we would have 2 exact answers. Without this correction, choice C would be an answer leading to 3 answers instead of 2. I'd check with the teacher on this potential typo.

volume of square pyramid = ((area of base)*(height))/3
volume of square pyramid = ((36)*(10))/3
volume of square pyramid = (360)/3
volume of square pyramid = 120

This value is NOT larger than 253.333333, so choice C can be eliminated
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Part D)

Volume of Sphere = (4/3)*pi*r^3
Volume of Sphere = (4/3)*pi*(4.5)^3
Volume of Sphere = (4/3)*pi*91.125
Volume of Sphere = (4/3)*91.125*pi
Volume of Sphere = 121.5*pi
Volume of Sphere = 381.70350741116

This value is larger than 253.333333 so choice D is the other answer.