This section evaluates square root expressions by calculating how far a person can see. Scratch paper and a graphing calculator are needed.

Note:

• Some problems will give all of the “inputs” in order to evaluate the square root expression.
• Some problems will give an “output” and ask for an “input” requiring a solution of a square root equation.  You can solve the equation either algebraically or graphically (or with a table).  Use the easiest method.  

How Far Can You See?

Source: Glittering Sea and Horizon, Alves, Flickr

1. Have you ever stared off at the horizon and wondered how far you are seeing?  Assuming there is nothing obstructing your view (e.g., trees or buildings), the distance (in miles) you can see is dependent on the height of your eyes above the level surface you are standing and can be approximated by the equation:

Where H (in feet) is the height of your eyes above the ground and R is the radius of the Earth (around 3963.192 miles).

Suppose you are standing at ground level and your eyes are 6 feet above the ground. How far can you see?

Solutions are given to you for each problem, but you should give an honest attempt at solving each problem first before looking at the solution. If you get stuck, try using the provided hints.

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Hint #1

Substitute the values given in problem into the formula and write the equation you need to solve.

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Hint #2

Use your calculator to evaluate the expression on the right side of the equation.

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Answer

A person whose eyes are 6 feet above level ground can see approximately 3 miles.

Sources: To The Sky, Suwaif, Flickr; Shanghai World Financial Center, Bert van Dijk, Flickr

2. The Burj Khalifa (shown on the left), a skyscraper in Dubai, is the tallest man-made structure. It has the world's second highest public observation deck at 446.5 m (1,465 ft).

The highest observation deck in the world is on the Shanghai World Financial Center (shown on the right) in China. Its observation deck is 474 m (1555 ft). How much further can you see from the Shanghai WFC than you can from the Burj Khalifa? (Assume your eyes are 6 feet above the observation deck.)

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Hint #1

Determine how far you can see from the observation deck of the Burj Khalifa observation deck. Don't forget to add 6 feet to the height of the observation deck to account for the eye height.

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Hint #2

Determine how far you can see from the observation deck of the Shanghai World Financial Center observation deck. Don't forget to add 6 feet to the height of the observation deck to account for the eye height.

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Answer

A person with eyes six feet above ground level can see approximately 1.4 miles further from the observation deck of the Shanghai World Financial Center than from the observation deck of the Burj Khalifa.

3. How high above the ground is a viewer’s eye if they can see 100 miles?

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Hint #1

Substitute the values given in problem into the formula and write the equation you need to solve.

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Hint #2

Enter the left side of the equation in Y1 and the right side in Y2 of your graphing calculator.

Set an appropriate window:

Graph the two equations and find their intersection.

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Answer

For a person to see 100 miles, their eye level would have to be approximately 6660 feet high.

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Solving the Problem Algebraically

Step 1: Substitute the values given in the problem into the formula and write the equation you need to solve.

Step 2: Simplify under the radical.

Step 3: Multiply both sides of the equation by 5280 to isolate the radical.

Step 4: Square both sides of the equation to eliminate the radical.

Step 5: Rewrite the equation moving all of the terms to one side in order to set the equation equal to 0.

Step 6: Use the quadratic formula to solve the equation. You might want to store each coefficient as constants in your calculator and use those constants in your quadratic formula. BE CAREFUL TO USE THE CORRECT PARENTHESES!

0 = H² + 41,851,307.52H - 2.78784x 10¹¹

(The negative solution is extraneous.)

For a person to see 100 miles, the eye level would have to be approximately 6660 feet high.