You want to determine how much it will cost for a dump truck to deliver the soil, versus using a borrowed pickup truck. Cost for Delivery It cosis $$ \$ 25 $$ for every 14 cubic yards for each truckload. Note: You can't order a partial truckload. Based on your Part 2 answers, how much will it cost to deliver the topsoil? $$ \$ $$ $$ \square $$ 175 Cost for Pickup You want to compare this to picking up the soil yourself. The truck you have available has a bed that is 80 inches long, 69 inches wide, and 20 inches tall. The store is 6 miles away and it takes about 20 minutes to drive there. The truck averages about 17 miles per gallon and gas costs $$ \$ 3.04 $$ per gallon. What is the volume of topsoil that you can fit in the truck without it spilling over? Round answers to two decimal places, if needed. $$ \square $$ 110400 cubic inches $$ \begin{array}{l} =63.89 \quad \text { cubic feet } \ =237 \quad \sim \text { cubic yards } \end{array} $$ How many trips will you need to make in your truck? $$ \square $$ trips in the truck What is the total cost (in gas) to delrver the soil in the pickup truck? 5 $$ \square $$ 103.36 Comparing Cost for Final Decision Which option is cheaper? $$ \square $$ use own truok How much cheaper is it? $$ \$ $$ $$ \square $$ 127.64

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Let's break down the problem step by step to find the costs associated with both delivery options: using a dump truck and using a pickup truck.

### Step 1: Cost for Delivery with Dump Truck

The cost for delivery using a dump truck is given as $$ \$25 $$ for every 14 cubic yards.

To find the total cost, we need to know how many truckloads are required based on the total volume of topsoil needed.

Assuming the total volume of topsoil needed is $$ V $$ cubic yards, the number of truckloads required can be calculated as:
$$
\text{Number of truckloads} = \lceil \frac{V}{14} \rceil
$$
The total cost for the dump truck delivery will then be:
$$
\text{Total Cost} = \text{Number of truckloads} \times 25
$$

### Step 2: Volume of Topsoil in Pickup Truck

The dimensions of the pickup truck bed are:
- Length: 80 inches
- Width: 69 inches
- Height: 20 inches

First, we calculate the volume in cubic inches:
$$
\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}
$$
Next, we convert this volume to cubic feet and then to cubic yards:
$$
\text{Volume in cubic feet} = \frac{\text{Volume in cubic inches}}{1728}
$$
$$
\text{Volume in cubic yards} = \frac{\text{Volume in cubic feet}}{27}
$$

### Step 3: Number of Trips Required

To find the number of trips required, we use the volume of topsoil that can fit in the truck and the total volume needed:
$$
\text{Number of trips} = \lceil \frac{V}{\text{Volume in cubic yards}} \rceil
$$

### Step 4: Total Cost for Gas

The distance to the store is 6 miles, and the truck averages 17 miles per gallon. The cost of gas is $$ \$3.04 $$ per gallon.

First, we calculate the total distance for a round trip:
$$
\text{Total Distance} = 2 \times 6 \text{ miles} = 12 \text{ miles}
$$
Next, we find out how many gallons of gas are needed for one trip:
$$
\text{Gallons needed per trip} = \frac{\text{Total Distance}}{17}
$$
Then, we calculate the cost of gas for one trip:
$$
\text{Cost of gas per trip} = \text{Gallons needed per trip} \times 3.04
$$
Finally, the total cost for gas for all trips is:
$$
\text{Total Gas Cost} = \text{Cost of gas per trip} \times \text{Number of trips}
$$

### Step 5: Comparing Costs

Now we can compare the total costs of using the dump truck and the pickup truck to determine which is cheaper and by how much.

Let's perform the calculations.

1. **Calculate the volume of the pickup truck:**
   - Volume in cubic inches:
   $$
   V_{\text{inches}} = 80 \times 69 \times 20
   $$
   - Convert to cubic feet:
   $$
   V_{\text{feet}} = \frac{V_{\text{inches}}}{1728}
   $$
   - Convert to cubic yards:
   $$
   V_{\text{yards}} = \frac{V_{\text{feet}}}{27}
   $$

2. **Calculate the number of trips:**
   - Assuming $$ V $$ is the total volume of topsoil needed in cubic yards.

3. **Calculate the total gas cost.**

Let's perform these calculations.

### Performing Calculations

1. **Volume of Pickup Truck:**
   $$
   V_{\text{inches}} = 80 \times 69 \times 20
   $$
   $$
   V_{\text{inches}} = 110400 \text{ cubic inches}
   $$
   $$
   V_{\text{feet}} = \frac{110400}{1728} \approx 63.89 \text{ cubic feet}
   $$
   $$
   V_{\text{yards}} = \frac{63.89}{27} \approx 2.37 \text{ cubic yards}
   $$

2. **Number of Trips:**
   Assuming $$ V = 175 $$ cubic yards (as per the problem statement):
   $$
   \text{Number of trips} = \lceil \frac{175}{2.37} \rceil \approx 74 \text{ trips}
   $$

3. **Total Gas Cost:**
   - Total distance for one trip:
   $$
   \text{Total Distance} = 12 \text{ miles}
   $$
   - Gallons needed per trip:
   $$
   \text{Gallons needed per trip} = \frac{12}{17} \approx 0.7059
   $$
   - Cost of gas per trip:
   $$
   \text{Cost of gas per trip} = 0.7059 \times 3.04 \approx 2.14
   $$
   - Total gas cost:
   $$
   \text{Total Gas Cost} = 2.14 \times 74 \approx 158.36
   $$

### Final Comparison

1. **Cost for Dump Truck:**
   - Assuming $$ V = 175 $$:
   $$
   \text{Number of truckloads} = \lceil \frac{175}{14} \rceil = 13
   $$
   $$
   \text{Total Cost} = 13 \times 25 = 325
   $$

2. **Cost for Pickup Truck:**
   $$
   \text{Total Gas Cost} \approx 158.36
   $$

3. **Cheaper Option:**
   - The pickup truck is cheaper.
   - Difference:
   $$
   \text{Difference} = 325 - 158.36 \approx 166.64
   $$

### Summary of Results
- Cost for Dump Truck: $$ \$325 $$
- Cost for Pickup Truck: $$ \$158.36 $$
- Cheaper Option: Use own truck
- How much cheaper: $$ \$166.64 $$

Now, let's finalize the calculations and present the results.

### Final Results
- Cost for Dump Truck: $$ \$325 $$
- Volume of Pickup Truck: $$ 2.37 $$ cubic yards
- Number of Trips: $$ 74 $$
- Total Gas Cost: $$ \$158.36 $$
- Cheaper Option: Use own truck
- How much cheaper: $$ \$166.64 $$
- **Cost for Dump Truck:** \$325 - **Volume of Pickup Truck:** 2.37 cubic yards - **Number of Trips Needed:** 74 trips - **Total Gas Cost for Pickup Truck:** \$158.36 - **Cheaper Option:** Use your own truck - **How Much Cheaper:** \$166.64

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