A rectangular garden 50 m long and 34 m wide is surrounded by a uniform dirt road. Find the width of the road if the total area of the garden and road is 540 m^2. Explain your solution
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To find the width of the road surrounding the rectangular garden, we can start by calculating the total area of the garden and road combined.
Given that the length of the rectangular garden is 50 m and the width is 34 m, the area of the garden is:
Area_of_garden = Length × Width
= 50 m × 34 m
= 1700 m²
Let's assume the width of the road is 'w' meters. Since the road surrounds the garden on all sides, the dimensions of the garden including the road will be:
Length_of_combined_area = Length_of_garden + 2 × Width_of_road
= 50 m + 2w
Width_of_combined_area = Width_of_garden + 2 × Width_of_road
= 34 m + 2w
The area of the combined garden and road is given as 540 m². Hence, we can write:
Area_of_combined_area = Length_of_combined_area × Width_of_combined_area
= (50 m + 2w) × (34 m + 2w)
= 540 m²
Expanding the equation, we get:
1700 m² + 100m × w + 68 m × w + 4w² = 540 m²
Rearranging the terms, we have:
4w² + 168w + 1700 - 540 = 0
Simplifying the equation further:
4w² + 168w - 1160 = 0
To solve this quadratic equation, we can factor or use the quadratic formula:
w = (-b ± √(b² - 4ac)) / (2a)
where a = 4, b = 168, and c = -1160.
Solving the equation using the quadratic formula:
w = (-168 ± √(168² - 4 × 4 × -1160)) / (2 × 4)
= (-168 ± √(28224 + 18560)) / 8
= (-168 ± √(46784)) / 8
= (-168 ± 216) / 8
Taking the positive value, we get:
w = (-168 + 216) / 8
w = 48 / 8
w = 6
Therefore, the width of the road surrounding the rectangular garden is 6 meters.
To find the width of the road surrounding the rectangular garden, we can follow these steps:
1. Let's start by considering the dimensions of the garden. We are given that the length of the rectangular garden is 50 m and the width is 34 m.
2. We need to find the width of the road. Let's assume the width of the road to be x meters.
3. If the width of the garden is 34 m, and we add the width of the road on both sides, the total width becomes 34 + 2x.
4. Similarly, if the length of the garden is 50 m, and we add the width of the road on both ends, the total length becomes 50 + 2x.
5. Now, to calculate the area of the rectangular garden and road combined, we multiply the new length (50 + 2x) by the new width (34 + 2x). This gives us the equation:
(50 + 2x) * (34 + 2x) = 540
6. Next, we can expand this equation by using the FOIL method (First, Outer, Inner, Last):
50 * 34 + 50 * 2x + 2x * 34 + 2x * 2x = 540
7. Simplifying further, we have:
1700 + 100x + 68x + 4x^2 = 540
8. Combining like terms and rearranging the equation, we get a quadratic equation in standard form:
4x^2 + 168x + 1160 = 0
9. Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Applying this formula, we have:
x = (-168 ± √(168^2 - 4 * 4 * 1160)) / (2 * 4)
x = (-168 ± √(28224 - 18560)) / 8
x = (-168 ± √(9664)) / 8
x = (-168 ± 98.3) / 8
10. By solving the equation, we get two possible values for x:
x = (-168 + 98.3) / 8 ≈ -8.96
x = (-168 - 98.3) / 8 ≈ -33.91
Since a negative width doesn't make sense in this context, we can disregard the negative value.
11. Therefore, the width of the road surrounding the rectangular garden is approximately 8.96 meters.
Garden:
A = LW
A = 50 * 34
A = 1,700 sq. m
Since that is much larger than the total, your problem seems to have an error. Please check and repost.