Posted by Rachel | Posted in Coffee and Accessories | Posted on 06-10-2009
Tags: coffe, coffee, coffee drip cone, coffee drip cone filter, coffee drip cone holder, coffee drip cone large, coffee drip cone maker, cone, drip, filter
Related Rates Questions! 10 POINTS!?
1. A conical paper cup (vertex down) is being filled with water at the rate of 3 cm^3/sec. If the depth of the water is always twice the radius at the surface, find the following:
a) How fast is the radius increasing when the water is 2 cm deep?
[I got 3 / (2pi) cm/sec]
b) How fast is the area of the surface of the water increasing when the water is 2 cm deep?
[I got -44.25 cm²/sec, which cannot be right]
2. A coffee maker has a filter holder and filter in the shape of a cone with radius 5 cm. 500 cm^3 of water are poured into the filter holder. Brewed coffee drips out of the cone at the rate of 20 cm^3/min into a cylindrical coffee pot that has the same radius as the filter holder.
a) Find a formula for the rate of change of the depth of the coffee in the coffee pot.
b) What is the final depth of the coffee in the coffee pot?
Please show work, whoever helps the most gets 10 points!
1.
Given:
dV/dt = 3 cm^3/s
h(t) = 2r(t)
a.) h = 2cm, r = 1cm
V(t)= 1/3(pi)*h(t)*r(t)^2
= 2pi*r^3
dV/dt = 6pi*r^2(dr/dt) = 3
= 6pi(dr/dt) = 3
dr/dt = 1/(2pi) cm/s
b.)
A = pi*r^2
dA/dt = 2pi*r*(dr/dt)
dr/dt = 1/(2pi)
dA/dt = r = 1 cm^2 / sec
2.
Given:
dVcone/dt = -20 cm^3/min
dVcyl/dt = 20 cm^3/min
R = 5 cm
Vcone = 500 cm^3
Vcyl = pi*r^2h = 25pih
a.) dVcyl/dt = 25pi* (dh/dt) = 20
dh/dt = 4/(5pi) cm/min
b.) Vcyl = 25pi*H = 500 cm^3
H = 20/pi cm
Coffee Obsession : Drip Cone
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