Deep Water Lighthouses
deep water lighthouses
need some help on this Calulus Problems..?
Hey guys..
can anyone help me on those problems ,
1-A spherical balloon is being inflated so that it's volume is increasing at the rate of 5m^3 /min.
At what rate is the Diameter increasing at the instant when the diameter is 12 m?
2-Water is Pouring into a conical cistern at the rate of 10 cubic feet per minutes. if the height of the inverted cone is 8 feet and the radius of it's circular opening is 4fee,how fast is the Radius of the water surface increasing when the waster is 4 feet deep?
3-A boat Sails parallel to to a Straight beach at a constant speed of 12 miles per hour, Staying 4 miles offshore . How fast is it approaching lighthouse on the shoreline at the instant it is exactly 5 miles from the lighthouse?
any help is appreciated thanks.
1)
Volume of the spherical balloon = (4/3) PI r^3, where 'r' is the radius.
Write this formula using the diameter.
(4/3) PI (2r)^3 / 8 = (1/6) PI d^3, where 'd' is the diameter.
dV/dt = 5 (given)
V= (1/6) PI d^3
differentiate both sides with respect to t
dV/dt = (1/6) PI (3d^2) dd/dt
5 = (3/6) PI d^2 dd/dt
when d=12
5 = (1/2) PI (12)^2 dd/dt
5 = 72 PI dd/dt
dd/dt = 5 / 72PI m/min
2)
Volume of a cone = (1/3) PI r^2 h
Using similar triangles, r/h = 4/8
r/h = 1/2
h=2r
r=h/2
Volume = (1/3) PI r^2 (2r)
V= (2/3) PI r^3
dV/dt =10 (given)
differentiate both sides with respect to t
dv/dt = (2/3) PI (3r^2) dr/dt
10 = 2 PI r^2 dr/dt
when h=4, r=2; r^2 = 4
10 = 2PI (4) dr/dt
10 =8PI dr/dt
dr/dt = (10/8PI) = 5/4PI ft/min
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