A rectangular grass area ub a park measures 50 yards by 100 yards. The city wishes to put a uniform sidewalk around the grass area which would increase the area by 459 yd ^2. What is the width of sidewalk
Im having a hard time setting up the problem.
To find the width of the sidewalk, we need to first calculate the current area of the grass area and then determine the new area including the sidewalk.
The current area of the grass area can be calculated by multiplying the length (50 yards) by the width (100 yards):
Current Area = Length x Width
= 50 yards x 100 yards
= 5000 yd^2
Now, we know that the city wants to add a uniform sidewalk around the grass area, which will increase the overall area by 459 yd^2.
New Area = Current Area + Increase in Area
= 5000 yd^2 + 459 yd^2
= 5459 yd^2
Next, we need to determine the dimensions of the new grass area with the added sidewalk. Let's assume the width of the sidewalk is 'x' yards. Since the sidewalk is added evenly on all sides, the new length and width of the grass area will increase by 2x yards.
New Length = Length + 2x
New Width = Width + 2x
Therefore, the new area of the grass area with the added sidewalk can be calculated as:
New Area = New Length x New Width
= (Length + 2x) x (Width + 2x)
= (50 yd + 2x) x (100 yd + 2x)
= 5459 yd^2
Setting this equation equal to the new area, we get:
(50 yd + 2x) x (100 yd + 2x) = 5459 yd^2
To solve this quadratic equation, we can expand and rearrange the equation:
(50 x 100) + (50 x 2x) + (2x x 100) + (2x x 2x) = 5459
5000 + 100x + 100x + 4x^2 = 5459
4x^2 + 200x - 459 = 0
Now, we can solve this quadratic equation using methods like factoring, completing the square, or using the quadratic formula. Once we find the value of x, it will give us the width of the sidewalk.