Answer to Question #198640 in Math for Dream

Question

You are planning on throwing a party after social distancing is lifted. A cube-shaped tank has edges that are choose a whole number between 11 in and 14 in. ___________ long. The tank is filled with punch that has a density of 0.02 lb./in³. 

Answer 1

The question provided was, perhaps, incomplete, I will solve it according to a more complete version found at the link above. It consists of 3 subquections (a, b, c).

a) Calculate the volume (V) and mass (m) of the tank.

These values depend on the length (denote it as L) and the density (r) of the tank.

The volume can be calculated as $V=L^3$ , and the mass $m = r * V$ .

As the edges length can vary between 11 in and 14 in, and the whole length of the tank is not specified, we can denote the whole length as L0, and the effective length of the liquid column as $L1 = L0 - E (edge)$.

So we can have $V1=(L0-E)^3$, it can be, for example $V1=(L0-11)^3 in^3.$

And if we want to calculate the volume of the tank and liquid together, it is $V0=L0^3$ .

And the mass of the liquid: $m1 = r * V1 = r * (L0-E)^3,$  we can have

$m1=0.02lb/in^3∗(L0−11)3in^3=0.02(L0−11)^3lb$

But if we want to calculate the mass of the liquid column and the tank together,such as in the point b, we should first calculate the mass of the tank as if it is entirely made from its material, presumably glass, but it is not given, so denote it as r2.

$Vt=L0^3,$ and then substract the liquid volume from it:

$V2 =V0-V1= L0^3-(L0-E)^3$ , use it to calculate the mass of the tank:

$mt= r2*V2=r2*(L0^3-(L0-E)^3) lb.$

The mass of the tank with the liquid:

$mtl= m0=r2*(L0^3-(L0-E)^3)+r * (L0-E)^3$ lb.

b) Should the tank be placed on a table that can support a maximum weight of 50?

It depends on the mass of the tank, which can be caltulated using the formulae given above, as a mass of the tank and punch together, m0. If the resulting m>=50 lb, it's obvious that we shoud not put that tank on that table, and even it is 2-3 lb lighter than the maximal weight because someone can put the other thing on the table such as cups or lean on it.

c) Using the same punch, would the density of the punch change if the tank was

only ¾ full?

No, the density of the liquids doesn't change with the volume change, it does not depend on the volume, but on the temperature and tensionif the liquid is not pressed, only if we put the less quantity of it.

This occurs due to the constancy of the gravitational effect between the molecules of the liquid at constant temperature and pressure, which makes the distance between the molecules remain unchanged on average, and the liquid maintains a constant volume.