To determine the measure of ΔBD⌢, the portion of the Earth visible from a hot air balloon 0.5 miles above the ground, use the tangent of an angle formula. The distance from the balloon to the horizon is the hypotenuse of a right triangle, and by rearranging the formula and substituting values, the measure of ΔBD⌢ can be found. Finally, round the answer to the nearest tenth.
To find the measure of ΔBD⌂, the portion of the Earth that you can see from a hot air balloon 0.5 miles above the ground, you can use the formula for the tangent of an angle. Since the radius of the Earth is approximately 4000 miles, you can consider the distance from the balloon to the horizon as the hypotenuse of a right triangle. The tangent of the angle can be calculated using the formula tan(θ) = opposite/adjacent, where the opposite side is the distance from the balloon to the horizon and the adjacent side is the radius of the Earth. Rearrange the formula to solve for the distance from the balloon to the horizon: distance = tan(θ) * radius of Earth. Substituting the values, we have distance = tan(θ) * 4000 miles.
Based on the question, the distance from the balloon to the horizon is 0.5 miles. Plugging this value into the formula, we have 0.5 miles = tan(θ) * 4000 miles. To solve for θ, divide both sides by 4000 miles: tan(θ) = 0.5/4000. Finally, take the inverse tangent (arctan) of both sides to find θ. Using a scientific calculator, compute arctan(0.5/4000) and round the answer to the nearest tenth to find the measure of ΔBD⌂.
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