A farmer has 120 metres of fencing.
He is going to make a rectangular enclosure PQRS with the fencing.
He is also going to divide the enclosure into two equal parts by fencing along MN.
SNR
The width of the enclosure is x metres. The length of the enclosure is y metres.
(a) (i) Show that y = 60 – 1.5x
The area of the enclosure PQRS is A m2 (ii) Show that A = 60x – 1.5x2
To solve this problem, we will use the given information and apply the principles of geometry to find the relationships between the variables.
(a) (i) To show that y = 60 - 1.5x, we need to consider the perimeter of the rectangular enclosure PQRS.
The perimeter of a rectangle is given by the formula:
Perimeter = 2(length + width)
Given that the total length of the fencing is 120 meters, we can write the equation as:
120 = 2(y + x)
Now, let's solve this equation for y:
Divide both sides by 2:
60 = y + x
Subtract x from both sides:
y = 60 - x
Therefore, we have shown that y = 60 - 1.5x.
(ii) To show that A = 60x - 1.5x^2, we need to find the area of the rectangular enclosure PQRS.
The area of a rectangle is given by the formula:
Area = length * width
Substituting the value of y from the previous part, we can rewrite the equation as:
Area = (60 - x) * x
Simplify the expression:
Area = 60x - x^2
Therefore, we have shown that A = 60x - 1.5x^2.