Question

a rectangular park is 65 yards wide and 110 yards long. give the length and width of another rectangular park that has the same perimeter but a larger area. width = \\(\square\\) yards length = \\(\square\\) yards

Explanation:

Step1: Calculate original perimeter

The perimeter of a rectangle is $P = 2(l + w)$. Substitute $l=110$, $w=65$:
$$P = 2(110 + 65) = 2(175) = 350 \text{ yards}$$

Step2: Find target sum of sides

For same perimeter, $l + w = \frac{P}{2} = \frac{350}{2} = 175$ yards.

Step3: Choose closer length/width

For maximum area, a rectangle approaches a square. Pick $w=80$, $l=175-80=95$ (both positive, sum to 175).

Step4: Verify area comparison

Original area: $110 \times 65 = 7150$ square yards.
New area: $95 \times 80 = 7600$ square yards, which is larger.

Answer:

width = 80 yards
length = 95 yards

Note: Any pair of positive numbers that add to 175 and are closer to each other than 65 and 110 is valid, e.g., 85 and 90, 75 and 100, etc.