Question

Distance to the Horizon The distance that a person can see to the horizon on a clear day from a point above the surface of Earth varies directly as the square root of the height at that point. If a person 144 m above the surface of Earth can see 18 km to the horizon, how far can a person see to the horizon from a point 64 m above the surface?

Accepted Answer

Identify the variables and the direct variation relationship: Let $$d$$ represent the distance to the horizon (in km) and $$h$$ represent the height above the Earth's surface (in meters). The problem states that $$d$$ varies directly as the square root of $$h$$, so we write the equation as $$d = k \sqrt{h}$$, where $$k is $$the constant of proportionality. Use the given information to find the constant $$k$$: Substitute $$d = 18 km $$and $$h = 144 m $$into the equation $$d = k \sqrt{h} to $$get $$18 = k \sqrt{144}. $$Since $$\sqrt{144} = 12$$, this simplifies to $$18 = 12k. $$Solve for $$k$$: Divide both sides of the equation $$18 = 12k by 12 to $$isolate $$k$$, giving $$k = \frac{18}{12}. $$Use the constant $$k to $$find the distance for the new height: Substitute $$k$$ and the new height $$h = 64 m $$into the original equation $$d = k \sqrt{h} to $$get $$d = k \sqrt{64}. $$Since $$\sqrt{64} = 8$$, this becomes $$d = 8k. $$Calculate the distance $$d by $$multiplying $$8 by $$the value of $$k$$ found in step 3. This will give the distance to the horizon from a point 64 m above the surface.

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Key Concepts

Direct Variation

Square Root Function

Solving Proportions