Measurement

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

Solve problems that require maximizing the area of a rectangle for a fixed perimeter or minimizing the perimeter of a rectangle for a fixed area