Final answer:
The width of the dirt road surrounding the rectangular garden is found to be 3 meters using the overall area given and the dimensions of the garden, by setting up and solving a quadratic equation.
Step-by-step explanation:
To find the width of the dirt road surrounding the rectangular garden, we will use the area information given. The area of just the garden is 50 meters by 34 meters, which is 1700 square meters. Since the total area of the garden and dirt road is 540 square meters more than that, the combined area is 1700 m2 + 540 m2 = 2240 m2.
Let the uniform width of the road be x meters. The overall dimensions of the garden and road together would then be (50 + 2x) meters in length and (34 + 2x) meters in width.
The equation for the total area is:
(50 + 2x)(34 + 2x) = 2240
1700 + 100x + 68x + 4x2 = 2240
4x2 + 168x + 1700 = 2240
4x2 + 168x - 540 = 0
Dividing the entire equation by 4 to simplify, we get:
x2 + 42x - 135 = 0
Solving this quadratic equation, we look for factors of -135 that add up to 42. The factors are 45 and -3. Therefore, x = -45 or x = 3.
We ignore the negative value because a width cannot be negative, so x = 3 meters is the width of the dirt road.