Water is being poured into a spherical bowl of radius 4 cm at a rate of 2 cm3/s. How fast is the water level rising when it is at 2 cm?
3 answers
A man is walking beside a 30-m street lamp. After walking 25 m away from the street lamp, he sunk into a quicksand. At the time when 2 m of him are above the ground, he is sinking at 1 m/s. At what rate is the length of his shadow changing at that moment?
draw a side view of the bowl. When the depth is y cm, then the surface of the water has radius r cm, such that
(4-y)^2 + r^2 = 16
and the volume of water is
v = π/6 y(3r^2 + y^2) = π/6 y(3(16-(4-y)^2)+y^2) = 4πy^2 - π/3 y^3
dv/dt = πy(8-y) dy/dt
So plug in your numbers and solve for dy/dt
(4-y)^2 + r^2 = 16
and the volume of water is
v = π/6 y(3r^2 + y^2) = π/6 y(3(16-(4-y)^2)+y^2) = 4πy^2 - π/3 y^3
dv/dt = πy(8-y) dy/dt
So plug in your numbers and solve for dy/dt
5cm/s