[RESOLVED] Problems with Extending Differentiation

[MathBaby]

1. A hollow cone of base radius a and height 3a is held vertex downwards. The cone is initially empty and liquid is poured into it at a rate of 4π cm³/s. Find the rate at which the depth of the liquid in the vessel is increasing 16 seconds after the pouring commenced. (answer: 0.25 cm/s)

2. A container is in the shape of a cone of semi-vertical angle 30° with its vertex downwards. Liquid flows into the container at the rate of √3π /4 cm³/s. At the instant when the radius of the circular surface of the liquid is 5cm, find the rate of increase in:
a) the radius of the circular surface of the liquid, (answer: 0.01 cm/s)
b) the area of the circular surface of the liquid. (answer: 0.1π cm/s)

3. A funnel has a circular top of diameter 20cm and a height of 30cm. When the depth of liquid in the funnel is 12cm, the liquid is dripping from the funnel at a rate of 0.2 cm³/s. At what rate is the depth of the liquid in the funnel decreasing at this instant? (answer: 0.0040 cm/s)


Sorry to keep posting differentiation questions, but I'm not good at differentiation... For question 2, I know how to do. The main problem for question 2 is the 30° that made me blur... As for question 1 and 3, I really have no idea how to solve it...

[Glaysher]

1. Let V be the volume of water in the cone and h be the height

Then question tells us dV/dt = 4π

For a cone V = (πr^2h)/3

In our case r = h/3

So V = (πh^3)/27

Find dV/dh. dh/dV = 1/(dV/dh)

dh/dt = (dh/dV) x (dV/dt) = 36/h^2

Find dV/dh

At t = 16 V = 16 x 4π

Find h and substitute the value into dh/dt

3. Treat funnel as cone. Use similar definitions to 1

Told when h = 12 dV/dt = -0.2 as volume decreasing

radius = r = h/3 again as radius is 10 when height 30

so V = (πh^3)/27

Find dV/dh

Find dh/dt in a similar way to 1. Will get answer that is negative as height is decreasing and dh/dt will measure amount it is increasing by

2. Hope the attached diagram helps

[MathBaby]

Thanks for question 1 and 3 :bigyello:

As for question 2, do I have to use tan 30° to find the radius or something? I still very blur bout question 2 :sick:

[Glaysher]

tan 30° = r/h

So h = r/tan 30°

Remember tan 30° is an exact trig ratio soyoucan simplify

Work out volume of the cone V in terms of r this time

a) Find dV/dr. Have dV/dt

Use similar method to 1 and 2 to find dr/dt

b) Let area be denoted by A

Find A in terms of r and find dA/dr

dA/dt = dA/dr x dr/dt

[MathBaby]

oo thanks!! I knew it has somethin to do with tan 30°, just cant figure out how :p Should memorize this technique in order to solve this kind of questions next time :bigyello:

[Glaysher]

Again no problem. You often come up with more interesting questions than I usually see.