Volume Comparison and Packing Problem
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Ava has some chocolate bars like the one shown below.
The cross-section of each prism is an isosceles triangle.
Ava is putting these chocolate bars into the cuboid-shaped box shown below.
a) How many times larger is the volume of the box than the volume of the chocolate bar?
What is the maximum number of chocolate bars that could fit inside the box? Draw a sketch to show how they would fit.
This question includes visual content: There are two 3D diagrams. The first is a triangular prism labeled 'CHOCO PRISM' with an isosceles triangle cross-section. The triangle base is $5\text{ cm}$ and its vertical height is $8\text{ cm}$. The length of the prism is $30\text{ cm}$. The second is an open cuboid-shaped box. The dimensions of the box are a height of $8\text{ cm}$, a width of $20\text{ cm}$, and a length of $30\text{ cm}$. Right-angle symbols are shown for the height measurements of both objects.
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Step by Step Written Solution
In this problem, we need to compare the volume of a triangular prism chocolate bar with a cuboid box, and then determine how many bars can actually fit inside.
Chocolate Bar vs. Box Volume
Let's start by calculating the volume of a single chocolate bar. It is a triangular prism, so the formula is the area of the triangular cross-section times the length.
The triangular face has a base of five centimeters and a height of eight centimeters. So the area is one half times five times eight.
Simplifying this, one half of forty is twenty, and twenty times thirty gives us a volume of six hundred cubic centimeters per bar.
Now, let's look at the box. It's a cuboid with dimensions twenty centimeters by eight centimeters by thirty centimeters.
Multiplying twenty by eight by thirty gives us a total box volume of four thousand eight hundred cubic centimeters.
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