Applications of Calculus to Physical World: Related Rates Questions (1 Viewer)

[Porcia]

Could someone please help me answer the following questions. Thanks!

Fitzpatrick 25 (a)

15. A boat is being pulled into a wharf by a rope at speed of 20 metres/min. If the rope is attached to a point on the boat, 7m vertically below the wharf, at what rate is the rope being drawn in, when the boat is 24m from wharf.

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

The answers should be:

15. 19.2 metres/min.

24. 4.5 cm/s.

 

[SoulSearcher]

It's always useful to draw a diagram in these types of questions.
15. We have to find dk/dt, where k is the length of the rope between the end of the boat closest to the wharf and the person pulling the rope at the top of the wharf.
Let x be the distance between the boat and the bottom of the wharf.
So dx/dt = 20 metres/min
Now we have to find the length of the rope in terms of x.
Using pythagoras, k² = x² + 7², and therefore k = rt(x² + 7²)
therefore dk/dx = x/rt(x² + 49]
therefore dl/dt = dk/dx * dx/dt
= x/rt(x² + 49) * 20, letting x = 24
= 24/rt(576 + 49) * 20
= 24/25 * 20
= 19.2 metres/min

 

[Porcia]

thank you, i can now see how to do it and appreciate it. any takers for the other one?

 

[word.]

same concept just draw the diagram and it should become clear what to do