An object detected or radar is 5 miles to the east, 4 miles to the north, and 1 mile above the tracking station. Among the following, which is the closet approximation to the distance, in miles, that the object is from the tracking station?

Options

A. `6.5`
B. `7.2`
C. `8.3`
D. `9.0`

Answer

To find the distance from the tracking station to the object detected by radar, we can use the Pythagorean theorem in a three-dimensional context. Here’s the step-by-step solution:

### Step 1: Identify the coordinates
The tracking station is at the origin point (0, 0, 0). The object is located:
- 5 miles to the east (x-axis)
- 4 miles to the north (y-axis)
- 1 mile above (z-axis)

Thus, the coordinates of the object are (5, 4, 1).

### Step 2: Calculate the distance in the horizontal plane
First, we need to find the distance from the tracking station to the projection of the object in the horizontal plane (ignoring the height). This forms a right triangle where:
- One leg (east) = 5 miles
- Another leg (north) = 4 miles

Using the Pythagorean theorem:
\[ d_{horizontal} = \sqrt{(5^2) + (4^2)} = \sqrt{25 + 16} = \sqrt{41} \]

### Step 3: Calculate the total distance
Now we need to find the distance from the tracking station to the object in three dimensions. We treat the distance we just calculated as one leg of a new right triangle where:
- One leg = \(d_{horizontal} = \sqrt{41}\)
- The other leg (height) = 1 mile

Using the Pythagorean theorem again:
\[ d_{total} = \sqrt{(\sqrt{41})^2 + (1^2)} = \sqrt{41 + 1} = \sqrt{42} \]

### Step 4: Approximate the distance
Now we need to approximate \(\sqrt{42}\):
\[ \sqrt{42} \approx 6.48 \]

Thus, the distance from the tracking station to the object is approximately 6.48 miles.

### Step 5: Final answer
The closest approximation to the distance in miles that the object is from the tracking station is **6.5 miles**.

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