Question

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?
b) what is the maximum number of chocolate bars that could fit inside the box?
draw a sketch to show how they would fit.

Explanation:

Step1: Calculate the volume of the chocolate bar

The chocolate bar is a triangular prism. The formula for the volume of a triangular prism is \(V = \text{Area of the base}\times\text{length}\). The base is an isosceles triangle with base \(b = 5\space\text{cm}\) and height \(h=8\space\text{cm}\). The area of the triangle \(A=\frac{1}{2}\times b\times h=\frac{1}{2}\times5\times8 = 20\space\text{cm}^2\). The length of the prism \(l = 30\space\text{cm}\). So the volume of the chocolate bar \(V_{chocolate}=20\times30=600\space\text{cm}^3\)

Step2: Calculate the volume of the box

The box is a cuboid. The formula for the volume of a cuboid is \(V=\text{length}\times\text{width}\times\text{height}\). Here, length \(l = 30\space\text{cm}\), width \(w = 20\space\text{cm}\) and height \(h = 8\space\text{cm}\). So the volume of the box \(V_{box}=30\times20\times8=4800\space\text{cm}^3\)

Step3: Find the ratio of the volume of the box to the volume of the chocolate bar

\(\frac{V_{box}}{V_{chocolate}}=\frac{4800}{600}=8\)

Step4: Determine the maximum number of chocolate bars

We need to consider the dimensions. Let's assume we place the triangular - faced side of the chocolate bar (base of the triangle \(b = 5\space\text{cm}\), height of the triangle \(h = 8\space\text{cm}\)) against the faces of the box.
If we consider the base of the triangle (\(b = 5\space\text{cm}\)) along the width of the box (\(20\space\text{cm}\)): \(\frac{20}{5}=4\). The height of the triangle (\(h = 8\space\text{cm}\)) matches the height of the box (\(8\space\text{cm}\)). The length of the chocolate bar (\(30\space\text{cm}\)) matches the length of the box (\(30\space\text{cm}\)). So the number of chocolate bars \(n = 4\)

Answer:

a) \(8\)
b) \(4\)