A rectangular grassy area in a park measures 50 yards by 100 yards. The city wishes to put a uniform sidewalk around the grassy area which would increase the area by 459 yd^2. What is the width of the sidewalk?
new length = 100 + 2 x
new width = 50 + 2 x
new area =(2x+100)(2x+50)
so
(2x+100)(2x+50) - 50*100 = 459
solve quadratic :)
What do you mean by solve quadratic?
To find the width of the sidewalk, let's first calculate the current area of the grassy area.
The current dimensions of the grassy area are:
Length = 100 yards
Width = 50 yards
So, the current area of the grassy area is given by:
Area = Length * Width
= 100 yards * 50 yards
= 5000 yd^2
Now, we are given that the city wishes to put a uniform sidewalk around the grassy area which would increase the area by 459 yd^2.
Let's assume the width of the sidewalk to be 'x' yards.
The new dimensions of the grassy area would be:
Length = original length + 2 * width of the sidewalk = 100 yards + 2x yards
Width = original width + 2 * width of the sidewalk = 50 yards + 2x yards
The new area of the grassy area plus the sidewalk would be:
New Area = (Length + 2 * width of the sidewalk) * (Width + 2 * width of the sidewalk)
= (100 yards + 2x) * (50 yards + 2x)
= 5000 yd^2 + 200x yd + 100x yd + 4x^2 yd^2
Given that the new area is the current area plus an additional 459 yd^2, we can set up the following equation:
New Area - Current Area = 459 yd^2
(5000 yd^2 + 200x yd + 100x yd + 4x^2 yd^2) - 5000 yd^2 = 459 yd^2
Simplifying the equation:
200x yd + 100x yd + 4x^2 yd^2 = 459 yd^2
Now, we can solve this quadratic equation to find the possible values of 'x' which represent the width of the sidewalk.