a)
i)
Let r be the radius of the oil spilled and C be its circumference.
As the oil spreads in a circular pattern,
Differentiating both sides of the above equation with respect to time t, we get
The radius increases at a constant rate of 6 ft/s which means
The rate of change of circumference is
Hence, the circumference is increasing at the rate of 12 ft/s.
ii)
Let r be the radius of the oil spilled and A be its area.
As the oil spreads in a circular pattern,
Differentiating both sides of the above equation with respect to time t, we get
The radius increases at a constant rate of 6 ft/s which means
The rate of change of area at r = 30 ft is
Hence, the area is increasing at a rate of 360/s when the radius of the spill is 30 ft.
b)
i)
The height(in meters) of the Football from the ground after t seconds of throwing is given by
The acceleration of the Football is the second derivative of the height function h(t).
Hence, the acceleration of the Football is -32 m/.
ii)
The height(in meters) of the Volleyball from the ground after t seconds of throwing is given by
The velocity of the Volleyball is the first derivative of the height function h(t).
At t = 3 seconds, the velocity is
Hence, the velocity of the Volleyball at t = 3 seconds is -66 m/s.
Note - I have answered 4 sub-parts as HOMEWORKLIB guidelines.
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