Final answer:
The width of the rectangular lawn can be represented as 'w = (1/2)t - 3'. By substituting the value of 'w' into the equation for the perimeter, we can find the value of 't'. The width in terms of the length is 'w = (1/2)t - 3'.
Step-by-step explanation:
Let's represent the length of the lawn as 't' and the width as 'w'. According to the given information, the width is 3 metres less than half the length. So we can write the equation: w = (1/2)t - 3.
The perimeter of a rectangle is given by the formula: P = 2(length + width). Since the perimeter is 42 metres, we can write the equation: 42 = 2(t + w).
To solve these two equations, we can substitute the value of 'w' into the second equation and solve for 't'. Once we find the value of 't', we can substitute it back into the equation for 'w' to find its value.
i. Show that w + 1 = 21:
Substituting the value of 'w' into the second equation: 42 = 2(t + (1/2)t - 3). Simplifying: 42 = 2((3/2)t - 3). Dividing both sides by 2: 21 = (3/2)t - 3. Adding 3 to both sides: 24 = (3/2)t. Multiplying both sides by 2/3: (2/3) * 24 = t.
Substituting the value of 't' into the equation for 'w': w = (1/2)(2/3 * 24) - 3. Simplifying: w = (1/2)(16) - 3 = 8 - 3 = 5.
Therefore, w + 1 = 5 + 1 = 6.
ii. Width in terms of the length:
Substituting the value of 't' into the equation for 'w': w = (1/2)t - 3 = (1/2)(2/3 * 24) - 3 = 8 - 3 = 5.