1. suppose you are drinking root beer from a conical paper cup. The cup has a diameter of 8 cm and a depth of 10 cm. As you suck on the straw, root beer leaves the cup at the rate of 7cm^3/sec. At what rate is the height of the liquid in the the cup changing
(a) when the liquid is 6 cm deep?
(B) at the instant when the last drop leaves the cup?
a math question
2007-10-18 3:12 pm
回答 (2)
2007-10-18 7:05 pm
✔ 最佳答案
Volume, V, of a cone of height h, radius rV=pi r^2 h /3
denote the radius/height ratio by
k=r/h
then
V=pi k^2 h^3 / 3
differentiate with respect to h,
dV/dh = pi k^2 3h^2/ 3
=pi k^2 h^2
or
dh=dV/( pi k^2 h^2)
For the given data,
r=8/2=4 (full cup)
h=10 (full cup)
r/h=4/10=0.4
dV/dt=7 cm^3
a.
h=6 cm
dh/dt=dV/dt / (pi k^2 h^2)
=7 / (pi 0.4^2 6^2)
= 0.3868 cm / sec.
b. Assume last drop means an infinitely small drop, and not 1/30 ml.
h=0
dh/dt=dV/dt / (pi k^2 h^2)
= 7 / (pi 0.4^2 0^2)
= infinite
2007-10-18 7:15 pm
The volume of the cup = 1/3r2h
The relation of r and h:
Diameter = 8cm, height=10cm, linearly in ratio
r=4h/10
The volume of the cup V= 4/75 h3
Then dV=4/25 h2 dh
dV=7cm3/s
h=6, dh=25/4 dV/ h2=25/4 (7)/(62)=1.215 cm/s
V=7, h=5.08cm
dh=25/4 dV/ h2=25/4 (7)/(5.082)=1.694 cm/s
The relation of r and h:
Diameter = 8cm, height=10cm, linearly in ratio
r=4h/10
The volume of the cup V= 4/75 h3
Then dV=4/25 h2 dh
dV=7cm3/s
h=6, dh=25/4 dV/ h2=25/4 (7)/(62)=1.215 cm/s
V=7, h=5.08cm
dh=25/4 dV/ h2=25/4 (7)/(5.082)=1.694 cm/s
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