What Are the Related Rates Problems About?

[Minusu]

I've tried these problems but I just can't wrap my head around them. Thanks to anyone that can help me out here. I have a couple more problems similar to these to solve, but if I can get to understand these then I'll have no problem with the others.
Here are some diagrams I drew: http://gyazo.com/bf78da97179aa8d62d5e44b484ffcdc3 and http://gyazo.com/65689d4a0de1f23ceeae3c236c1c891e
1. Homework Statement :
A conical paper cup is 30cm tall with a radius of 10cm. The cup is being filled with water a rate of 16*pi/3 cm^3/sec. How fast is the water level rising when the water level is 6cm?
dV/dt = 16*pi/3
r=10
h=30
hw(height of water)=6
dh/dt=?
2. Homework Equations :
V=1/3*pi*r^2*h


3. The Attempt at a Solution :
V=1/3*pi*r^2h
dV/dt=1/3*pi*r^2*dh/dt
16*pi/3=1/3*pi*10^2*dh/dt
3/100*pi *16*pi/3=100*pi/3dh/dt *3/100*pi
16*pi/100*pi=dh/dt
16/100=dh/dt



1. Homework Statement :
A 7 ft tall person is walking away from a 20 ft tall lamppost at a rate of 6/x ft/sec, where x is the distance from the person to the lamppost. Assume the scenario can be modeled with right triangles. At what rate is he length of the person's shadow changing when the person is 15 ft from the lamppost?
y= height of lamppost, 20ft
x= distance from lamppost to person, 15 ft
dx/dt= 6/15
2. Homework Equations :
a^2 + b^2 = c^2


3. The Attempt at a Solution :
x^2 + y^2 = c^2
2x*dx/dt + 2y*dy/dt = 0
2*15*(6/15) + 2*20*dy/dt = 0
12+40dy/dt=0
dy/dt = -12/40

 

[Saitama]

I've tried these problems but I just can't wrap my head around them. Thanks to anyone that can help me out here. I have a couple more problems similar to these to solve, but if I can get to understand these then I'll have no problem with the others.
Here are some diagrams I drew: http://gyazo.com/bf78da97179aa8d62d5e44b484ffcdc3 and http://gyazo.com/65689d4a0de1f23ceeae3c236c1c891e
1. Homework Statement :
A conical paper cup is 30cm tall with a radius of 10cm. The cup is being filled with water a rate of 16*pi/3 cm^3/sec. How fast is the water level rising when the water level is 6cm?
dV/dt = 16*pi/3
r=10
h=30
hw(height of water)=6
dh/dt=?
2. Homework Equations :
V=1/3*pi*r^2*h


3. The Attempt at a Solution :
V=1/3*pi*r^2h
dV/dt=1/3*pi*r^2*dh/dt
16*pi/3=1/3*pi*10^2*dh/dt
3/100*pi *16*pi/3=100*pi/3dh/dt *3/100*pi
16*pi/100*pi=dh/dt
16/100=dh/dt

Radius of the cone filled with water varies as the water is poured in. At any instant, if the height of the water filled is h, the radius of the filled portion isn't r. You will have to relate r and h somehow. (Hint: Using similar triangles. )

 

[Minusu]

http://gyazo.com/62a49a2ec489bb6662afb35a72f981b9 . So how about r^2+h^2 = c^2? But if that is the case, I'm not sure what I would do with c^2. Or should I rearrange the equation in order to combine r^2+h^2=c^2 and V=1/3*pi*r^2*h?

 

[Saitama]

http://gyazo.com/62a49a2ec489bb6662afb35a72f981b9 . So how about r^2+h^2 = c^2? But if that is the case, I'm not sure what I would do with c^2. Or should I rearrange the equation in order to combine r^2+h^2=c^2 and V=1/3*pi*r^2*h?

(see attachment)
The angle θ always remains constant. You can find two values of tan θ from here and compare them to find a relation between r and h.

 

[Dick]

http://gyazo.com/62a49a2ec489bb6662afb35a72f981b9 . So how about r^2+h^2 = c^2? But if that is the case, I'm not sure what I would do with c^2. Or should I rearrange the equation in order to combine r^2+h^2=c^2 and V=1/3*pi*r^2*h?

You are confusing constants with variables. You can't say r=10 and then let r also be variable. Relabel some of them. Suppose R=10 is the radius of the cone and the height of the cone is H=30. Now suppose the height of the water is h. What is r, the radius of the surface at height h? As Pranav-Arora said use similar triangles.