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.
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.
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.
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.