You fly a hot air balloon 1.5 miles above the ground. What is the measure of BD⌢, the portion of Earth that you can see? Round your answer to the nearest tenth. (Earth's radius is approximately 4000 miles.)

The horizon is the apparent boundary where the Earth's surface seems to meet the sky. It's the line that separates the visible portion of the Earth's surface from the portion that is not visible due to the curvature of the Earth. The horizon appears to be a straight line, but it is actually a curved line due to the spherical shape of the Earth. The distance to the horizon depends on the observer's height above the ground, as well as the radius of the Earth. The higher the observer, the farther away the horizon appears.

The distance from the balloon to the horizon is the same as the radius of the Earth plus the height of the balloon. So in this case, it's 1.5 + 4000 = 4001.5 miles.

Now we need to find the distance AC, which is the distance from the center of the Earth to point C. To do this, we can use the Pythagorean theorem:

AC² + BC² = (radius of Earth)²

AC² + (4001.5)² = (4000)²

AC² = (4000)² - (4001.5)²

AC ≈ 68.2 miles

Now we can find the distance BD using similar triangles. In the triangle ABD, we know that AB is the radius of the Earth, which is 4000 miles. We also know that AC is 68.2 miles, and we want to find BD. So we can set up the following proportion:

BD / AB = AC / AD

Solving for BD, we get:

BD = AB * (AC / AD)

To find AD, we can use the Pythagorean theorem again:

AD² = AB²+ BD²

AD² = (4000)²+ (BD)²

AD ≈ 4000.3 miles

Now we can plug in the values we have:

BD = AB * (AC / AD)

BD = 4000 * (68.2 / 4000.3)

BD ≈ 68.1 miles

So the portion of the Earth that you can see from the hot air balloon is approximately 68.1 miles. Rounded to the nearest tenth, the answer is 68.1 miles.

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BenicioBrody BenicioBrody · Answer 2 · 2025-02-05T05:02:43-05:00

To find the measure of BD⌢, the portion of Earth that you can see from a hot air balloon 1.5 miles above the ground, calculate the distance to the horizon using a specific formula based on the Earth's radius and the observer's height, resulting in approximately 109.5 miles as the answer.

To find the measure of BD⌢, the portion of Earth that you can see from the hot air balloon 1.5 miles above the ground:

Calculate the distance to the horizon using the formula D = √(2 imes r imes h), where r is the radius of the Earth (approximately 4000 miles) and h is the observer's height (1.5 miles).

Substitute the values into the formula: D = √(2 imes 4000 imes 1.5) = √12000 = 109.5 miles.

Therefore, the measure of BD⌢, the portion of Earth that you can see, is approximately 109.5 miles when rounded to the nearest tenth.