AP calculus
posted by Alannah .
The base of a coneshaped tank is a circle of radius 5 feet, and the vertex of the cone is 12 feet above the base. The tank is being filled at a rate of 3 cubic feet per minute. Find the rate of change of the depth of water in the tank when then depth is 7 feet.

when the water is at depth x, the radius of the surface of the water is 5/12 (12x)
so, the volume of the air space is 1/3 pi r^2 h
= 1/3 pi (5/12 (12x))^2 (12x)
= 25pi/432 (12x)^3
the volume of water is thus the tank volume less the air space:
v = pi/3 * 25^2 * 12  25pi/432 (12x)^3
= 100 pi  25pi/432 (12x)^3
dv/dt = 25/144 pi (12x)^2 dx/dt
3 = 25/144pi * 25 dx/dt
dx/dt = 432/(625pi) = 0.22 ft/min
Respond to this Question
Similar Questions

calculus
A hollow cone has height 5 feet and base diameter 4 feet. The vertex of the cone is pointed down so that it can serve as a container. Water is poured into the cone at the rate 3/2 cubic feet per second. At what rate (in feet per second) … 
math
a vessel comtaining water has the shape of an inverted circular cone of base radius 5 feet and height 10 feet.the water flow from the apex of the cone at a constant rate of 3 cubic feet per minute.how fast is the water lavel rising … 
Calculus
A water tank is shaped like an inverted right circular cone with a base radius of 14 feet and a height of 25 feet high. If water flows into the tank at a rate of 20 ft^3/min, how fast is the depth of the water increasing when the water … 
Math
An oil tank is the shape of an inverted right circular cone with the pointed end down. The tank is 15 feet tall and is 12 feet in diameter at the top. At 1:00pm, the tank has oil 5 feet deep in it. Oil is pouring in at 5 cubic feet … 
Calculus
Water is draining at a rate of 2 cubic feet per minute from the bottom of a conically shaped storage tank of overall height 6 feet and radius 2 feet . How fast is the height of water in the tank changing when 8 cubic feet of water … 
Calculus
A conical tank has a height that is always 3 times its radius. If water is leaving the tank at the rate of 50 cubic feet per minute, how fast if the water level falling in feet per minute when the water is 3 feet high? 
Please check my calculus
A conical tank has a height that is always 3 times its radius. If water is leaving the tank at the rate of 50 cubic feet per minute, how fast if the water level falling in feet per minute when the water is 3 feet high? 
Calculus
Water is running into an open conical tank at the rate of 9 cubic feet per minute. The tank is standing inverted, and has a height of 10 feet and a base diameter of 10 feet. At what rate is the radius of the water in the tank increasing … 
Calculus
Water is running into an open conical tank at the rate of 9 cubic feet per minute. The tank is standing inverted, and has a height of 10 feet and a base diameter of 10 feet. At what rate is the exposed surface area of the water changing … 
Math
The base of a pyramidshaped tank is a square with sides of length 12 feet, and the vertex of the pyramid is 10 feet above the base. The tank is filled to a depth of 4 feet, and water is flowing into the tank at the rate of 2 cubic …