- Analyze the relationship between distance and time to form the proportion \(\frac{54}{1.2} = \frac{135}{x}\).
- Cross-multiply to obtain the equation \(54x = 135 \cdot 1.2\).
- Simplify the product \(135 \cdot 1.2 = 162\) and divide by 54 to isolate \(x\).
- Conclude that the truck will take \(\boxed{3}\) hours to travel 135 miles.
Explanation
- Problem Analysis and Setup
To determine the time required for the delivery truck to travel 135 miles, we start with the given information that the truck travels 54 miles in 1.2 hours at a constant speed. This relationship between distance and time is expressed as:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{54 \text{ miles}}{1.2 \text{ hours}} \]
Assuming the truck continues at this constant speed, we want to find the time \(x\) in hours required to cover a distance of 135 miles, leading to the equation:
\[ \frac{54}{1.2} = \frac{135}{x} \]
This is a standard proportion problem where the ratio of distance to time remains constant.
- Solving the Proportion
To solve for \(x\), we use the property of proportions: if \(\frac{a}{b} = \frac{c}{d}\), then \(a \cdot d = c \cdot b\). Applying this to our equation \(\frac{54}{1.2} = \frac{135}{x}\), we get:
\[ 54 \cdot x = 135 \cdot 1.2 \]
First, we calculate the product on the right side:
\[ 135 \cdot 1.2 = 162 \]
Now the equation is:
\[ 54 \cdot x = 162 \]
To isolate \(x\), we divide both sides by 54:
\[ x = \frac{162}{54} \]
Dividing 162 by 54 yields exactly 3. Therefore, it will take the truck 3 hours to travel 135 miles.
Examples
This type of calculation is common in logistics and travel planning. For instance, if you are planning a trip and know your average speed from previous segments of the journey, you can accurately estimate your arrival time for longer distances. Whether you are driving a car, taking a train, or managing a fleet of delivery trucks, understanding constant rate relationships helps in optimizing routes and managing delivery schedules efficiently.