I have several problems that I don't know how to do. I will post one now and hope that someone can help me.
The problem is:
A cylinder made of playing dough with dimensions h=10cm, and r=1cm, is being stretched from both ends.
(a) If V is the volume of the dough and during this stretching process the dough remains cylinder, show that (dh/dt)=(-2V)/(pi r cubed)*(dr/dt)=(-2h)/(r)*(dr/dt)
(b) If the rate of stretching is (dh/dt)=0.1 mm/s, find the rate of change of its base diameter r when h=12.1 cm.
I have no clue where to start and what to do with the problem. All I have written is a drawn image of the cylinder and the Volume equation of a cylinder which is V=(pi)*r(squared)*h.
Can someone please help me?
Assume the total volume of the playdough stays the same. As the cylinder lengthens, it must get narrower.
pi r^2 h = constant = V
h = (V/pi) r^-2
dh/dt = - 2(V/pi)r^-3 (dr/dt)
= -2 (pi r^2 h/pi) r^-3 (dr/dt
= -2 (h/r) (dr/dt)
How did you get h?
Thank you very much, although I would like to know how did you get h?
h is related to the volume V (which is constant) and the variable r. When h increases, r decreases. The formula which relates h to r was provided. A formula that relates dr/dt to dh/dt was derived.
To get the function h(r), I used the constant-volume equation
pi r^2 h = constant = V
and a bit of algebra.
Alright! Thank you! And for part B. Would I just plug in the numbers?
For Part b, yes, just plug in the numbers.
dh/dt = -2 (h/r) (dr/dt)
You are given dh/dt and h, and have to first solve for r to use that formula.
Use the fact that hr^2 = constant = 10 cm^3. So when h=12.1 cm,
r = sqrt (10/12.1) = 0.9091 cm
dr/dt = (-1/2)(r/h)(dh/dt)
= (-0.5)(0.9091/12.1)(0.01 cm/s)
I made a mistake in the calculation. Let's fix it.
To find the rate of change of the base diameter r, we can use the formula:
dr/dt = (-0.5)(r/h)(dh/dt)
Given that dh/dt = 0.1 mm/s and h = 12.1 cm, we need to find the value of r.
To find r, we can use the equation for the volume of the cylinder:
V = πr^2h
In this case, the volume V is constant, so we can write:
V = πr^2h = π(1 cm)^2(10 cm)
Simplifying, we have:
π(1 cm)^2(10 cm) = πr^2(12.1 cm)
Dividing both sides by π and 12.1 cm, we get:
r^2 = (1 cm^2 × 10 cm) / 12.1 cm ≈ 0.826 cm^2
Taking the square root of both sides, we find:
r ≈ 0.909 cm
Now we can substitute the given values into the formula for dr/dt:
dr/dt = (-0.5)(0.909 cm / 12.1 cm)(0.1 mm/s)
Simplifying, we have:
dr/dt ≈ -0.0376 mm/s
Therefore, the rate of change of the base diameter r when h = 12.1 cm is approximately -0.0376 mm/s.