A rectangular lawn measures 14cm by 8cm.it is surrounded by a path of uniform width.If the area of the path is 48cm²,find it's width
14x8 cm? That must be the lawn for a Polly Pocket house!
If the width of the path is x, then
(14+2x)(8+2x) - 14*8 = 48
x = 1
suspect you mean meters not centimeters
length = 14+ 2 w
width = 8 +2 w
total area to outside of path = (8+ 2w )(14 + 2w) = 4 (4+w)(7+w)
= 4 ( 28 +11 w + w^2)
area inside path = 14*8
so
48 = 4 ( 28 +11 w + w^2) - 14 * 8
12 = ( 28 +11 w + w^2) - 28
w^2 + 11 w - 12 = 0
(w+12)(w-1) = 0
w = 1
To find the width of the path, we need to calculate the difference between the dimensions of the rectangular lawn and the dimensions of the overall area (including the path).
Let's assume the width of the path is denoted by "x".
The length of the overall area including the path would be:
14 cm (length of the lawn) + 2x (width of the path on both sides)
Similarly, the width of the overall area including the path would be:
8 cm (width of the lawn) + 2x (width of the path on both ends)
The area of the overall area including the path can be calculated by subtracting the area of the lawn from the total area:
(14 cm + 2x) * (8 cm + 2x) - (14 cm * 8 cm)
Given that the area of the path is 48 cm², we can set up the following equation:
(14 cm + 2x) * (8 cm + 2x) - (14 cm * 8 cm) = 48 cm²
Now let's solve this equation step by step:
Step 1: Expand the equation
(112 cm² + 28 cmx + 16 cmx + 4x²) - 112 cm² = 48 cm²
Step 2: Simplify the equation
4x² + 44 cmx = 48 cm²
Step 3: Move all terms to one side to set the equation equal to zero
4x² + 44 cmx - 48 cm² = 0
Step 4: Divide through by 4 to simplify the equation
x² + 11 cmx - 12 cm² = 0
Step 5: Factorize the equation
(x + 12 cm)(x - cm) = 0
Step 6: Set each factor equal to zero and solve for x
x + 12 cm = 0 or x - cm = 0
Solving the first equation: x + 12 cm = 0
x = -12 cm
Since the width cannot be negative, we discard this solution.
Solving the second equation: x - cm = 0
x = cm
Therefore, the width of the path is cm.
To find the width of the path surrounding the rectangular lawn, we need to understand the relationship between the areas of the lawn, the path, and the total rectangle.
1. Start by determining the total area of the rectangle, including both the lawn and the path. This can be calculated by multiplying the dimensions of the rectangle: 14cm * 8cm = 112cm².
2. Next, subtract the area of the lawn from the total area to find the area of the path alone. In this case, the area of the path is given as 48cm², so subtracting this gives: 112cm² - 48cm² = 64cm².
3. Since the path is a uniform width on all sides, the width of the path will be the same for both the length and the width of the rectangle. Let's call this width "w".
4. To calculate the width of the path, we can divide the area of the path by the length of the rectangle. Since the path has the same width on both sides, we divide the area of the path by the length of the rectangle + twice the width of the path. In this case, the equation becomes:
w * (8cm + 2w) = 64cm².
5. We can now solve this equation to find the value of "w". Simplifying the equation gives us the following quadratic equation:
8w + 2w^2 = 64.
6. Rearranging the equation to standard quadratic form:
2w^2 + 8w - 64 = 0.
7. Factoring the equation, we get:
(2w - 8)(w + 8) = 0.
8. Solving for the possible values of "w" gives us:
w = 8cm (ignoring the negative value).
Therefore, the width of the path surrounding the rectangular lawn is 8cm.