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andrezito [222]
3 years ago
8

A rectangular metal tank with an open top is to hold 171.5 cubic feet of liquid. what are the dimensions of the tank that requir

e the least material to build?
Physics
1 answer:
Semmy [17]3 years ago
4 0
To minimize the material usage we have to have the volume requested with the minimum surface area.
The volume is:
171.5 =xyz
And the surface is:
S=xy+2xz+2yz
From the first equation we get:
z=\frac{171.5}{xy} ; k=171.5\\ z=\frac{k}{xy}\\
I will use k instead of a number just for the conveince.
We plug this into the second equation and we get:
S=xy+2k\frac{1}{x}+2k\frac{1}{y}
To find the minimum of this function we have to find the zeros of its first derivative.
Sx will denote the first derivative with respect to x and Sy will denote the first derivative with respect to Sy.
S_x=y-2k\frac{1}{x^2}\\ S_y=x-2k\frac{1}{y^2}
Now let both derivatives go to zero and solve the system (this will give us the so-called critical points).
0=y-2k\frac{1}{x^2}\\
0=x-2k\frac{1}{y^2}\\
y=2k\frac{1}{x^2}\\
x=2k\frac{1}{y^2}\\
Now we plug in the first equation into the other and we get:
x=\frac{\frac{2k}{1}}{\frac{4k^2}{x^4}}\\
x^3=2k\\
x=(2\cdot171.5)^{1/3}\\
x=7

Now we can calculate y:
y=2k\frac{1}{x^2}\\
y=2\cdot 171.5\frac{1}{7^2}=7
And finaly we calculate z:
z=\frac{171.5}{xy}\\
z=\frac{171.5}{7\cdot7}=3.5
And finaly let's check our result:
V=xyz=7\cdot7\cdot3.5=171.5
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