Etydhv

By a calculation of the size of the cubic cm which its sizes are: 3.8330 * 3.8330 * 5.17455, I got a volume of 76.02. How can I get precisely '''76.04''' cubic cm, by changing the first two mentioned factors (i.e. 3.8330 * 3.8330 * 5.17455) equally?

N.b. I tried many ways and I couldn't find it, always I got more or less but not precisely.

You have $3.833 times 3.833 times5.174=76.02 .$

You can change it by multiplying both sides by $displaystyle frac76.0476.02$.

We then have $3.833 times 3.833 times5.174 times displaystyle frac76.0476.02=76.02 times displaystyle frac76.0476.02$.

This comes out to $3.833 times 3.833 times5.17564=76.04 $

Actually $3.833 cdot 3.833 cdot 5.174 = 76.0158$ so the added volume will be $.0242$

One way is to think of it as adding a sheet $3.833 cdot 3.833$ with a volume of $0.0242 textcm^3$. How thick does it have to be to equal that volume?

Hence, $frac0.02423.833^2 = .00165$

So the dimensions will be $3.833 cdot 3.833 cdot (5.174 + .00165)$

$3.833 cdot 3.833 cdot 5.17565 = 76.040$

To avoid getting caught-up in specific numbers ...

Suppose you have
$$acdot b cdot c = d$$
but you want $d$ to become $e$. You can make this happen by multiplying both sides by $e/d$:
$$left(acdot b cdot c right)cdot fraced = dcdotfraced = e$$

Now, you can use the left-hand side's factor of $e/d$ to make adjustments to $a$, $b$, and/or $c$. If you just wanted to adjust one factor, you could write, say,

$$left( acdot fracedright)cdot bcdot c ;=; e tag1$$

If you wanted to adjust two factors proportionally (as is specifically requested in the question), you can "split" $e/d$ equally across the factors using a square root:

$$fraced = sqrtfracedcdotsqrtfraced qquadtoqquadleft(acdot sqrtfracedright)cdotleft(bcdot sqrtfracedright)cdot c ;=; e tag2$$

Finally, if you later decide you actually want to adjust your entire box proportionally, you can use cube roots:

$$left(acdotsqrt[3]fracedright)cdotleft(bcdotsqrt[3]fracedright)cdot left(ccdotsqrt[3]fracedright) ;=; e tag3$$

Naturally, the same type of thing works with any number of overall factors and desired adjustments, using higher-level roots as needed.