A lighthouse is located on a small island 3 km 3 km away from the nearest point P P on a straight shoreline and its light makes 4 4 revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km 1 km from P P? (Round your answer to one decimal place.)

The light house is moving at the speed of 83.8Km/minute.

Step-by-step explanation:

The question forms a right angle triangle where point O to P is 3km.and P to Q is 1km . The angle between point O and Q which is the light house is thetha.

The lighthousevmakes 4rev/min

dtheta/dt = 4× 2pi/1min.= 8piRad/min

From the right angle ,tan theta = x/3

Differenting both sides with respect to t

d(tantheta)/dtheta = 1/3×dx/dt

d(tantheta)/dtheta × dtheta/ dt = 1/3×dx/dt

Using chain rule

Sec^2theta× dtheta/dt = 1/3×dx/dt

(1 + tan^2 theta)×dtheta/dt = 1/3×dx/dt

When x = 1, we have tan theta= 1/3

(1 + 1/9)×dtheta/dt = 1/3×dx/dt

Substituting dthata/dt = 8 pi

10/9 × 8pi = 1/3×dx/dt

10/3 × 8pi = dx/dt

80pi/3 = dx/dt

dx/dt = (80 × 3.142)/3

dx/dt = 251.36/ 3 = 83.79km/hr

Therefore the lighthouse is moving at a speed of 83.8km/minute

answer: 1) twenty-seven times y

              2) twenty-seven multiplied by y

2,160 π

step-by-step explanation: