A lighthouse, L, is on an island 8 mi away from the closest point, P, on the beach. If the lighthouse light rotates clockwise at a constant rate of 19 revolutions/min, how fast (in mi/min) does the beam of light move across the beach 4 mi away from the closest point on the beach? =___________ (mi/min ### Understanding Distance and Geometry#### Description of the Diagram:The diagram presents a visual representation of a right-angled triangle formed by two locations: point **L** on an island and point **P** on the mainland.#### Key Elements:1. **Triangle Components**: - The hypotenuse of the triangle extends from point **P** on the mainland directly to point **L** on the island, and it measures 8 miles. - One leg of the right triangle runs horizontally from point **P** for 4 miles. - The other leg extends vertically from the end of the 4-mile leg to point **L** on the island, representing the direct distance across the water.2. **Points & Distance**: - **Point L** represents the location of interest on the island. - **Point P** represents the starting point on the mainland. - The direct distance across the water from the mainland at point **P** to the island at point **L** is 8 miles. - The horizontal distance from point **P** inland is 4 miles.#### Mathematical Example:The diagram is an excellent illustration of the Pythagorean theorem, which is fundamental in geometry for calculating distances and understanding the relationships between the sides of right-angled triangles. According to the Pythagorean theorem:\[ \text{Hypotenuse}^2 = \text{Base}^2 + \text{Perpendicular}^2 \]Using the values provided:- Hypotenuse (\(c\)) = 8 miles- Base (\(b\)) = 4 milesWe can calculate the Perpendicular (\(a\)):\[ 8^2 = 4^2 + a^2 \]\[ 64 = 16 + a^2 \]\[ a^2 = 64 - 16 \]\[ a^2 = 48 \]\[ a = \sqrt{48} \approx 6.93 \text{ miles}\]Thus, the vertical distance across the water is approximately 6.93 miles.

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Answered: A lighthouse, L, is on an island 8 mi…

Holt Mcdougal Larson Pre-algebra: Student Edition 2012

1st Edition

ISBN:9780547587776

Author:HOLT MCDOUGAL

Publisher:HOLT MCDOUGAL

Chapter12: Angle Relationships And Transformations

Section12.5: Reflections And Symmetry

Problem 20E

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A lighthouse, L, is on an island 8 mi away from the closest point, P, on the beach. If the lighthouse light rotates clockwise at a constant rate of 19 revolutions/min, how fast (in mi/min) does the beam of light move across the beach 4 mi away from the closest point on the beach?

=___________ (mi/min