A rectangular lawn measuring 8m by 4m is surrounded by a flower bed of uniform width. The combined area of the lawn and the flower bed is 165cm^2. What is the width of the flower bed
Would you check the numbers? The area of the lawn is 800 x 400 cm which is larger than the flower bed so the flower bed can't surround the lawn. Or am I missing something?
To find the width of the flower bed, follow these steps:
Step 1: Convert the dimensions of the lawn from meters to centimeters.
The width of the lawn is 8 meters, which is equal to 800 centimeters.
The length of the lawn is 4 meters, which is equal to 400 centimeters.
Step 2: Set up an equation to represent the combined area of the lawn and the flower bed.
The combined area of the lawn and the flower bed is given as 165 cm^2.
The area of the lawn is the product of its width and length, which is 800 cm * 400 cm = 320,000 cm^2.
The area of the flower bed is equal to the total combined area minus the area of the lawn, which is 320,000 cm^2 - 165 cm^2 = 319,835 cm^2.
Step 3: Calculate the width of the flower bed.
Let's assume the width of the flower bed is 'w' centimeters.
The width of the entire area (lawn + flower bed + flower bed) can be expressed as 800 cm + 2w cm.
The length of the entire area (lawn + flower bed + flower bed) can be expressed as 400 cm + 2w cm.
The area of the entire area is equal to the product of its width and length:
(800 cm + 2w cm) * (400 cm + 2w cm) = 319,835 cm^2.
Expanding and rearranging the equation:
320,000 cm^2 + 1600w cm + 1600w cm + 4w^2 cm^2 = 319,835 cm^2.
Simplifying the equation:
4w^2 cm^2 + 3200w cm + 320,000 cm^2 = 319,835 cm^2.
Step 4: Solve the equation to get the width of the flower bed.
Rearrange the equation to be a quadratic equation:
4w^2 cm^2 + 3200w cm + 320,000 cm^2 - 319,835 cm^2 = 0.
Simplifying the equation:
4w^2 cm^2 + 3200w cm + 165 cm^2 = 0.
To solve the quadratic equation, you can factorize or use the quadratic formula.
Using the quadratic formula: w = (-b ± √(b^2 - 4ac))/(2a)
where a = 4, b = 3200, and c = 165.
Solving the quadratic equation using the quadratic formula:
w = (-3200 ± √(3200^2 - 4 * 4 * 165))/(2 * 4).
Calculating: w ≈ (-3200 ± √(10,240,000 - 2640))/8.
Simplifying: w ≈ (-3200 ± √(10,237,360))/8.
Performing the square root calculation: w ≈ (-3200 ± 3200.8167)/8.
Calculating two possible values for w: w ≈ (-3200 + 3200.8167)/8 or w ≈ (-3200 - 3200.8167)/8.
Simplifying: w ≈ (0.8167)/8 or w ≈ (-6400.8167)/8.
Calculating the values of w: w ≈ 0.102 cm or w ≈ -800.102 cm.
Since the width cannot be negative, the width of the flower bed is approximately 0.102 cm.
Therefore, the width of the flower bed is approximately 0.102 cm.
To find the width of the flower bed, we need to subtract the area of the lawn from the combined area of the lawn and the flower bed.
1. Calculate the area of the lawn: Area of the lawn = Length × Width = 8m × 4m = 32m^2.
2. Convert the area of the lawn to square centimeters (cm^2): 1m^2 = 10,000cm^2. Therefore, 32m^2 = 32 × 10,000cm^2 = 320,000cm^2.
3. Subtract the area of the lawn from the combined area to find the area of the flower bed: Area of the flower bed = Combined area - Area of the lawn.
Area of the flower bed = 165cm^2 - 320,000cm^2.
4. Since the area of the lawn is greater than the combined area, the result will be negative. This indicates that there is an error, and there might be a mistake in the problem statement or its calculations. Please double-check the given information and ensure it is correct.
If you still have any questions, feel free to ask!