Application of Calculus - Optimisation

[Choi20]

An open top rectangular tank has a volume 9 cm³.

i) If the base of the tank has a dimension where the length is twice its width, show that the total surface area, A, of the tank is A = 2x² + 27 / x.

ii) Find the dimensions of the tank for the surface area A to be minimum.
I'm confused with this question. I'm not sure, do i need to find one more equation then compare it with A = 2x² + 27 / x equation. Please help me?

 

[Deleted member 4993]

An open top rectangular tank has a volume 9 cm³.

i) If the base of the tank has a dimension where the length is twice its width, show that the total surface area, A, of the tank is A = 2x² + 27 / x.

ii) Find the dimensions of the tank for the surface area A to be minimum.
I'm confused with this question. I'm not sure, do i need to find one more equation then compare it with A = 2x² + 27 / x equation. Please help me?

You are given A(rea) = 2x² + \(\displaystyle \frac{27}{x}\)

Do you know what 'x' is?

 

[Choi20]

Is it the minimum value for the first derivatives of A = 2x² + 27 / x.

 

[HallsofIvy]

No, x is NOT a formula involving x! But what Subhotosh Kahn was asking was if you know what x means.

The given information is "the base of the tank has a dimension where the length is twice its width" There is NO "x" in that. "\(\displaystyle A= 2x^2+ 27/x\)" is meaningless until you have said what x means.

 

[JeffM]

An open top rectangular tank has a volume 9 cm³.

i) If the base of the tank has a dimension where the length is twice its width, show that the total surface area, A, of the tank is A = 2x² + 27 / x.

ii) Find the dimensions of the tank for the surface area A to be minimum.
I'm confused with this question. I'm not sure, do i need to find one more equation then compare it with A = 2x² + 27 / x equation. Please help me?

This word problem is unusual because some variables have already been labeled, in one case explicitly and in one case implicitly. In the usual word problem, the very first thing to do is to identify and to label in writing each relevant variable with a unique letter. One relevant variable is surface area, which has explicitly been assigned the letter A. But another variable has been assigned the letter x, without specifying what the variable is. The question that has been asked is

\(\displaystyle x = what?\)

Moreover three relevant variables have not been assigned a letter at all. One of those is volume.

So V = volume. What are the two missing variables?

The second step in a word problem is to use the specific information in the problem and general information to formulate in mathematical form relations among the variables. General information that you are expected to know for this problem are how to calculate volume and area from the linear measures of the rectangular tank. Write those down. What are they?

Does any of this make sense?

 

[JeffM]

Is it the minimum value for the first derivatives of A = 2x² + 27 / x.

This is not quite right.

After you have identified your variables and the relations among them, you will have to find the value or values of x that make the first derivative of A with respect to x equal to zero (not minimum). But that in and of itself will not tell you what the dimensions of the tank are.