c) Is the amount of AP Caleulus Integration App AP Type (Chapter 38) Name A scie

Question

c) Is the amount of AP Caleulus Integration App AP Type (Chapter 38) Name A scientific study using ice is performed in a special laboratory. Ice chips are dropped from a chute onto platform made up of a strong screen. The ice chips take the form of a right sircular cone with both its and height changing with time. The volume of the cone increases at the radius rate of 600 cubic inches per minute. (Note: the volume V of a right cincular cone with radius r and height his given by V-h). a) At the instant when the radius of the cone is 10 inches and its height is 2 inches, the radius is increasing at the rate of 4 inches per minute. Using correct units, what is the rate of change of the beight of the cone with respect to time? r 10 h 2 r2 dh olt 4 32. 4423 04.7 d b) At a certain point in time, a heater under the screened platform turns on. It melts the ice at the rate of M()-100i cubie inches per minute, where t is the time in minutes since the heater is turned on. Ice still continues to pour down onto the platform at the rate of 600 cubic inches per minute Find t time t when the volume of the cone reaches its maximum. Justify your answer avv c) By the time the melting process be eters of ice had already formed on the platform. Write and evaluate an expression involving an integral that gives the maximum amount of ice on the platform. 303. 9be

Explanation / Answer

a)
Volume incre at rate 600 cubic inches per min

So, dV/dt = +600 in^3/min

dr/dt = 4 in/min when r = 10 and h = 2

V = 1/3 * pir^2 * h

Now, product rule is applied....

dV/dt = pi/3 * d/dt(r^2*h)

dV/dt = pi/3 * [r^2*dh/dt + h*d/dt(r^2)]

= pi/3 * [r^2*dh/dt + 2rh*dr/dt] ----> this is how we get this

Now, plug in values is all, yo!

600 = pi/3 * [10^2*dh/dt + 2*10*2*4]

Solve for dh/dt now

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b)
M = 100sqrt(t)

Rate of ice pouring in : 600 in^3/min
Rate of melting : 100sqrt(t) in^3/min

So, rate of change of ice is :

dV/dt = 600 - 100sqrt(t)

Now, for max volume,
dV/dt = 0

600- 100sqrt(t) = 0

sqrt(t) = 6

t = 6^2

t = 36

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