Ontario 09 Academic (MPM1D)

Minimising perimeter with a fixed area (Investigation)

Lesson

Whether it is cost, time or distance, we are often trying to minimise things. Usually we have restrictions which are key to determining the minimum possible value.

- Explore the impact of changing the dimension of a rectangle with a fixed area
- Make connections between area and perimeter of rectangles

With a partner, come up with at least two scenarios where you would have a fixed area, but are flexible with the dimensions (perimeter). It does not need to be a rectangle.

Consider the possible scenario below which requires minimising the perimeter for a particular area.

A rectangular wedding cake needs to feed $80$80 people. One standard serving of cake should have a top area of about $20$20 cm^{2}. It has an expensive decoration around the edge of the cake which costs $\$4.35$$4.35/cm. We want to find the dimensions of the cake that will minimise the cost of the decoration. Work through the questions below to find the best dimensions for the cake.

- If $20$20 cm
^{2}feeds one person, what area of cake is required to feed $80$80 people? - If the length of the cake was $100$100 cm, what would the width need to be? What would this make the perimeter? What would the cost of the decoration be?
- If the length of the cake was $80$80 cm, what would the width need to be? What would this make the perimeter? What would the cost of the decoration be?
- Through investigation, find the length and width of the cake which would minimise the perimeter and then find the cost of the cake decoration.

**Possible strategies**

- Use a table of values with trial and error
- Use manipulatives to construct various rectangles
- Use technology to investigate by setting the area and then changing the slider for the length until the perimeter is as small as possible.

- In general, what special type of rectangle minimises the perimeter?
- One bag of grass seed can cover an area of $400$400 m
^{2}. A parks employee bought $9$9 bags of grass seed and this will perfectly cover the area of a new rectangular off leash dog park. She needs to minimise the cost of the fencing around the park. If fencing costs $\$25$$25 per metre and includes $4$4 posts, what dimensions will minimise the area and hence what is the minimum possible cost for the fencing? -
We typically see circular or rectangular cakes, but not triangular or hexagonal, why do you think that is? Consider both practical and mathematical reasons.

Determine the minimum perimeter of a rectangle with a given area by constructing a variety of rectangles, using a variety of tools and by examining various values of the side lengths and the perimeter as the area stays constant