Equations Investigation – Peter’s Gardening Service.

1. Equations Investigation – Peter’s Gardening Service. By Clare Watts 8K

2. Introduction • Peter is a professional gardener who tends the gardens and laws of his many clients. One of his jobs is to fertilise the lawns regularly. He is trying a new fertiliser which, according to the directions, states that when the contents of the container are mixed with water the solution covers 64m² of lawn. To decide how many containers of fertiliser he will need, Peter needs to have some idea of possible sizes of lawn that one container will cover.

3. Square Lawn 1. If the length of the square is represented by x, write an expression for the area of the square. 2. Since the area needs to be 64m², write an equation which shows that the area of the square is equal to 64m² using x. 3. Can you solve this equation? What is the size of a square lawn which can be covered by one tin of fertiliser? x 1. x² = Area. 2. √64m² = x 3. √64m² = 8m²

4. Rectangular Lawn 1. If the length of the rectangle is represented by l, and the width is represented by w, write an expression for the area of the rectangle 2. Since the area needs to be 64m², write an equation which shows the area of the rectangle is equal to 64m² using l and w. 3. Can you solve this equation? What are 5 different sizes of a rectangle lawn which can be covered by one tin of fertiliser? l 1. l x w = Area 2. 64m² ÷ l = w 3. a. 64m² ÷ 1m² = 64m² b. 64m² ÷ 2m² = 32m² c. 64m² ÷ 4m² = 16m² d. 64m² ÷ 5m² = 12.8m² w

5. Triangle Lawn 1. If the height of the triangle is represented by h, and the base is represented by b, write an expression for the area of the triangle. 2. Since the area needs to be 64m², write an equation which shows the area of the triangle is equal to 64m² using h and b. 3. Can you solve this equation? What are 5 different sizes of a triangle lawn which can be covered by one tin of fertiliser? 1. 0.5 x h x b = Area 2. 64m² ÷ (0.5 x h) = b 3. a. 64m² ÷ (0.5 x 2m²) = 64m² b. 64m² ÷ (0.5 x 10m²) = 12.6m² c. 64m² ÷ (0.5 x 20m²) = 6.4m² d. 64m² ÷ (0.5 x 8m²) = 8m² e. 64m² ÷ (0.5 x 32m²) = 4m² h b

6. Circular Lawn 1. If the radius of the circle is represented by r, write an expression for the area of the circle. 2. Since the area needs to be 64m², write an equation which shows that the area of the circle is equal to 64m² using r. 3. Can you solve this equation? What is the size of a circle lawn which can be covered by one tin of fertiliser? 1. pi x r² = Area 2. 64m²  pi = r² √(64m²  pi) = r 3. 64m²  pi = r² √(64m²  pi) = r 4.51m² = r r

7. L-Shaped Lawn 1. If the length of the l-shape is represented by l, write an expression for the area of the l-shape. 2. Since the area needs to be 64m², write an equation which shows that the area of the l-shape is equal to 64m² using l. 3. Can you solve this equation? What is the size of a l-shaped lawn which can be covered by one tin of fertiliser? l l 1. l² x 3= Area 2. √(64m² 3) = l 3. √21.33m² = l 4.62m² = l l l

8. Trapezium Lawn 1. If the height of the trapezium is represented by h, the base is represented by b, and the top is represented by t, write an expression for the area of the trapezium. 2. Since the area needs to be 64m², write an equation which shows that the area of the trapezium is equal to 64m² using h, b and t. 3. Can you solve this equation? What is the size of a trapezium lawn which can be covered by one tin of fertiliser? t 1. 0.5 x (t +b) x h = Area 2. 64m²  (0.5 x (t + b)) = h 3. 64m²  (0.5 x (10 +12)) = 5.82m² h b