Math Snap

STEP 1

1. The rectangle has a perimeter of $48$ meters.
2. The area of the rectangle is $140$ square meters.
3. Let the length of the rectangle be $l$ meters and the width be $w$ meters.

STEP 2

1. Set up the equations for perimeter and area.
2. Solve the system of equations.
3. Verify the solution.

STEP 3

Set up the equations for perimeter and area:
1. Perimeter equation:
$$ 2l + 2w = 48
\] 2. Area equation:
$$ lw = 140
\]

STEP 4

Solve the system of equations:
1. Simplify the perimeter equation:
$$ l + w = 24
\] 2. Express $l$ in terms of $w$:
$$ l = 24 - w
\] 3. Substitute $l$ in the area equation:
$$ (24 - w)w = 140
\] 4. Expand and rearrange the equation:
$$ 24w - w^2 = 140
\] 5. Rearrange to form a quadratic equation:
$$ w^2 - 24w + 140 = 0
\] 6. Solve the quadratic equation using the quadratic formula:
$$ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\] where $a = 1$, $b = -24$, $c = 140$.
7. Calculate the discriminant:
$$ b^2 - 4ac = 576 - 560 = 16
\] 8. Calculate the roots:
$$ w = \frac{24 \pm \sqrt{16}}{2}
\] $$ w = \frac{24 \pm 4}{2}
\] 9. The possible values for $w$ are:
$$ w = \frac{28}{2} = 14 \quad \text{or} \quad w = \frac{20}{2} = 10
\] 10. Corresponding $l$ values:
- If $w = 14$, then $l = 24 - 14 = 10$.
- If $w = 10$, then $l = 24 - 10 = 14$.

Verify the solution:
1. Check the perimeter:
$$ 2(14) + 2(10) = 28 + 20 = 48
\] 2. Check the area:
$$ 14 \times 10 = 140
\] Both conditions are satisfied.
The dimensions of the rectangle are $14$ m by $10$ m.