There's A Lighttt....

Calculus Level pending

A light shines from the top of a pole 5 m high. A ball is dropped from a point 4 m away from the pole. Find how fast the shadow of the ball is moving when the ball is 2 m above the ground and moving at a speed of 9 m/sec.

(Bilkent University, 1st Midterm Question)


The answer is -20.

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1 solution

Haries Ajrun
Dec 7, 2014

This is the Image This is the Image

Finding the equation of x and y

5 y = x + 4 x \frac { 5 }{ y } =\frac { x+4 }{ x } \quad thus x = 4 y 5 y \quad x=\frac { 4y }{ 5-y }

Derivating x regarding to time using quotient rule

d x d t = ( 4 5 y + 4 y ( 5 y ) 2 ) d y d t \frac { dx }{ dt } =\left( \frac { 4 }{ 5-y } +\frac { 4y }{ { (5-y) }^{ 2 } } \right) \frac { dy }{ dt }

Thus at the moment when y = 2 y=2 m and d y d t = 9 m / s e c \frac { dy }{ dt } =9m/sec

d x d t = ( 4 5 2 + 4.2 ( 5 ) 2 ) . ( 9 ) = 20 m / s e c \frac { dx }{ dt } =\left( \frac { 4 }{ 5-2 } +\frac { 4.2 }{ { (5-) }^{ 2 } } \right) .(-9)=-20m/sec

The shadow of the ball is moving at a speed of -20 m/sec at this moment

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