Respuesta :
I've worked the problem and I also get the cost equation of
c(r) = 16000/r + 26/3*pi*r^2
which is basically identical to what you've gotten. So I'm not going to bother explaining how to get to this point.
But since you're looking for a minimum, that just screams FIRST DERIVATIVE. And for a simple function such as this, that's easy to get. So let's do it.
c(r) = 16000/r + 26/3*pi*r^2
Rewrite first term to make power explicit, so
c(r) = 16000*r^(-1) + 26/3*pi*r^2
Now for simple functions such as this, just multiply each term's coefficient by the exponent, then subtract 1 from the exponent. So we get
c'(r) = -16000*r^(-2) + 52/3*pi*r
c'(r) = -16000/r^2 + 52/3*pi*r
And that's the first derivative of your cost function. Your desired radius will be the value r where the value of the first derivative is 0. So
c'(r) = -16000/r^2 + 52/3*pi*r
0 = -16000/r^2 + 52/3*pi*r
16000/r^2 = 52/3*pi*r
16000=52/3*pi*r^3
48000/(52*pi) = r^3
12000/(13*pi) = r^3
cuberoot(12000/(13*pi)) = r
r is approximately 6.648076481
c(r) = 16000/r + 26/3*pi*r^2
which is basically identical to what you've gotten. So I'm not going to bother explaining how to get to this point.
But since you're looking for a minimum, that just screams FIRST DERIVATIVE. And for a simple function such as this, that's easy to get. So let's do it.
c(r) = 16000/r + 26/3*pi*r^2
Rewrite first term to make power explicit, so
c(r) = 16000*r^(-1) + 26/3*pi*r^2
Now for simple functions such as this, just multiply each term's coefficient by the exponent, then subtract 1 from the exponent. So we get
c'(r) = -16000*r^(-2) + 52/3*pi*r
c'(r) = -16000/r^2 + 52/3*pi*r
And that's the first derivative of your cost function. Your desired radius will be the value r where the value of the first derivative is 0. So
c'(r) = -16000/r^2 + 52/3*pi*r
0 = -16000/r^2 + 52/3*pi*r
16000/r^2 = 52/3*pi*r
16000=52/3*pi*r^3
48000/(52*pi) = r^3
12000/(13*pi) = r^3
cuberoot(12000/(13*pi)) = r
r is approximately 6.648076481
The dimensions to be used if the volume is fixed at 8,000 cubic units and the cost of construction shows that the radius is 6.6 units.
How to calculate the height?
The volume of the silo is 8000. The volume is calculated as:
V = πr²h + 2/3πr³
8000 = πr²h + 2/3πr³
h = 8000/πr² - 2/3r
The cost function is given as;
C = 16000x/r + 26/3πr²x
After the cost is minimized, the value of r, the radius is 6.6.
Learn more about more about dimensions on:
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