How Far Can You See? Because of the curvature of the earth,...

How Far Can You See? Because of the curvature of the earth, the maximum distance $D$ that you can see from the top of a tall building or from an airplane at height $h$ is given by the function $$ D(h)=\sqrt{2 r h+h^{2}} $$ where $r=3960 \mathrm{mi}$ is the radius of the earth and $D$ and $h$ are measured in miles. a. Find $D(0.1)$ and $D(0.2)$ b. How far can you see from the observation deck of Toronto's CN Tower, 1135 ft above the ground? c. Commercial aircraft fly at an altitude of about 7 mi. How far can the pilot see? d. Find the net change in the value of distance $D$ as $h$ changes from $1135 \mathrm{ft}$ to $7 \mathrm{mi}$.

How Far Can You See? Because of the curvature of the earth, the maximum distance $D$ that you can see from the top of a tall building or from an airplane at height $h$ is given by the function
$$
D(h)=\sqrt{2 r h+h^{2}}
$$
where $r=3960 \mathrm{mi}$ is the radius of the earth and $D$ and $h$ are measured in miles.
a. Find $D(0.1)$ and $D(0.2)$
b. How far can you see from the observation deck of Toronto's CN Tower, 1135 ft above the ground?
c. Commercial aircraft fly at an altitude of about 7 mi. How far can the pilot see?
d. Find the net change in the value of distance $D$ as $h$ changes from $1135 \mathrm{ft}$ to $7 \mathrm{mi}$.

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Key Concepts

Function Evaluation

Evaluating a function involves substituting specific values into the function to determine its corresponding outputs. In mathematical analysis, this process allows one to understand the behavior of a function at various points, such as calculating the distance one can see for given values of height.

Geometry of a Sphere and Earth's Curvature

The concept explores how the shape of the Earth, approximated as a sphere, influences the line of sight from an elevated position. The formula D(h) = ?(2rh + h²) derives from the geometric relationship between the radius of a circle and a tangent drawn from a point outside the circle, representing the distance to the horizon.

Unit Conversion

Ensuring consistency in measurement units is crucial when applying mathematical formulas. This involves converting quantities, such as feet to miles, so that the formula can be accurately applied without discrepancies arising from mixed measurement units.

Net Change Calculation

Net change refers to the overall difference in the output of a function when the input variable changes from one value to another. This concept is used to compare how the function's result evolves over an interval, providing insights into the cumulative effect of changing the input parameter.

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