A rectangle lawn has perimeter of 34m. Another rectangular lawn is 2cm narrower but 2m longer than the first lawn. If A(one) and A(two) are the areas of the first and second lawn respectively and A(one):A(two)=7:6, find

Question

i)the dimension of the lawn.
II)the area of each lawn.
Please have tried but i couldn't get it

Answer 1

Let the length of the first lawn be L meters and the width be W meters. Then, the perimeter is given by:
2L + 2W = 34m. Equation [1].

The dimensions of the second lawn are (W - 0.02) meters in width and (L + 2) meters in length.

A(one) = L * W
A(two) = (W - 0.02) * (L + 2)

Given that A(one) : A(two) = 7:6, we can write:

6 * A(one) = 7 * A(two).
6 * (L * W) = 7 * ((W - 0.02) * (L + 2)). Equation [2].

Now we will solve systems of equations [1] and [2].

From equation [1]: L = 17 - W.
Substitute into equation [2]:
6 * (17 - W) * W = 7 * ((W - 0.02) * (17 - W + 2))

Expanding and simplifying:
6 * (17 - W) * W = 7 * (W - 0.02) * (19 - W)
102W - 6W^2 = 133W - 7W^2 - 2.54 + 0.14W
W^2 - 31W + 2.54 = 0

Solve for W using the quadratic formula:
W = (31 ± sqrt((-31)^2 - 4 * 1 * 2.54)) / 2 = (31 ± sqrt(937.84)) / 2

Since W represents the width of the rectangle, we will take the positive square root.
W = (31 + sqrt(937.84)) / 2 = 27.44 / 2 = 13.72 meters

Now find L: L = 17 - W = 17 - 13.72 = 3.28 meters.

Now that we have the dimensions of the first lawn, we can find the area of each lawn:
A(one) = L * W = 3.28 * 13.72 = 45.02 square meters.
A(two) = (W - 0.02) * (L + 2) = (13.72 - 0.02) * (3.28 + 2) = 13.7 * 5.28 = 72.34 square meters.

So the dimensions of the first lawn are:
Length = 3.28 meters
Width = 13.72 meters

And the areas of each lawn are:
A(one) = 45.02 square meters.
A(two) = 72.34 square meters.