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You are flying in a hot air balloon about 1.2 miles above the ground. Find the measure of the arc that represents the part of Earth you can see. Round your answer to the nearest tenth.

(The radius of the Earth is about 4000 miles.)

Answer :

Final Answer:

The measure of the arc that represents the part of Earth you can see from a hot air balloon 1.2 miles above the ground is approximately 18.1 miles when rounded to the nearest tenth.

Explanation:

To find the measure of the arc, you can use the formula for the circumference of a circle, [tex]\(C = 2\pi r\)[/tex], where [tex]\(r\)[/tex] is the radius of the Earth. In this case, the radius of the Earth is given as 4000 miles. However, you are flying in a hot air balloon 1.2 miles above the ground, so the effective radius to consider is [tex]\(4000 + 1.2 = 4001.2\)[/tex] miles.

Substitute this radius into the formula:

[tex]\[C = 2\pi \times 4001.2 \approx 25119.04 \text{ miles}.\][/tex]

This is the circumference of the circle representing the Earth. To find the length of the arc corresponding to your line of sight, you can use the formula for the length of an arc, which is a fraction of the circumference based on the central angle formed by the arc.

Since you are viewing the Earth from above, the central angle is 360 degrees [tex](or \(2\pi\) radians)[/tex]. The fraction of the circumference corresponding to your line of sight is [tex]\(\frac{1.2}{4001.2}\)[/tex] because your height is 1.2 miles above the Earth's surface.

[tex]\[ \text{Arc length} = \frac{1.2}{4001.2} \times 25119.04 \approx 18.1 \text{ miles (rounded to the nearest tenth).}\][/tex]

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