To solve this problem, let’s define the variables and set up an equation based on the information provided.

Problem Breakdown:

1. Let the length of the field be $L$ (in feet).
2. The width of the field is 20 feet less than the length, so the width is $L - 20$.
3. The area of the field is given as 12,000 square feet.

Setting up the Equation:

The area of a rectangle is given by: $\text{Area} = \text{Length} \times \text{Width}$ Using the given values, we have: $12,000 = L \times (L - 20)$

Expanding this equation: $12,000 = L^2 - 20L$

Rearrange to form a quadratic equation: $L^2 - 20L - 12,000 = 0$

Solving the Quadratic Equation

We can solve for $L$ using the quadratic formula: $L = \frac{-(-20) \pm \sqrt{(-20)^2 - 4 \cdot 1 \cdot (-12,000)}}{2 \cdot 1}$ $L = \frac{20 \pm \sqrt{400 + 48,000}}{2}$ $L = \frac{20 \pm \sqrt{48,400}}{2}$ $L = \frac{20 \pm 220}{2}$

This gives two potential solutions for $L$:

1. $L = \frac{20 + 220}{2} = 120$
2. $L = \frac{20 - 220}{2} = -100$ (which is not feasible, as length cannot be negative)

Therefore, the length of the field is 120 feet.

Answer:

The correct answer is:

• 120 ft

Would you like a more detailed explanation or have any questions?

Related Questions:

1. How would you calculate the width of the field in this scenario?
2. What if the width was 30 feet less than the length instead of 20 feet? How would that change the solution?
3. Can this quadratic equation be solved by factoring instead of using the quadratic formula?
4. What are the dimensions if the area was 15,000 square feet but the width remained 20 feet less than the length?
5. How would this problem change if the field were triangular with a given base and height?

Tip:

When solving area-based problems involving rectangles, always set up an equation using the area formula and rearrange it into a solvable quadratic form if necessary.