Measurement

Ontario 10 Applied (MFM2P)

Maximising volume (Investigation)

Lesson

What do you think will hold more water? A really big flat container or a really small tall container?

You can test this by trying the following activity with your class.

Get an A4 piece of paper.

Using a ruler, measure out equal squares on each corner of the sheet. The size of these squares is up to you. Mark these squares with dotted lines using a pencil.

Now, draw solid lines across the page as shown below.

Using scissors, cut out the squares along the dotted lines.

Now, fold the paper along the lines to form the container as shown below. You can use tape to hold the container together.

Remember to pick your squares carefully. What will the shape of the container be if the squares are small? What will the shape be if the squares are large?

Measure the length, width and height of your container using a ruler.

Recall from our previous chapter that we can find the volume of a prism by the following formula.

$\text{Volume of any Prism }=\text{Area of Base }\times\text{Height }$Volume of any Prism =Area of Base ×Height

Calculate the volume of your container and compare it with the volume of your classmates' containers.

Who made the container with the largest volume? Who made the container with the smallest volume? What shapes did these containers have? Were there any containers that had the same volume but different shapes?

Use the GeoGebra applet below to change the cut-out size of the squares and observe how it affects the shape and volume of the container.

How large do the squares have to be for the volume to be as large as it can be?

Notice that if the squares are too large or too small, then the volume won't be very big either way. The largest container is somewhere between these two cases.

As it turns out, on an A4 sheet of paper, the largest volume is attained by a square with sides of roughly $40$40 mm!

Did you know?

Eventually you will be able to solve this problem exactly using calculus.

Solve problems involving the surface areas of prisms, pyramids, and cylinders, and the volumes of prisms, pyramids, cylinders, cones, and spheres, including problems involving combinations of these figures, using the metric system or the imperial system, as appropriate