When a tsunami forms over deep ocean, the wavefront will form a region that is i

Question

When a tsunami forms over deep ocean, the wavefront will form a region that is initially circular. As the tsunami moves at near supersonic speeds, Earth's curvature comes into play and the region through which the wavefront has traveled forms a spherical cap, the area of which is given by

A=2R^2(1cos(r/R))

Where R is the radius of the Earth (about 3960 miles) and r is the distance that the tsunami has traveled. Find the rate at which the area of the impacted region is growing if the tsunami has traveled 100 miles from its point of origin and is currently traveling at 500 mph.

Rate of growth: square miles per hour

Explanation / Answer

given area

A=2R2(1cos(r/R))

diferentiate with respect to time t

dA/dt = 2R2(0(-sin(r/R))(1/R)(dr/dt))

dA/dt = 2R(sin(r/R))(dr/dt)

R=3960,r=100,dr/dt =500

dA/dt = 2*3960(sin(100/3960))(500)

dA/dt = 2*3960(sin(100/3960))(500)

dA/dt =314126square miles per hour

area increasing at 314126square miles per hour

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