Application of calculus (1 Viewer)

[Saturn WY15]

Q24 ) Circular cylinder of height 6cm and base radius of 4cm sits on a table with its axis vertical. A point source of light moves vertically upward at a speed of 3cm/s above the central axis of the cylinder, thus creating a circular shadow on the table. Find the rate at which the radius of the shadow is decreasing when light is at a distance 4cm above the top of cylinder .

 

[bleakarcher]

one second mate, working on it for ya. why are the forums so inactive here lately?

 

[bleakarcher]

alright here is a picture I drew of the situation

 

[Drongoski]

one second mate, working on it for ya. why are the forums so inactive here lately?

Too hard - can't do it.

 

[bleakarcher]

Let the height of the light source above the top of the cylinder after some time t from when the light source began moving be h. Let the radius of the shadow after this time t be r. By similar triangles it follows that:
=> r/4=(h+6)/h
i.e. r=4+(24/h)
Differentiating implicitly with respect to time t:
=> dr/dt=(-24/h^2)*(dh/dt)
Since dh/dt=3,
=> dr/dt=-72/h^2
When h=4,
=> dr/dt=-72/4^2=-4.5 cm/s
Hence, the radius of the shadow is decreasing at a rate of 4.5 cm/s when h=4.

Hope this helped

 

[bleakarcher]

Too hard - can't do it.

you're an expert lol, one problem with this question that may benefit the OP as well.

When the line hits the cylinder how can you assume that it doesn't bend?

 

[Drongoski]

=> dr/dt=-72/4^2=-4.5 cm/s
Hence, the radius of the shadow is decreasing at a rate of 4.5 cm/s when h=4.

Answer correct. Another great effort from bleak.

 

[Saturn WY15]

Let the height of the light source above the top of the cylinder after some time t from when the light source began moving be h. Let the radius of the shadow after this time t be r. By similar triangles it follows that:
=> r/4=(h+6)/h
i.e. r=4+(24/h)
Differentiating implicitly with respect to time t:
=> dr/dt=(-24/h^2)*(dh/dt)
Since dh/dt=3,
=> dr/dt=-72/h^2
When h=4,
=> dr/dt=-72/4^2=-4.5 cm/s
Hence, the radius of the shadow is decreasing at a rate of 4.5 cm/s when h=4.

Hope this helped

Thanks man you are absolute life-saver !!! Keep up the good work