A cylinder is to be constructed so that its height plus the perimeter of its base must equal to 2pi cm. Find the ratio of the radius of this cylinder to its height if it is to have a maximum volume.
My work:
SA = 2pirh + 2pir^2
2pi = 2pirh + 2pir^2
0 = 2pirh + 2pir^2 - 2pi
h = (-r^2 + 1) / r
V = pir^2((-r^2 + 1) / r)
V = pi(-r^2 + 1)
And now I don't know what to do? I don't even know if I've done the process in the beginning correctly...
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Text answer is (1 : pi)
You started with surface area, but nowhere is surface area involved.
it said "its height plus the perimeter of its base must equal to 2pi cm"
---> h + 2pir = 2pi
so h = 2pi - (2pi)r
since we want to maximize volume we need a volume equation.
V = pi(r^2)h
= pi(r^2)(2pi - 2pi(r))
= 2(pi)^2(r^2 - r^3)
V' = 3(pi)^2(2r - 3r^2) = 0 for a max of V
it is easy to show that r = 2/3 (r=0 is not possible)
so h = 2pi - 2pi(2/3)
= ....
then find r : h
where I said :
V' = 3(pi)^2(2r - 3r^2) = 0 for a max of V
should obviously have been
V' = 2(pi)^2(2r - 3r^2) = 0 for a max of V
2 pi r + h = 2 pi (cm)
Let a = r/h be the unknown ratio
2 pi a*h + h = 2 pi
h (2 pi a + 1) = 2 pi
V = pi r^2 h = pi a^2 h^3
= pi a^2 / [a + (1/2 pi)]^3
Set dV/da = 0 and solve for a.
But check my math first.
If the machine produces 20,000 straws in 8 hours, how many straws can it produce in 50 hours?
To find the ratio of the radius of the cylinder to its height for maximum volume, you have correctly set up the equation for the volume of the cylinder. Now, to find the maximum volume, we need to determine the critical points of the function.
Taking the derivative of the volume function with respect to r:
dV/dr = -2pir
Setting this derivative equal to zero to find the critical point:
-2pir = 0
From this, we can see that r = 0, but since the radius cannot be zero, we can disregard this value.
Next, we need to check the endpoints of the possible range for r. The perimeter of the base is given by 2pi * r, and this combined with the height h must equal 2pi cm. Thus, the maximum possible value for r is 1 cm.
Now, we need to evaluate the volume at the critical point and the endpoints:
V(0) = pi(-0^2 + 1) = pi
V(1) = pi(-1^2 + 1) = 0
Comparing these values, we can see that the maximum volume occurs when r is 1 cm. Thus, the ratio of the radius to the height for maximum volume is:
1 cm : h
Therefore, the ratio is 1 : h, and it is not specifically 1 : pi as mentioned in the text answer.