Water is poured into a vessel in the shape of a right circular cone of vertical angle [tex]$60^{\circ}$[/tex] with the axis vertical at a rate of [tex]$8 cm^3 s^{-1}$[/tex]. At what rate is the water surface rising when the depth of the water is 4 cm?
Answer (1)
Express the volume V of the water in the cone in terms of its height h , using the relationship between the radius and height derived from the cone's angle: V = 9 1 π h 3 .
Differentiate the volume equation with respect to time t to relate the rates of change of volume and height: d t d V = 3 1 π h 2 d t d h .
Substitute the given values d t d V = 8 and h = 4 into the differentiated equation.
Solve for d t d h to find the rate at which the water surface is rising: 2 π 3 .
Explanation
Problem Setup We are given a right circular cone with a vertical angle of 6 0 ∘ . Water is poured into the cone at a rate of 8 cm 3 / s . We want to find the rate at which the water surface is rising, d t d h , when the depth of the water is 4 cm .
Volume and Radius Relationship Let V be the volume of the water, h be the depth, and r be the radius of the water surface. The volume of a cone is V = 3 1 π r 2 h . Since the semi-vertical angle is 3 0 ∘ , we have tan ( 3 0 ∘ ) = h r , so r = h tan ( 3 0 ∘ ) = 3 h .
Volume in terms of height Substitute r = 3 h into the volume formula: V = 3 1 π ( 3 h ) 2 h = 9 1 π h 3 .
Differentiating with respect to time Differentiate both sides with respect to time t : d t d V = d t d ( 9 1 π h 3 ) = 3 1 π h 2 d t d h .
Substituting given values We are given d t d V = 8 and h = 4 . Substitute these values: 8 = 3 1 π ( 4 2 ) d t d h = 3 16 π d t d h .
Solving for dh/dt Solve for d t d h : d t d h = 16 π 8 ⋅ 3 = 2 π 3 .
Final Answer The rate at which the water surface is rising when the depth of the water is 4 cm is 2 π 3 cm/s.
Examples
Imagine you're designing a conical reservoir for a water purification system. Knowing how fast the water level rises as you pump water in is crucial for calibrating sensors and preventing overflows. This calculus problem provides the mathematical foundation to predict the water level change, ensuring your system operates efficiently and safely.
