Question

the construction crew for a new house will dig a hole in the shape of a rectangular prism for the house’s 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.
( \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 isn’t full.
trips
(c) if it costs $1075 for each trip that the truck makes, how much will it cost to haul away all the dirt?
$

Explanation:

Part (a)

Step 1: Find the 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, and \( h = 6 \) ft.
\( V = 45 \times 33 \times 6 \)
\( V = 45 \times 198 \)
\( V = 8910 \) cubic feet.

Step 2: Convert cubic feet to cubic yards

We know that \( 1 \) yard \( = 3 \) feet, so \( 1 \) cubic yard \( = 3^3 = 27 \) cubic feet.
To convert cubic feet to cubic yards, divide the volume in cubic feet by \( 27 \).
\( \text{Volume in cubic yards} = \frac{8910}{27} \)
\( \text{Volume in cubic yards} = 330 \)

Step 1: Determine the number of trips

Each trip the truck can haul \( 14 \) cubic yards. The total volume of dirt is \( 330 \) cubic yards.
We use the formula \( \text{Number of trips} = \lceil \frac{\text{Total Volume}}{\text{Volume per trip}}
ceil \) (since we can't have a fraction of a trip and need to haul all the dirt).
\( \frac{330}{14} \approx 23.57 \)
Since we need to take all the dirt, we round up to the next whole number.

Step 1: Calculate the total cost

Each trip costs \( \$1075 \) and the number of trips is \( 24 \).
Total cost \( = 1075 \times 24 \)
\( 1075 \times 24 = (1000 + 75) \times 24 = 1000 \times 24 + 75 \times 24 = 24000 + 1800 = 25800 \)

Answer:

\( 330 \) cubic yards

Part (b)