Microsoft Word - MTH 251_lab_manual_cover.doc 75 178 Problem 46.1 Slushy is flow
ID: 3210071 • Letter: M
Question
Microsoft Word - MTH 251_lab_manual_cover.doc 75 178 Problem 46.1 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 36 cm3 of slushy. Determine the rate at which the height of the remaining slushy changes at the instant there are exactly 8 cm3 of slushy remaining in the cup 46.1.1 The volume formula for a right circular cylinder is V--r2 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 Vand 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 46.1.2 What is the radius at the top of the cup? Use the concept of similar triangle to express r in terms of h. Substitute this expression into the volume formula and simplify. The resultant equation is the relation equation for the problem 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 46.1.3Explanation / Answer
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