Let's break down the problem step by step based on the image provided.

(a) Find the volume of dirt that will be hauled away in cubic yards.

The hole is in the shape of a rectangular prism, and the dimensions are:

• Length: 54 feet
• Width: 45 feet
• Depth: 12 feet

The volume $V$ of a rectangular prism is calculated by the formula: $V = \text{length} \times \text{width} \times \text{height}$ This will give us the volume in cubic feet. However, we need to convert this to cubic yards.

1 yard = 3 feet, so: $1 \text{ cubic yard} = (3 \, \text{feet})^3 = 27 \, \text{cubic feet}$

Now, calculate the volume in cubic feet: $V = 54 \times 45 \times 12 = 29,160 \, \text{cubic feet}$

Convert cubic feet to cubic yards: $\text{Volume in cubic yards} = \frac{29,160}{27} = 1,080 \, \text{cubic yards}$

So, the volume of dirt is 1,080 cubic yards.

(b) How many trips are needed to haul all of the dirt away?

Each trip, the truck can haul 17 cubic yards. To find the number of trips needed: $\text{Number of trips} = \frac{\text{Total volume of dirt}}{\text{Volume per trip}} = \frac{1,080}{17} \approx 63.53$ Since the crew cannot make a fraction of a trip, we round up to 64 trips.

So, 64 trips are needed.

(c) Total cost to haul away the dirt?

The cost per trip is $1,125. To find the total cost, multiply the number of trips by the cost per trip: $\text{Total cost} = 64 \times 1,125 = 72,000 \, \text{dollars}$

So, the total cost is $72,000.

Would you like further details on any part of this solution? Here are some related questions to explore:

1. How would the volume change if the depth of the hole increased by 3 feet?
2. What would the cost be if the truck could carry 20 cubic yards per trip instead of 17?
3. If the cost per trip increased by 10%, how much more would it cost in total?
4. How many trips would be needed if the hole dimensions were doubled?
5. What is the impact of rounding up trips on the total cost?

Tip: Always ensure to convert units correctly when dealing with different measurement systems to avoid calculation errors.