Question 1084033
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Answer: <font color=red>choice B and choice D</font>
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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 <a href = "https://en.wikipedia.org/wiki/Frustum">frustum</a>. 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
<img src = "https://i.imgur.com/Q6SmuyA.png">
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
<img src = "https://i.imgur.com/ZcqI9fa.png">
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 <a href="http://mathworld.wolfram.com/PyramidalFrustum.html">this page</a> to find the volume of the frustum.


That volume formula is
{{{V = (1/3)*h*(A[1]+A[2]+sqrt(A[1]*A[2]))}}}


First we need to calculate {{{A[1]}}} and {{{A[2]}}}, which are the areas of the parallel bases
{{{A[1]}}} = area of smaller square base (with side length 4)
{{{A[2]}}} = area of larger square base (with side length 6)


If you refer back to figure 3a, {{{A[1]}}} is the red square and {{{A[2]}}} is the green square.


Area of the smaller square
{{{A = s^2}}}
{{{A[1] = 4^2}}}
{{{A[1] = 16}}}


Area of the larger square
{{{A = s^2}}}
{{{A[2] = 6^2}}}
{{{A[2] = 36}}}


Now we can use the volume formula to get...
{{{V = (1/3)*h*(A[1]+A[2]+sqrt(A[1]*A[2]))}}}
{{{V = (1/3)*10*(16+36+sqrt(16*36))}}} Plug in the areas (found above) and the height h = 10.
{{{V = 253.333333333333}}} Use a calculator
{{{V = 253.333333}}}


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.


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