A balloon in the form of a right cone surmiunted by a hemispere,having a diameter equal to height of the cone,is being inflated.how fast is its volume changing w.r.t its total height 'h' when h
I assume that h is the height of the cone from the floor to the base of the hemisphere. Thus the diameter of the cone at the floor is 2 h and the height of the cone if it went to the tip would be 2 h
volume of cone of base diameter 2 h and height 2 h:
(1/3)(2h)(pi/4)(4 h^2)= (2/3)pi h^3
volume of cut off tip of cone:
(1/3)(h)(pi/4)h^2 = (1/12) pi h^3
so
volume of cone base = (7/12)pi h^3
now
volume of hemisphere = (1/2)(4/3)pi(h/2)^3 = pi h^3/12
so
total balloon volume = (2/3) pi h^3
dV/dh = 2 pi h^2
dV/dt = dV/dh * dh/dt
so
dV/dt = 2 pi h^2 dh/dt
To find how fast the volume of the balloon is changing with respect to its height, we need to use the concept of related rates.
Let's denote the cone height as h (the total height of the balloon), and let the radius of the hemisphere be r. Given that the diameter of the hemisphere is equal to the height of the cone, we have r = h/2.
The volume V of the balloon can be calculated by summing the volume of the cone and hemisphere:
V = V_cone + V_hemisphere
The volume of the cone, V_cone, is given by the formula:
V_cone = (1/3) * π * r^2 * h
Substituting r = h/2 into the equation, we have:
V_cone = (1/3) * π * (h/2)^2 * h
= (1/3) * π * (h^2/4) * h
= (1/12) * π * h^3
The volume of the hemisphere, V_hemisphere, is given by the formula:
V_hemisphere = (2/3) * π * r^3
Substituting r = h/2 into the equation, we have:
V_hemisphere = (2/3) * π * (h/2)^3
= (2/3) * π * (h^3/8)
= (1/4) * π * h^3
Adding the volumes together, we get the total volume V of the balloon:
V = V_cone + V_hemisphere
= (1/12) * π * h^3 + (1/4) * π * h^3
= (1/12 + 1/4) * π * h^3
= (1/6) * π * h^3
Now, let's differentiate V with respect to h to find how fast its volume is changing:
dV/dh = d/dh [(1/6) * π * h^3]
= (1/6) * π * (3h^2)
So, the rate of change of the volume with respect to the height is given by:
dV/dh = (1/2) * π * h^2
Therefore, the volume of the balloon is changing at a rate of (1/2) * π * h^2 with respect to its total height h.