I need help with calculus. Please?

Question

I missed a day and now I'm a little behind. I have an idea of how to do these, but I just want to make sure I'm right before I spend 2 hours doing a whole page of them.

1) A balloon rises at the rate of 8 feet per second from a point on the ground 60 feet from an observer. Find the rate of the angle of elevation when the balloon is 25 feet above the ground.

2) The height of a cylinder with a radius of 4 cm is increasing at a rate of 2 cm/min. Find the rate of change of the volume of the cylinder when h=10 cm.

Answer 1

1. Okay, let a, be distance from ground to balloon and b be angle.
So, da/dt=8.
So, b=arctan(a/60).
And d(b/a) = 1/(60((a^2/3600)+1)).
And d(b/a)=1/(60((25^2/3600)+1)) =.0142.
And (d(b/a))((da/dt) = db/dt = .0142*8= .1136 approximately.

2.V=pi*r^2*h=4^2*pi*h=16pi*h..
dh/dt=2.
dV/dh =16pi
dV/dt=(dV/dh)(dh/dt)=(16pi)(2) =32pi.

Answer 2

#1: First consider what the angle of elevation actually is: we know the horizontal distance from the observer to the balloon is 60 ft, and the vertical distance is the height of the balloon. If h is the height of the balloon, then this forms a right triangle with adjacent side 60 ft and opposite side h. So the angle of elevation θ is given by the relation:

tan θ = h/(60 ft)
θ = arctan (h/(60 ft))

Now, we want to know the rate of change of θ with respect to time. Knowing that the derivative of arctan x is 1/(1+x²) and using the chain rule, we find:

dθ/dt = 1/(1+(h/(60 ft))²) * d(h/(60 ft))/dt = 1/(1+(h/(60 ft))²) * 1/(60 ft) * dh/dt

Simplifying this a bit:

dθ/dt = 3600 ft²/(3600 ft² + h²) * 1/(60 ft) * dh/dt
dθ/dt = 60 ft/(3600 ft² + h²) * dh/dt

Now, we want to find dθ/dt when h=25 ft. We know the balloon is rising at a rate of 8 ft/s (that is, dh/dt = 8 ft/s), so plugging this in:

dθ/dt = 60 ft/(3600 ft² + 625 ft²) * 8 ft/s

Simplifying:

dθ/dt = 480 ft²/(4225 ft²) s⁻¹
dθ/dt = 480/4225 s⁻¹
dθ/dt = 96/845 s⁻¹ ≈ 0.11361 s⁻¹

So the angle of elevation is changing at about 0.11361 radians per second (noting that radians are a dimensionless unit, so it was expected that radians/second would appear as s⁻¹ in dimensional analysis).

2: Again, first find the volume of the cylinder:

V = πr²h = 16π cm² h

Now, using the chain rule:

dV/dt = 16π cm² dh/dt

And plugging in the known value for dh/dt:

dV/dt = 16π cm² * (2 cm/min)
dV/dt = 32π cm³/min ≈ 100.53 cm/min

Answer 3

Well at least you didn't miss out on anything complicated.

1) is just trigonometry arctangent (25/60)=23 DEGREES

2) The base(top) of a cylinder stays constant as the height
increases. Therefore dv/dt also constant
V=4*pi*h
dV/dt=4*pi*dh/dt
=25.13cm^3/min. ans