1. A gardener has 30 feet by 20 feet rectangular plot of ground. He wants to build a brick of walkway of uniform width on the border of the plot. If the gardener wants to have 400 square feet of ground left for planting, how wide should he build the walkway?
2. A rectangular lawn is 60 meters by 80 meters. A part of the lawn is torn out to install a pool, leaving a strip of lawn of uniform width around the pool. The area of the pool is 1/6 of the old lawn areas. How wide is the trip of lawn?
Step 1: Calculate the area of the rectangular plot.
The area of the rectangular plot is calculated by multiplying the length and width together. Given that the length is 30 feet and the width is 20 feet, the area of the rectangular plot is:
Area = length * width
Area = 30 ft * 20 ft
Area = 600 square feet
Step 2: Calculate the area of the desired planting area.
The gardener wants to have 400 square feet of ground left for planting, so we subtract this area from the total area of the rectangular plot to find the area of the walkway.
Area of walkway = Area of rectangular plot - Desired planting area
Area of walkway = 600 sq ft - 400 sq ft
Area of walkway = 200 sq ft
Step 3: Calculate the width of the walkway.
To determine the width of the walkway, we need to divide the area of the walkway by the width of the rectangular plot. This will give us the uniform width of the walkway along each side.
Width of walkway = Area of walkway / Length of rectangular plot
Width of walkway = 200 sq ft / 30 ft
Width of walkway = 6.67 ft or approximately 6.7 feet
Therefore, the gardener should build the walkway to be approximately 6.7 feet wide.
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Step 1: Calculate the area of the rectangular lawn.
The area of the rectangular lawn is calculated by multiplying the length and width together. Given that the length is 60 meters and the width is 80 meters, the area of the rectangular lawn is:
Area = length * width
Area = 60 m * 80 m
Area = 4800 square meters
Step 2: Calculate the area of the pool.
The area of the pool is 1/6 of the old lawn area. To calculate the area of the pool, we divide the area of the rectangular lawn by 6.
Area of pool = Area of rectangular lawn / 6
Area of pool = 4800 sq m / 6
Area of pool = 800 square meters
Step 3: Calculate the area of the strip of lawn.
The area of the strip of lawn is the remaining area after the pool is installed. To find this, we subtract the area of the pool from the area of the rectangular lawn.
Area of strip of lawn = Area of rectangular lawn - Area of pool
Area of strip of lawn = 4800 sq m - 800 sq m
Area of strip of lawn = 4000 square meters
Step 4: Calculate the width of the strip of lawn.
To determine the width of the strip of lawn, we need to divide the area of the strip of lawn by the sum of the length and width of the rectangular lawn and pool. This will give us the uniform width of the strip of lawn around the pool.
Width of strip of lawn = Area of strip of lawn / (Length of rectangular lawn + Width of rectangular lawn + Length of pool + Width of pool)
Width of strip of lawn = 4000 sq m / (60 m + 80 m)
Width of strip of lawn = 4000 sq m / 140 m
Width of strip of lawn = 28.57 m or approximately 28.6 meters
Therefore, the strip of lawn around the pool should be approximately 28.6 meters wide.
To solve these questions, we need to use the concept of areas and proportions.
1. To find the width of the walkway, we can start by finding the area of the rectangular plot. The formula for calculating the area of a rectangle is length * width. In this case, the length is 30 feet and the width is 20 feet, so the total area is 30 * 20 = 600 square feet.
Since the gardener wants to leave 400 square feet for planting, we subtract that area from the total area to find the area of the walkway. So, the area of the walkway is 600 - 400 = 200 square feet.
Since the walkway has a uniform width, we can let the width be x feet. To find the dimensions of the walkway, we set up the following equation: (30 + 2x)(20 + 2x) = 200.
This equation represents the area of the walkway as the product of the length and width of the rectangle formed by the walkway, which is equal to 200 square feet.
Expanding the equation and simplifying gives us: 4x^2 + 100x - 200 = 0.
Now, we can solve this quadratic equation for x. Using factoring, the equation can be rewritten as: 4(x^2 + 25x - 50) = 0.
The quadratic expression inside the parentheses can be factored as (x + 50)(x - 1) = 0.
This means that the solutions to the equation are x = -50 and x = 1. Since the width cannot be negative, the width of the walkway is 1 feet.
Therefore, the gardener should build the walkway with a width of 1 foot.
2. To find the width of the strip of lawn around the pool, we first need to find the area of the lawn and the area of the pool.
The area of a rectangle is calculated by multiplying its length by its width. In this case, the length is 80 meters and the width is 60 meters, so the area of the lawn is 80 * 60 = 4800 square meters.
The area of the pool is given as 1/6 of the old lawn areas. Therefore, the area of the pool is (1/6) * 4800 = 800 square meters.
Let's assume the width of the strip of lawn around the pool is x meters. To find the dimensions of the strip, we subtract the area of the pool from the area of the lawn, and set up an equation: (60 - 2x)(80 - 2x) = 800.
Expanding the equation and simplifying gives us: 4x^2 - 280x + 3200 = 0.
To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, the quadratic equation doesn't factor nicely, so we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.
For the equation 4x^2 - 280x + 3200, the values of a, b, and c are 4, -280, and 3200 respectively. Plugging these values into the quadratic formula gives us: x = (-(-280) ± √((-280)^2 - 4*4*3200)) / (2*4).
Simplifying further gives us: x = (280 ± √(78400 - 51200)) / 8.
x = (280 ± √(27200)) / 8.
x = (280 ± 164.97) / 8.
Calculating the positive and negative solutions gives us two possible values for x: x ≈ 53.12 meters and x ≈ -4.97 meters. Since the width cannot be negative, the width of the strip of lawn is approximately 53.12 meters.
Therefore, the strip of lawn around the pool should be approximately 53.12 meters wide.
1.
Total area = 30*20 ft²
Let x=width of walkway,
final area = (30-2x)(20-2x)=400
Solve for x.
2.
Lawn area = 60*80 m²
Final lawn width = x
Pool area = (60-2x)(80
2x)=(1/6)(60*80)
Solve for x.