Please Show all work and give a detailed explanation Ill rate Sometimes the rela

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Please Show all work and give a detailed explanation Ill rate

Sometimes the relation equation is based upon a given or "known" formula. Slushy is flowing out of the bottom of a cup at the constant rate of 0.1 cm3/s. The cup is the shape of a right circular cone. The height of the cup is 12 cm and the cup (when full) holds a total of 36pi cm3 of slushy. Determine the rate at which the height of the remaining slushy changes at the instant there are exactly 8pi cm3 of slushy remaining in the cup. The volume formula for a right circular cylinder is V = pi/3r2 h where V is the volume of the cone, h is the height of the cone, and r is the radius at the top of the cone. We ultimately are going to define two of these variables in terms of the amount of slushy remaining in the cup. Which are the two variables relevant to the question at hand? That is, which quantity's rate of change are you given and which quantity's rate of change are you trying to determine? Hopefully you determined that V and h are the relevant variables. This means that r needs to be eliminated from the volume formula. A cross section of the cup and slushy is shown in Figure 46.2. Explicitly define h and V (including units) in terms of the amount of slushy remaining in the cup. Make sure that you communicate that each variable is dependent upon time and that you explicitly define your time variable. The rate variables will emerge when you differentiate the height and volume variables with respect to time. Go ahead and complete the problem in a manner consistent with that implied in problems 45.1 and 45.2.

Explanation / Answer

h =12 V = 36 pi

V = pi/3 r^2 h

36 *pi = pi/3 * r^2 *h = pi/3*r^2*12

36*pi/pi = 4r^2

4r^2 = 36

r^2 = 9

r = 3

r/h = 3/12 = 1/4

r = h/4

now   V = pi/3 *h^2/16*h   = pi/48*h^3

when V = 8picm^3

8*pi = pi/48 * h^3

h^3 = 8*48 = 384

h = 7.268482371328558

now

dV/dt = pi/48*3h^2*dh/dt

0.1 = pi/16*3*7.26^2 * dh/dt

dh/dt = 0.0032208879294704

height changing at rate of 0.00322 cm/sec

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