Question

a rectangular lot is 75 yards wide and 120 yards long. give the length and width of another rectangular lot that has the same perimeter but a smaller area. width = yards length = yards

Explanation:

Step1: Calculate the perimeter of the original rectangle

The formula for the perimeter of a rectangle is \(P = 2(l + w)\). For the given rectangle with \(l = 120\) yards and \(w=75\) yards, \(P=2(120 + 75)=2\times195 = 390\) yards.

Step2: Let the new width be \(x\) and new length be \(y\), so \(2(x + y)=390\), which simplifies to \(x + y=195\), or \(y = 195 - x\).

The area of the original rectangle \(A_{1}=120\times75 = 9000\) square yards. The area of the new rectangle \(A_{2}=x(195 - x)=195x - x^{2}\).
We can try values. If \(x = 100\), then \(y=195 - 100=95\).
The area of the new rectangle \(A_{2}=100\times95 = 9500\) (too big). If \(x = 50\), then \(y = 195 - 50=145\).
The area of the new rectangle \(A_{2}=50\times145=7250\) square yards.

Answer:

width = 50 yards, length = 145 yards