A rectangular park measuring 35 yards by 24 yards is surrounded by a trail of uniform width. If the area of the park and the trail combined is 2420 square yards, what is the width of the path?

To find the width of the path, we need to subtract the area of the park from the total area of the park and trail combined.

The area of the park is given by length × width, which is 35 yards × 24 yards = 840 square yards.

Therefore, the area of the trail is the total area minus the park area: 2420 square yards - 840 square yards = 1580 square yards.

Since the trail is uniform in width, we can let the width of the trail be x yards.

The total length (including the park and trail) is 35 yards + 2x yards (one width of trail on each side), and the total width is 24 yards + 2x yards.

The area of the trail is then calculated as the difference between the total area and the park area: (35 + 2x) yards × (24 + 2x) yards = 1580 square yards.

Expanding the equation, we have:
(35 + 2x) × (24 + 2x) = 1580

Simplifying the equation further, we get:
840 + 98x + 70x + 4x^2 = 1580
4x^2 + 168x + 840 - 1580 = 0
4x^2 + 168x - 740 = 0

To solve this quadratic equation, we can either factor it or use the quadratic formula.

Factoring the equation, we can write it as:
4(x^2 + 42x - 185) = 0

Now, we solve the quadratic factor:
x^2 + 42x - 185 = 0

This equation does not easily factor, so we can apply the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)

For the equation x^2 + 42x - 185 = 0, the coefficients are:
a = 1, b = 42, and c = -185.

Substituting these values into the quadratic formula, we get:
x = (-42 ± √(42^2 - 4(1)(-185))) / (2(1))
x = (-42 ± √(1764 + 740)) / 2
x = (-42 ± √2504) / 2
x = (-42 ± 50.04) / 2

Simplifying further:
x = (8.04 or -92.04) / 2
x ≈ 4.02 or -46.02

Since the width cannot be negative, the width of the path is approximately 4.02 yards.