The Empire State Building in New York City is 1250 ft high. The observation deck, located on the

The Empire State Building in New York City is 1250 ft high....

The Empire State Building in New York City is $1250 \mathrm{ft}$ high. The observation deck, located on the 102 nd floor, is at a height of $1050 \mathrm{ft.}$ How far could we see on a clear day from the observation deck?

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Key Concepts

Earth's Curvature

This concept refers to the fact that the Earth is roughly spherical in shape, which means that the surface curves away from an observer. As a result, there is a limit to how far one can see along the surface because the Earth’s surface eventually falls out of the observer’s line of sight. This curvature fundamentally determines the distance to the horizon for any given observation height.

Distance to the Horizon

This key concept involves calculating how far an observer can see before the curvature of the Earth blocks their line of sight. The distance is typically determined using the formula d = ?(2Rh), where R is the radius of the Earth and h is the observer's height above ground level. This formula stems from the geometry of a circle (or sphere) and provides an approximation under the assumption of a clear atmosphere and negligible atmospheric refraction.

Observer Height

The observer's height is a critical factor in limiting the distance to the horizon. As the height increases, the distance to the horizon also increases because the observer's line of sight extends further before intersecting with the curvature of the Earth. This concept is essential to understanding why elevated observation points, like towers or buildings, allow for a broader view of the surrounding area.

Transcript

00:01 Okay, so to find how far a person could see on a clear day from the observation deck of the empire state building, we're going to find the maximum distance.

00:12 We're going to call this distance d, and then to find the maximum site distance d, we use the formula that d is equal to 111 .7 times the square root of h.

00:26 Now, to use this formula, all distances must be in kilometers.

00:31 So we're given the height of the observation deck as 1 ,050 feet.

00:38 So therefore, we have that h is equal to 1 ,050 feet.

00:47 Okay.

00:48 So to convert h to kilometers, we first convert h to meters using the conversion rate, that one foot is equal to 0 .3048 meters.

00:59 So we have that h is equal to 1 ,050 feet.

01:09 And then we multiply that by our conversion of 0 .3048 meters over one foot.

01:20 Okay, so our units of feet cancel out.

01:24 And we get that h is equal to 320.

01:31 0 .04 meters.

01:34 Okay.

01:36 So this height h must now be converted to kilometers.