Question
IThe three locations that student was interested in were: Adventureland (Point A), Frontierland (Point F), and Tomorrowland (Point T).
Using the Pythagorean Theorem, the student calculated whether Frontierland or Tomorrowland was closer to Adventureland.
Adventureland is (-2,0)
Frontierland: (0,4)
Tomorrowland : (3,3)
(This is placed on a graph, where each unit in the graph is 40 feet)
The distance from Adventureland to Frontierland is about ? Units or ? Feet
The distance from Adventureland to Tomorrowland is about ? units or ? feet
Using the Pythagorean Theorem, the student calculated whether Frontierland or Tomorrowland was closer to Adventureland.
Adventureland is (-2,0)
Frontierland: (0,4)
Tomorrowland : (3,3)
(This is placed on a graph, where each unit in the graph is 40 feet)
The distance from Adventureland to Frontierland is about ? Units or ? Feet
The distance from Adventureland to Tomorrowland is about ? units or ? feet
Answers
GPT-4o mini
To find the distances between Adventureland and the other two locations using the Pythagorean Theorem, we'll first calculate the distances in unit coordinates and then convert those distances to feet.
**Step 1: Calculate the distance from Adventureland (Point A) to Frontierland (Point F).**
1. Coordinates:
\[ A = (-2, 0), \quad F = (0, 4) \]
2. Use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
\[ d = \sqrt{(0 - (-2))^2 + (4 - 0)^2} \]
\[ d = \sqrt{(2)^2 + (4)^2} \]
\[ d = \sqrt{4 + 16} \]
\[ d = \sqrt{20} \]
\[ d = 2\sqrt{5} \approx 4.47 \text{ units} \]
3. Convert to feet (1 unit = 40 feet):
\[ 4.47 \text{ units} \times 40 \text{ feet/unit} \approx 178.8 \text{ feet} \]
**Distance from Adventureland to Frontierland:**
- Approximately **4.47 units** or **178.8 feet**.
**Step 2: Calculate the distance from Adventureland (Point A) to Tomorrowland (Point T).**
1. Coordinates:
\[ A = (-2, 0), \quad T = (3, 3) \]
2. Use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
\[ d = \sqrt{(3 - (-2))^2 + (3 - 0)^2} \]
\[ d = \sqrt{(5)^2 + (3)^2} \]
\[ d = \sqrt{25 + 9} \]
\[ d = \sqrt{34} \approx 5.83 \text{ units} \]
3. Convert to feet (1 unit = 40 feet):
\[ 5.83 \text{ units} \times 40 \text{ feet/unit} \approx 233.2 \text{ feet} \]
**Distance from Adventureland to Tomorrowland:**
- Approximately **5.83 units** or **233.2 feet**.
**Summary of Distances:**
- **Adventureland to Frontierland:**
- Approximately **4.47 units** or **178.8 feet**.
- **Adventureland to Tomorrowland:**
- Approximately **5.83 units** or **233.2 feet**.
**Step 1: Calculate the distance from Adventureland (Point A) to Frontierland (Point F).**
1. Coordinates:
\[ A = (-2, 0), \quad F = (0, 4) \]
2. Use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
\[ d = \sqrt{(0 - (-2))^2 + (4 - 0)^2} \]
\[ d = \sqrt{(2)^2 + (4)^2} \]
\[ d = \sqrt{4 + 16} \]
\[ d = \sqrt{20} \]
\[ d = 2\sqrt{5} \approx 4.47 \text{ units} \]
3. Convert to feet (1 unit = 40 feet):
\[ 4.47 \text{ units} \times 40 \text{ feet/unit} \approx 178.8 \text{ feet} \]
**Distance from Adventureland to Frontierland:**
- Approximately **4.47 units** or **178.8 feet**.
**Step 2: Calculate the distance from Adventureland (Point A) to Tomorrowland (Point T).**
1. Coordinates:
\[ A = (-2, 0), \quad T = (3, 3) \]
2. Use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
\[ d = \sqrt{(3 - (-2))^2 + (3 - 0)^2} \]
\[ d = \sqrt{(5)^2 + (3)^2} \]
\[ d = \sqrt{25 + 9} \]
\[ d = \sqrt{34} \approx 5.83 \text{ units} \]
3. Convert to feet (1 unit = 40 feet):
\[ 5.83 \text{ units} \times 40 \text{ feet/unit} \approx 233.2 \text{ feet} \]
**Distance from Adventureland to Tomorrowland:**
- Approximately **5.83 units** or **233.2 feet**.
**Summary of Distances:**
- **Adventureland to Frontierland:**
- Approximately **4.47 units** or **178.8 feet**.
- **Adventureland to Tomorrowland:**
- Approximately **5.83 units** or **233.2 feet**.