You and your friends are designing and constructing a cone-shaped funnel. Whatever you use this funnel for is up to you. You decide that the height of the funnel will be 6 inches.(a) What does the radius of the base of the funnel need to be if you want the volume of it to be 69 inches cubed? (V = 1/3 πr 2h, where r is the radius of the circular base, and h is the height.)(b) You finally construct it, with the correct height and radius, and you pour the liquid into it to test it out. Again, the choice of liquid and the reason you are doing this is up to you. In measuring the effectiveness of the funnel, you determine that as the liquid flows out of the bottom, the radius of the liquid as it’s in the funnel is decreasing at a rate of 1 inch per second, and the height is decreasing at a rate of 1.75 inches per second. Find the rate at which the volume of the liquid (in inches cubed per second) is decreasing.(c) How many seconds will it take for the funnel to be empty at this rate?

Question
Asked Dec 3, 2019
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You and your friends are designing and constructing a cone-shaped funnel. Whatever you use this funnel for is up to you. You decide that the height of the funnel will be 6 inches.

(a) What does the radius of the base of the funnel need to be if you want the volume of it to be 69 inches cubed? (V = 1/3 πr 2h, where r is the radius of the circular base, and h is the height.)

(b) You finally construct it, with the correct height and radius, and you pour the liquid into it to test it out. Again, the choice of liquid and the reason you are doing this is up to you. In measuring the effectiveness of the funnel, you determine that as the liquid flows out of the bottom, the radius of the liquid as it’s in the funnel is decreasing at a rate of 1 inch per second, and the height is decreasing at a rate of 1.75 inches per second. Find the rate at which the volume of the liquid (in inches cubed per second) is decreasing.

(c) How many seconds will it take for the funnel to be empty at this rate?

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Expert Answer

Step 1

Part (a):

 

Let r is the radius and h is height of the funnel and it is given that height of the funnel is 6 inches:

 

Volume of the cone shaped funnel:

Volume = rr’h
3
69 = ar" (6)
ar` (6)
3
69
= r?
r' = 10.98
r =V10.98
r z 3.313 inches
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Volume = rr’h 3 69 = ar" (6) ar` (6) 3 69 = r? r' = 10.98 r =V10.98 r z 3.313 inches

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Step 2

Part (b):

 

It is given that the height is decreasing at a rate of 1.75 inches per second. The radius of the liquid is decreasing at a ...

Differentiate with resect to t:
dr
+ 2r
dt
dV
dh
1
T| r? .
..h
dt
dt
dr
dh
:-1.75 and
dt
Given that
dt
dV
T7 (3.3) ·(-1.75)+2(3.3)·(-1)·(6)
dt
3
=
7((-19.05)+(-62.75))
3
dV
z -62.75 (inches)' per second
dt
help_outline

Image Transcriptionclose

Differentiate with resect to t: dr + 2r dt dV dh 1 T| r? . ..h dt dt dr dh :-1.75 and dt Given that dt dV T7 (3.3) ·(-1.75)+2(3.3)·(-1)·(6) dt 3 = 7((-19.05)+(-62.75)) 3 dV z -62.75 (inches)' per second dt

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Math

Calculus