Question

the construction crew for a new house will dig a hole in the shape of a rectangular prism for the houses foundation. the hole will be 45 ft long, 33 ft wide, and 6 ft deep. to haul away all of the dirt from the hole, a dump truck will take trips to a nearby landfill. the crew knows the amount of dirt the truck can hold each trip in cubic yards.
(a) find the volume of dirt that will be hauled away in cubic yards. use the table of conversion facts, as needed.
(square) (\text{yd}^3)
(b) each trip, the truck can haul away (14 \text{yd}^3) of dirt. to save gas, the crew wants to take as few trips as necessary to haul all of the dirt away. since all of the dirt needs to be hauled away, the crew might have to take a partial load for one trip. how many trips are needed to haul away all of the dirt? count each trip it makes the same, even if the truck isnt full.
(square) trips
(c) if it costs $1075 for each trip that the truck makes, how much will it cost to haul away all the dirt?
$square)
conversion facts for length
1 foot (ft) = 12 inches (in)
1 yard (yd) = 3 feet (ft)
1 yard (yd) = 36 inches (in)

Explanation:

Part (a)

Step 1: Find volume in cubic feet

The volume \( V \) of a rectangular prism is given by \( V = l \times w \times h \), where \( l = 45 \) ft, \( w = 33 \) ft, \( h = 6 \) ft.
\( V = 45 \times 33 \times 6 \)
\( V = 45\times198 = 8910 \) cubic feet.

Step 2: Convert cubic feet to cubic yards

Since \( 1 \) yard \( = 3 \) feet, \( 1 \) cubic yard \( = 3^3 = 27 \) cubic feet.
To convert cubic feet to cubic yards, divide by \( 27 \).
\( \text{Volume in cubic yards} = \frac{8910}{27} \)
\( \frac{8910}{27}= 330 \)

Step 1: Divide total volume by truck capacity

Total volume is \( 330 \) cubic yards, truck capacity is \( 14 \) cubic yards per trip.
Number of trips \( = \frac{330}{14} \approx 23.57 \)

Step 2: Round up for partial trip

Since we can't take a fraction of a trip and need to haul all dirt, we round up to the next whole number.
\( 23.57 \) rounded up is \( 24 \).

Step 1: Multiply number of trips by cost per trip

Number of trips is \( 24 \), cost per trip is \( \$1075 \).
Total cost \( = 24 \times 1075 \)
\( 24\times1075 = 25800 \)

Answer:

\( 330 \)

Part (b)