Question

a rectangular park is 80 meters wide and 115 meters long. give the length and width of another rectangular park that has the same perimeter but a larger area. width = meters length = meters

Explanation:

Step1: Calculate the perimeter of the original park

The formula for the perimeter of a rectangle is \(P = 2\times(l + w)\). For the given park with \(w = 80\) meters and \(l=115\) meters, \(P=2\times(80 + 115)=2\times195 = 390\) meters.

Step2: Recall the relationship between perimeter, length, width and area

For a rectangle with perimeter \(P = 2\times(l + w)\) (constant), the area \(A=l\times w\). Using the fact that for a given perimeter, the area is maximized when \(l = w\) (a square). But we can also use the identity \(l=\frac{P}{2}-w\), and \(A = w\times(\frac{P}{2}-w)=\frac{P}{2}w-w^{2}\). Another approach is to note that if we make the length and width closer to each other (since \(A=l\times w\) and for a fixed \(l + w=\frac{P}{2}\), \(A\) is larger when \(|l - w|\) is smaller).
Since \(l + w=\frac{390}{2}=195\) (from \(P = 390\)), we can choose \(w = 90\) meters. Then \(l=195 - 90=105\) meters.
Check the area of the original park: \(A_{1}=80\times115 = 9200\) square - meters.
Check the area of the new park: \(A_{2}=90\times105=9450\) square - meters.

Answer:

width = \(90\) meters, length = \(105\) meters