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Sabrina lives in Milton and works in Chester. There are 2 routes she can take to get to work. She can travel on the highway or travel on the town roads that connect Milton, Bedford, Warren, and Chester, as shown in the diagram.



A map of 2 routes.Short description, A map of 2 routes.,Long description,
The map shows 2 routes from Milton to Chester. Together, the routes form a right trapezoid with Milton on the top left corner and Chester on the bottom left corner. The town roads route forms the 3 straight sides of the right trapezoid. The highway route forms the diagonal side.

The town roads route first goes right 15 miles from Milton to Bedford. Next, it goes down 14 miles to Warren. Finally, it goes left 25 miles to Chester.
The highway route slants down and to the left an unknown distance from Milton to Chester.



Sabrina can drive at a constant rate of 65 miles per hour on the highway and 45 miles per hour on the town roads.

Question
How much time, in hours, does Sabrina save by driving on the highway instead of driving on the town roads? Round the answer to the nearest tenth. Enter the answer in the box.

Response area with 1 text input box
hours

Answer 1

To calculate how much time Sabrina saves by driving on the highway instead of the town roads, we first need to determine the total distance for each route.

Distance Calculation for Town Roads Route:

1. From Milton to Bedford: 15 miles
2. From Bedford to Warren: 14 miles
3. From Warren to Chester: 25 miles

Total distance for the town roads route: \[ \text{Total Distance (Town Roads)} = 15 + 14 + 25 = 54 \text{ miles} \]

Time Calculation for Town Roads Route: Sabrina drives at 45 miles per hour on the town roads. Using the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \): \[ \text{Time (Town Roads)} = \frac{54 \text{ miles}}{45 \text{ miles per hour}} = 1.2 \text{ hours} \]

Distance Calculation for Highway Route: The highway route directly connects Milton to Chester. To find this distance, we can visualize the situation or use the right trapezoid properties. We can use the Pythagorean theorem since the route forms a right triangle.

• The horizontal distance from Milton to Chester can be thought of as:
  • From Milton to Bedford: 15 miles (horizontal)
  • From Bedford to Warren: 14 miles (vertical)
• To find the hypotenuse (highway distance, \(d\)), we need to calculate: \[ d = \sqrt{(15 \text{ miles})^2 + (14 \text{ miles})^2} = \sqrt{225 + 196} = \sqrt{421} \] Calculating this gives us approximately: \[ d \approx 20.5 \text{ miles} \]

Time Calculation for Highway Route: Sabrina drives at 65 miles per hour on the highway. Using the formula: \[ \text{Time (Highway)} = \frac{20.5 \text{ miles}}{65 \text{ miles per hour}} \approx 0.315385 \text{ hours} \]

Time Saved Calculation: To find the time saved by using the highway instead of the town roads: \[ \text{Time Saved} = \text{Time (Town Roads)} - \text{Time (Highway)} \] Substituting the values: \[ \text{Time Saved} = 1.2 \text{ hours} - 0.315385 \text{ hours} \approx 0.884615 \text{ hours} \] Rounding to the nearest tenth: \[ \text{Time Saved} \approx 0.9 \text{ hours} \]

Thus, Sabrina saves 0.9 hours by driving on the highway instead of the town roads.