i am really stuck on this application problem in my quadratic equations unit... find the width of a uniform concrete path placed around a 30m by 40m rectangular lawn if the concrete has an area that is 1/4 of the lawn.
Draw rectangle, label the width and length. The length is 30 and the width is 40. Divide this rectangle into fourths. Now you have four squares. Shade in one of the squares. That square will represent your concrete because its 1/4 of the lawn.
Do you think you are able to figure out the width of the concrete now?
To solve this application problem, we can start by setting up an equation based on the given information.
Let's assume that the width of the uniform concrete path is 'x' meters. Since the concrete path surrounds the rectangular lawn, both the length and width of the lawn will be decreased by twice the width of the concrete path. Therefore, the length of the lawn will be (30 - 2x) meters, and the width will be (40 - 2x) meters.
The area of the rectangular lawn is given by its length multiplied by its width:
Area of the lawn = Length × Width = (30 - 2x) × (40 - 2x)
According to the problem, the area of the concrete is 1/4 of the lawn's area:
Area of the concrete = (1/4) × Area of the lawn
Substituting the values, we have:
(1/4) × (30 - 2x) × (40 - 2x) = Area of the concrete
Now, we can solve this equation to find the value of 'x', which represents the width of the concrete path.
To solve the quadratic equation, we can simplify it to the standard form:
(1/4) × (30 - 2x) × (40 - 2x) = x²
Expanding the equation, we get:
(30 - 2x) × (40 - 2x) = 4x²
Next, we can distribute and simplify further:
(1200 - 100x - 80x + 4x²) = 4x²
Combining like terms, we have:
1200 - 180x + 4x² = 4x²
To isolate the 'x' term, let's subtract 4x² from both sides:
1200 - 180x = 0
Now, move all the terms to one side by subtracting 1200 from both sides:
-180x = -1200
Dividing both sides of the equation by -180, we find the value of 'x':
x = -1200 / -180
Simplifying further, we get:
x = 6.67
Hence, the width of the uniform concrete path is approximately 6.67 meters.