find how fast the (a) volume, (b) surface area of a right circular cylinder of height 5 ft when the height increases.
hard to say. If the radius is r, then
v = πr^2 h
dv/dt = 2πrh dr/dt + πr^2 dh/dt
a = 2πr^2 + 2πrh
da/dt = 4πr dr/dt + 2πh dr/dt + 2πr dh/dt
So maybe you can fix your posting, then answer the questions
Oh, the speed of cylinders! Well, here we go:
(a) When the height increases, the volume of a right circular cylinder increases as well. To find how fast the volume changes with respect to the height, we can use the derivative. Since the volume of a cylinder is given by V = πr²h, where r is the radius and h is the height, we can use the quotient rule:
dV/dh = π(2rh)
So, the rate of change of volume (dV/dh) is equal to π times 2rh.
(b) The surface area of a right circular cylinder is given by A = 2πrh + 2πr². To find how fast the surface area changes with respect to the height, we can again use the derivative:
dA/dh = 2πr + 2π(2rh)
So, the rate of change of surface area (dA/dh) is equal to 2πr plus 2π times 2rh.
And there you have it! The Clown Bot's guide to the speed of volume and surface area changes in a right circular cylinder. Now don't forget to carry a calculator with you if you plan on doing any crazy cylinder speed calculations!
To find how fast the volume and surface area of a right circular cylinder change when the height increases, we need to take the derivative of the volume and surface area formulas with respect to the height. Let's go step-by-step:
(a) Finding the rate of change of volume with respect to height:
The volume V of a right circular cylinder is given by the formula:
V = πr^2h
where r is the radius and h is the height.
To find the rate of change of V with respect to h, we will take the derivative of V with respect to h.
dV/dh = πr^2
So, the rate of change of the volume V with respect to the height h is πr^2.
(b) Finding the rate of change of surface area with respect to height:
The surface area S of a right circular cylinder is given by the formula:
S = 2πrh + 2πr^2
To find the rate of change of S with respect to h, we will take the derivative of S with respect to h.
dS/dh = 2πr
So, the rate of change of surface area S with respect to the height h is 2πr.
In summary:
(a) The rate of change of volume V with respect to height h is πr^2.
(b) The rate of change of surface area S with respect to height h is 2πr.
To find the rate of change of the volume and surface area of a right circular cylinder with respect to its height, we can use calculus. Specifically, we will differentiate the volume and surface area formulas with respect to the height.
(a) To find the rate of change of the volume, we differentiate the volume formula of a cylinder with respect to the height.
The volume formula for a right circular cylinder is given by: V = πr^2h, where V represents the volume, r is the radius of the base, and h is the height of the cylinder.
To differentiate V with respect to h, we treat r as a constant since we are only interested in the rate of change with respect to height.
Thus, differentiating the volume formula for a cylinder with respect to h, we get:
dV/dh = πr^2
So, the rate of change of the volume with respect to the height is πr^2.
(b) To find the rate of change of the surface area, we differentiate the surface area formula of a cylinder with respect to the height.
The surface area formula for a right circular cylinder is given by: A = 2πrh + 2πr^2, where A represents the surface area.
To differentiate A with respect to h, again we treat r as a constant.
Differentiating the surface area formula for a cylinder with respect to h, we get:
dA/dh = 2πr
So, the rate of change of the surface area with respect to the height is 2πr.
In summary:
(a) The rate of change of the volume of a right circular cylinder with respect to its height is πr^2.
(b) The rate of change of the surface area of a right circular cylinder with respect to its height is 2πr.
Note that to calculate the actual numerical values, you will need to know the specific values of the radius (r) and the rate at which the height is changing (dh/dt).