A rectangular field has a perimeter of 500 m and an area of 14400 m². Find the lengths of the sides.

Steps to Solve

1. Write Formulas for Perimeter and Area

The formulas for the perimeter and area of a rectangle are given as follows:

• Perimeter: ( P = 2(l + w) )
• Area: ( A = l \times w )

2. Substitute Known Values

Substituting the known values for perimeter and area into the formulas:

From the perimeter, we have: $$ 500 = 2(l + w) $$

From the area, we have: $$ 14400 = l \times w $$

3. Simplify the Perimeter Equation

To isolate ( l + w ), divide the perimeter equation by 2:

$$ l + w = 250 $$

4. Express One Variable in Terms of the Other

From the equation ( l + w = 250 ), we can express ( w ) in terms of ( l ):

$$ w = 250 - l $$

5. Substitute into the Area Equation

Substituting ( w ) in the area equation gives us:

$$ 14400 = l \times (250 - l) $$

6. Form a Quadratic Equation

Expanding the equation results in:

$$ 14400 = 250l - l^2 $$

Rearranging this gives:

$$ l^2 - 250l + 14400 = 0 $$

7. Solve the Quadratic Equation

To solve for ( l ), use the quadratic formula ( l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ), where ( a = 1, b = -250, c = 14400 ):

First, calculate the discriminant:

$$ b^2 - 4ac = (-250)^2 - 4 \times 1 \times 14400 $$

Calculating:

$$ 62500 - 57600 = 4900 $$

Now apply the quadratic formula:

$$ l = \frac{250 \pm \sqrt{4900}}{2} $$

8. Calculate the Length of l

Calculating ( \sqrt{4900} ):

$$ \sqrt{4900} = 70 $$

So we have:

$$ l = \frac{250 \pm 70}{2} $$

Calculating both possible values:

1. ( l = \frac{320}{2} = 160 )
2. ( l = \frac{180}{2} = 90 )

Thus, we find ( l = 160 ) or ( l = 90 ).

9. Find Corresponding Widths

For each length, find the corresponding width using ( w = 250 - l ):

• If ( l = 160 ), $$ w = 250 - 160 = 90 $$

• If ( l = 90 ), $$ w = 250 - 90 = 160 $$

So the dimensions of the rectangular field are 160 m by 90 m.