Paving cost is $10.30 per square yard. find the total cost of paving the shaded area in figure 27-21. Round the answers to nearest ten dollars.
170'-0" Radius
92'0 is the degrees
256'-0" is the shaded area
The sum total is 14780
please help
To find the total cost of paving the shaded area, we need to calculate the area of the shaded region first.
Given:
Radius of the circular region: 170 ft
Degrees of the shaded area: 92°
Total area of the shaded region: 256 ft^2
To find the area of the shaded region, we need to find the area of the full circle first.
1. Calculate the area of the full circle:
The formula for the area of a circle is A = πr^2, where A represents the area and r represents the radius.
Plugging in the values: A_full = π(170)^2
2. Calculate the area of the shaded region:
To find the area of the shaded region, we can use the proportion of the central angle to the total angle of a circle. Since the total angle of a circle is 360°, we can calculate the area of the shaded region as follows:
A_shaded = A_full * (degrees of shaded area/360°)
3. Calculate the total cost of paving the shaded area:
The cost of paving is given as $10.30 per square yard. Since we have the area in square feet, we need to convert it to square yards.
1 yard = 3 feet, so 1 square yard = 3^2 = 9 square feet.
The total cost can be calculated as follows:
Total Cost = A_shaded (in square yards) * Cost per square yard.
Let's calculate step by step:
1. Calculate the area of the full circle:
A_full = π(170)^2
2. Calculate the area of the shaded region:
A_shaded = A_full * (92°/360°)
3. Convert the area to square yards:
A_shaded_yard = A_shaded / 9
4. Calculate the total cost:
Total Cost = A_shaded_yard * $10.30
Given the sum total of 14780, let's solve the equation:
14780 = Total Cost (rounded to the nearest ten dollars)
Therefore, we need to find the Total Cost rounded to the nearest ten dollars.
To find the total cost of paving the shaded area, we need to calculate the area of the shaded region first. Given that the radius is 170 feet and the shaded area is 256 square feet, we can use the formula for the area of a sector:
Area of a sector = (θ/360) * (π * r^2)
θ represents the angle of the sector, and r represents the radius.
In this case, the angle θ is given as 92 degrees, but we need to convert it to radians before plugging it into the formula. To convert degrees to radians, we use the conversion factor: π/180.
θ (in radians) = (92 * π) / 180
Now we can substitute the values into the formula:
Area of the shaded sector = ((92 * π) / 180) * (π * (170^2))
To simplify the calculation, we will approximate π as 3.14 and use the given radius of 170 feet:
Area of the shaded sector ≈ ((92 * 3.14) / 180) * (3.14 * 170^2)
Next, we need to convert the area from square feet to square yards. Since 1 yard is equal to 3 feet, 1 square yard is equal to 9 square feet. Therefore:
Area of the shaded region (in square yards) = (Area of the shaded sector) / 9
Finally, to find the total cost, we multiply the area of the shaded region (in square yards) by the cost per square yard:
Total cost = (Area of the shaded region in square yards) * ($10.30 per square yard)
By following these calculations, you should be able to determine the total cost of paving the shaded area in figure 27-21.