An engineer’s plan shows a canal with a trapezoidal cross section that is 8 ft deep and 14 ft across at the bottom with walls sloping outward at an angle of 45°. The canal is 620 ft long. A contractor bidding on the job estimates the cost to excavate the canal at $175/yd3. If the contractor adds 10% profit, what should the bid be?
$12800
To find out what the bid should be, we need to calculate the volume of the canal and then calculate the cost of excavating that volume.
First, let's calculate the cross-sectional area of the canal. Since the canal has a trapezoidal shape, we can split it into a rectangle and two triangles. The rectangle has a width of 14 ft (across the bottom of the canal) and a height of 8 ft (depth of the canal).
The area of the rectangle part is therefore:
Area_rectangle = width * height = 14 ft * 8 ft = 112 ft^2
Since the two triangles are congruent, we need to calculate the area of only one triangle and then double it.
The base of the triangle is the width of the rectangular part of the cross-section, which is 14 ft. The height of the triangle can be calculated using the angle given. Since the walls slope outward at an angle of 45°, the height of the triangle is equal to the width of the rectangle, which is also 14 ft.
The area of one triangle is:
Area_triangle = 1/2 * base * height = 1/2 * 14 ft * 14 ft = 98 ft^2
The total area of both triangles is:
Area_triangles = 2 * Area_triangle = 2 * 98 ft^2 = 196 ft^2
Now we can calculate the total cross-sectional area of the canal by adding the areas of the rectangle and the triangles:
Total_area = Area_rectangle + Area_triangles = 112 ft^2 + 196 ft^2 = 308 ft^2
To calculate the volume of the canal, we multiply the cross-sectional area by the length of the canal:
Volume = Total_area * Length = 308 ft^2 * 620 ft = 190,960 ft^3
Now we can calculate the cost of excavating this volume. The cost is given as $175 per cubic yard, so we need to convert the volume from cubic feet to cubic yards.
Since 1 cubic yard is equal to 27 cubic feet, we divide the volume by 27 to get the volume in cubic yards:
Volume_yards = Volume / 27 = 190,960 ft^3 / 27 = 7,077.04 yd^3
Now we can calculate the cost of excavating this volume:
Cost = Volume_yards * Cost per yard = 7,077.04 yd^3 * $175/yd^3 = $1,237,293.13
Finally, we need to add 10% profit to the cost:
Bid = Cost + 10% * Cost = $1,237,293.13 + 0.1 * $1,237,293.13 = $1,361,022.44
Therefore, the bid for the job should be $1,361,022.44.