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The deck of a ship is 90 feet above sea level. How far can a person see from the deck? How far can a person see from the bridge, which is 180 feet above sea level?

Note: Use 3960 miles as the radius of the Earth.

The distance a person can see from the deck is approximately _____ miles. (Round to two decimal places as needed.)

Sagot :

GinnyAnsweridn
Certainly! Let's solve this step by step.

### Given Data:
- Radius of the Earth ([tex]\(r\)[/tex]): 3960 miles
- Height of the deck above sea level ([tex]\(h_d\)[/tex]): 90 feet
- Height of the bridge above sea level ([tex]\(h_b\)[/tex]): 180 feet

#### Convert Heights from Feet to Miles:
1 mile = 5280 feet

[tex]\[ h_d = \frac{90 \text{ feet}}{5280 \text{ feet/mile}} \approx 0.017045454545454544 \text{ miles}\][/tex]
[tex]\[ h_b = \frac{180 \text{ feet}}{5280 \text{ feet/mile}} \approx 0.03409090909090909 \text{ miles}\][/tex]

### Distance to Horizon Formula:
The formula to calculate the distance to the horizon ([tex]\(d\)[/tex]) is:

[tex]\[ d = \sqrt{2 \cdot r \cdot h} \][/tex]

where [tex]\(r\)[/tex] is the radius of the Earth, and [tex]\(h\)[/tex] is the height above the sea level.

#### Calculate the Distance from the Deck:
Using the height of the deck ([tex]\(h_d\)[/tex]):

[tex]\[ d_d = \sqrt{2 \cdot 3960 \text{ miles} \cdot 0.017045454545454544 \text{ miles}} \approx 11.62 \text{ miles} \][/tex]

#### Calculate the Distance from the Bridge:
Using the height of the bridge ([tex]\(h_b\)[/tex]):

[tex]\[ d_b = \sqrt{2 \cdot 3960 \text{ miles} \cdot 0.03409090909090909 \text{ miles}} \approx 16.43 \text{ miles} \][/tex]

### Summary:
- From the deck of the ship (90 feet above sea level), a person can see approximately 11.62 miles.
- From the bridge of the ship (180 feet above sea level), a person can see approximately 16.43 miles.

These results are rounded to two decimal places as required.
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