The rate of change of the surface area when the radius is 6 inches and the height is 18 inches is 979.68 in² per minute.
Given:
radius of a right circular cylinder is increasing at a rate of 6 inches per minute, and the height is decreasing at a rate of 4 inches per minute.
we are asked to find the rate of change of the surface area when the radius is 6 inches and the height is 18 inches.
we have dr/dt = 6 in/min and dh/dt = -4 in/min
Volume, V = πr²h
Differentiate the above expression to get:
dV/dt = 2πrh dr/dt + πr² dh/dt
SUbstitute the value we get:
dV/dt = 2π(6)(18)(6) + π(6)²(-4)
= 4069.44 - 452.16
= 3617.28 in³ per minute.
The volume is increasing at a rate of 3617.28 in³ per minute.
Area = 2πr(h+r)
Differentiate the above expression:
dA/dt = 2π {(h+r) dr/dt + r (dh/dt + dr/dt)}
dA/dt = 2π {(18+6)(6) + 6(-4+6)}
= 2π(144 + 12)
= 312π
= 979.68 in² per minute.
The area is increasing at a rate of 979.68 in² per minute.
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The rate of change of the surface area when the radius is 6 inches and the height is 18 inches is 979.68 in² per minute.
Given:
The radius of a right circular cylinder is increasing at a rate of 6 inches per minute, and the height is decreasing at a rate of 4 inches per minute.
We are asked to find the rate of change of the surface area when the radius is 6 inches and the height is 18 inches.
We have dr/dt = 6 in/min and dh/dt = -4 in/min
In geometry, it is defined as a three-dimensional shape having two circular shapes at a distance called the height of the cylinder.
We know the volume of the cylinder is given by:
Volume, V = πr²h
Differentiate the above expression to get:
dV/dt = 2πrh dr/dt + πr² dh/dt
Substitute the value we get:
dV/dt = 2π*6*18*6 + π*6² * -4
= 4069.44 - 452.16
= 3617.28 in³ per minute.
The volume is increasing at a rate of 3617.28 in³ per minute.
Area = 2πr(h+r)
Differentiate the above expression:
dA/dt = 2π {(h+r) dr/dt + r (dh/dt + dr/dt)}
dA/dt = 2π {(18+6)(6) + 6(-4+6)}
= 2π(144 + 12)
= 312π
= 979.68 in² per minute.
The area is increasing at a rate of 979.68 in² per minute.
To learn more about the cylinder here visit:
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