a rotating sprinkler is placed in one corner of a rectangular garden 12cm by 8cm and rotates through an angle of 30°. Calculate the area of the garden not being watered
To calculate the area of the garden that is not being watered, we first need to find the area of the sector that is being watered by the rotating sprinkler.
1. Find the radius of the garden:
The radius of the garden is the distance from the corner where the sprinkler is placed to the opposite corner of the rectangular garden. We can use the Pythagorean theorem to find this distance:
r = √(12^2 + 8^2)
r = √(144 + 64)
r = √208
r ≈ 14.42 cm
2. Calculate the area of the sector:
The sector is a fraction of a circle formed by the angle that the sprinkler rotates (30°). The formula to calculate the area of a sector is:
Area of sector = (θ/360°) × πr^2
Area of sector = (30/360) × π(14.42^2)
Area of sector ≈ (30/360) × 3.1416 × (208)
Area of sector ≈ 2.0944 × 208
Area of sector ≈ 435.51 cm^2
3. Calculate the area of the garden not being watered:
To find the area not being watered, we subtract the area of the sector from the total area of the garden:
Area not being watered = (Total area of garden) - (Area of sector)
Area not being watered = (Length × Width) - (Area of sector)
Area not being watered = (12 cm × 8 cm) - (435.51 cm^2)
Area not being watered ≈ 96 cm^2 - 435.51 cm^2
Area not being watered ≈ -339.51 cm^2
Since we can't have a negative area, there is no area of the garden that is not being watered.
To calculate the area of the garden not being watered, we need to find the area of the sector being watered and subtract it from the total area of the garden.
Step 1: Calculate the area of the sector being watered
The formula to calculate the area of a sector is A = (θ/360) * π * r^2, where θ is the angle of the sector and r is the radius.
Given that the sprinkler rotates through an angle of 30°, we can use this information to calculate the area of the sector being watered.
θ = 30°
r = the shorter side of the rectangle (12cm)
Converting the angle to radians:
θ_rad = (30/180) * π
Calculating the area of the sector:
A_sector = (θ_rad/360) * π * r^2
Step 2: Calculate the total area of the garden
The total area of the garden is the product of the length and width of the rectangle.
Total_area = length * width
Given that the length of the garden is 12cm and the width is 8cm:
Total_area = 12cm * 8cm
Step 3: Calculate the area of the garden not being watered
To find the area of the garden not being watered, subtract the area of the sector being watered from the total area of the garden.
Area_not_watered = Total_area - A_sector
Now let's calculate the values and find the area of the garden not being watered:
θ_rad = (30/180) * π = 0.5236 radians
A_sector = (0.5236/360) * π * (12cm)^2
Total_area = 12cm * 8cm
Area_not_watered = Total_area - A_sector
Calculating these values will give you the area of the garden not being watered.