A rectangular field is 50 yards wide and 100 yards long.
Give the length and width of another rectangular field that has the same perimeter but a smaller area.
width yards
length yards

Ask by Stuart Pearson. in the United States

Mar 17,2025

Question

A rectangular field is 50 yards wide and 100 yards long.
Give the length and width of another rectangular field that has the same perimeter but a smaller area.
width yards
length yards

Ask by Stuart Pearson. in the United States

Mar 17,2025

UpStudy AI · Accepted Answer

We start with the original field dimensions:

- Width $$ w_1 = 50 $$ yards
- Length $$ l_1 = 100 $$ yards

The perimeter is given by

$$
P = 2(w + l)
$$

For the original field, the perimeter is

$$
P = 2(50 + 100) = 2(150) = 300 \text{ yards}
$$

Any other rectangle with the same perimeter must satisfy

$$
2(w + l) = 300 \quad \Longrightarrow \quad w + l = 150
$$

The area of a rectangle is

$$
A = w \times l
$$

The original field has an area

$$
A_1 = 50 \times 100 = 5000 \text{ square yards}
$$

To find another rectangular field with the same perimeter but a smaller area, we can choose any pair $$ (w, l) $$ such that

$$
w + l = 150
$$

and

$$
w \times l < 5000.
$$

One possible choice is to let

$$
w = 40 \text{ yards}.
$$

Then

$$
l = 150 - 40 = 110 \text{ yards}.
$$

The area of this new field is

$$
A_2 = 40 \times 110 = 4400 \text{ square yards},
$$

which is indeed smaller than $$ 5000 $$ square yards.

Thus, one acceptable answer is:

- Width $$ = 40 $$ yards
- Length $$ = 110 $$ yards.
Width = 40 yards Length = 110 yards