Answer :
Sure, let's solve this step-by-step.
Given:
1. The perimeter of the rectangular terrain is 580 meters.
2. If the length decreases by 70 meters, and the width increases by 40 meters, the area remains unchanged.
We need to determine the area of the terrain.
### Step-by-step solution:
1. Define the variables:
Let's denote:
- [tex]\( l \)[/tex] as the length of the rectangle,
- [tex]\( w \)[/tex] as the width of the rectangle.
2. Perimeter equation:
The formula for the perimeter of a rectangle is:
[tex]\[
2l + 2w = 580 \implies l + w = 290 \quad (1)
\][/tex]
3. Area equation:
The area of the rectangle will remain the same even if the length decreases by 70 meters and the width increases by 40 meters.
Thus:
[tex]\[
l \times w = (l - 70) \times (w + 40) \quad (2)
\][/tex]
4. Expanding and simplifying the area equation:
We expand the right side of the equation (2):
[tex]\[
l \times w = l \times w - 70w + 40l - 2800
\][/tex]
Thus:
[tex]\[
l \times w = l \times w + 40l - 70w - 2800
\][/tex]
Simplify by subtracting [tex]\( l \times w \)[/tex] from both sides:
[tex]\[
0 = 40l - 70w - 2800 \quad (3)
\][/tex]
5. Solving for [tex]\( l \)[/tex] and [tex]\( w \)[/tex]:
Use equation (1):
[tex]\[
l = 290 - w \quad (from\ equation\ 1)
\][/tex]
Substitute [tex]\( l \)[/tex] in equation (3):
[tex]\[
0 = 40(290 - w) - 70w - 2800
\][/tex]
Simplify:
[tex]\[
0 = 11600 - 40w - 70w - 2800
\][/tex]
Combine like terms:
[tex]\[
0 = 11600 - 110w - 2800
\][/tex]
Simplify further:
[tex]\[
0 = 8800 - 110w
\][/tex]
Solving for [tex]\( w \)[/tex]:
[tex]\[
110w = 8800
\][/tex]
[tex]\[
w = \frac{8800}{110} = 80
\][/tex]
6. Find [tex]\( l \)[/tex]:
Using [tex]\( w = 80 \)[/tex] in equation (1):
[tex]\[
l + 80 = 290
\][/tex]
[tex]\[
l = 290 - 80 = 210
\][/tex]
7. Calculate the area:
[tex]\[
\text{Area} = l \times w = 210 \times 80 = 16800 \, \text{square meters}
\][/tex]
So, the area of the terrain is:
[tex]\[
\boxed{16800 \, \text{m}^2}
\][/tex]
Given:
1. The perimeter of the rectangular terrain is 580 meters.
2. If the length decreases by 70 meters, and the width increases by 40 meters, the area remains unchanged.
We need to determine the area of the terrain.
### Step-by-step solution:
1. Define the variables:
Let's denote:
- [tex]\( l \)[/tex] as the length of the rectangle,
- [tex]\( w \)[/tex] as the width of the rectangle.
2. Perimeter equation:
The formula for the perimeter of a rectangle is:
[tex]\[
2l + 2w = 580 \implies l + w = 290 \quad (1)
\][/tex]
3. Area equation:
The area of the rectangle will remain the same even if the length decreases by 70 meters and the width increases by 40 meters.
Thus:
[tex]\[
l \times w = (l - 70) \times (w + 40) \quad (2)
\][/tex]
4. Expanding and simplifying the area equation:
We expand the right side of the equation (2):
[tex]\[
l \times w = l \times w - 70w + 40l - 2800
\][/tex]
Thus:
[tex]\[
l \times w = l \times w + 40l - 70w - 2800
\][/tex]
Simplify by subtracting [tex]\( l \times w \)[/tex] from both sides:
[tex]\[
0 = 40l - 70w - 2800 \quad (3)
\][/tex]
5. Solving for [tex]\( l \)[/tex] and [tex]\( w \)[/tex]:
Use equation (1):
[tex]\[
l = 290 - w \quad (from\ equation\ 1)
\][/tex]
Substitute [tex]\( l \)[/tex] in equation (3):
[tex]\[
0 = 40(290 - w) - 70w - 2800
\][/tex]
Simplify:
[tex]\[
0 = 11600 - 40w - 70w - 2800
\][/tex]
Combine like terms:
[tex]\[
0 = 11600 - 110w - 2800
\][/tex]
Simplify further:
[tex]\[
0 = 8800 - 110w
\][/tex]
Solving for [tex]\( w \)[/tex]:
[tex]\[
110w = 8800
\][/tex]
[tex]\[
w = \frac{8800}{110} = 80
\][/tex]
6. Find [tex]\( l \)[/tex]:
Using [tex]\( w = 80 \)[/tex] in equation (1):
[tex]\[
l + 80 = 290
\][/tex]
[tex]\[
l = 290 - 80 = 210
\][/tex]
7. Calculate the area:
[tex]\[
\text{Area} = l \times w = 210 \times 80 = 16800 \, \text{square meters}
\][/tex]
So, the area of the terrain is:
[tex]\[
\boxed{16800 \, \text{m}^2}
\][/tex]
