Question

- The volume of a right circular cylinder is given by V=
π
*r*^{2}h, where r is the radius of its circular base and h is its height.

- Differentiate the volume formula with respect to t to determine an equation relating the rates of change dV/dt , dr/dt , dh/dt.

- At a certain instant, the height is 6 inches and increasing at 1 in/sec and the radius is 10 inches and decreasing at 1 in/sec. How fast is the volume changing at that instant? Is the volume increasing or decreasing? How can you tell?

Answer #1

Let V be the volume of a cylinder having height h and radius r,
and assume that h and r
vary with time.
(a) How are dV /dt, dh/dt, and dr/dt related?
(b) At a certain instant, the height is 18 cm and increasing at 3
cm/s, while the radius is 30
cm and decreasing at 3 cm/s. How fast is the volume changing at
that instant? Is the
volume increasing or decreasing at that instant?

Suppose the radius, height and volume of a right circular
cylinder are denoted as r, h, and V . The radius and height of this
cylinder are increasing as a function of time. If dr/dt = 2 and
dV/dt = 10π when r = 1, h = 2, what is the value of dh/dt at this
time?

The
radius of a right circular cone is decreasing at a rate of
1.5cm/sec and the height is increasing at a rate of 5cm/sec. At
what rate is the volume changing when the height is 12cm and the
radius 2cm? Leave your answer in terms of pi.

The radius of a circular cylinder is increasing at rate of 3
cm/s while the height is decreasing at a rate of 4 cm/s.
a.) How fast is the surface area of the cylinder changing when
the radius is 11 cm and the height is 7 cm? (use A =2 pi r2 +2 pi
rh )
b.) Based on your work and answer from part (a),is the surface
area increasing or decreasing at the same moment in time? How do...

A grain silo has the shape of a right circular cylinder
surmounted by a hemisphere. If the silo is to have a volume of
516π ft3, determine the radius and height of
the silo that requires the least amount of material to build.
Hint: The volume of the silo is
πr2h
+
A grain silo has
the shape of a right circular cylinder surmounted by a hemisphere.
If the silo is to have a volume of 516π ft3,
determine the...

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Asking for both radius and height.

A potter forms a piece of clay into a right circular cylinder.
As she rolls it, the height hh of the cylinder increases and the
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Suppose the height of the cylinder is increasing by 0.4 centimeters
per second. What is the rate at which the radius is changing when
the radius is 4 centimeters and the height is 11 centimeters?

1.) A rock is thrown into a still pond. The circular ripples move
outward from the point of impact of the rock so that the radius of
the circle formed by a ripple increases at a rate of 5
ft./min.
Find the rate at which the area is changing at the instant the
radius is 7 feet.
when the radius is 7 feet, the area is changing at approximately __
Square feet per minute
2.)
The radius of a spherical...

The volume of a cylinder can be computed as: v = π * r * r * h
Write a C++ function that computes the volume of a cylinder given r
and h. Assume that the calling function passes the values of r and
h by value and v by reference, i.e. v is declared in calling
function and is passed by reference. The function just updates the
volume v declared in calling function. The function prototype is
given by:...

Suppose water is leaking from a tank through a circular hole of
area
Ah
at its bottom. When water leaks through a hole, friction and
contraction of the stream near the hole reduce the volume of water
leaving the tank per second to
cAh
2gh
, where
c (0 < c < 1)
is an empirical constant.
A tank in the form of a right-circular cone standing on end,
vertex down, is leaking water through a circular hole in its...

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