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By a calculation of the size of the cubic $\text {cm}$ which its sizes are: $3.8330 \times 3.8330 \times 5.17455$, I got a volume of $76.02 cm^3$. How can I get precisely "$76.04$" $\text {cm}^3$, by changing the first two mentioned factors (i.e. $3.8330 \times 3.8330 \times 5.17455$) equally?

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

For the love of maths
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Arithmetic-Enthusiast
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3 Answers3

12

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

Suppose you have $$a\cdot b \cdot c = d$$ but you want $d$ to become $e$. You can make this happen by multiplying both sides by $e/d$: $$\left(a\cdot b \cdot c \right)\cdot \frac{e}{d} = d\cdot\frac{e}{d} = e$$

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,

$$\left( a\cdot \frac{e}{d}\right)\cdot b\cdot c \;=\; e \tag{1}$$

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:

$$\frac{e}{d} = \sqrt{\frac{e}{d}}\cdot\sqrt{\frac{e}{d}} \qquad\to\qquad\left(a\cdot \sqrt{\frac{e}{d}}\right)\cdot\left(b\cdot \sqrt{\frac{e}{d}}\right)\cdot c \;=\; e \tag{2}$$

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

$$\left(a\cdot\sqrt[3]\frac{e}{d}\right)\cdot\left(b\cdot\sqrt[3]\frac{e}{d}\right)\cdot \left(c\cdot\sqrt[3]\frac{e}{d}\right) \;=\; e \tag{3}$$

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

Blue
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4

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

You can change it by multiplying both sides by $\displaystyle \frac{76.04}{76.02}$.

We then have $3.833 \times 3.833 \times5.174 \times \displaystyle \frac{76.04}{76.02}=76.02 \times \displaystyle \frac{76.04}{76.02}$.

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

Saketh Malyala
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    +1 Note that you can multiply any one of the three factors by $76.04/75.02$, although there's an aesthetic argument for keeping the first two equal. – Ethan Bolker Apr 28 '19 at 21:36
  • Thank you very much for you answer (1+^). Is there a way to add change the first two parameters (3.833*3.833)? I mean even-though it'll be the same size, but if I want to change these dimensions is important. – Arithmetic-Enthusiast Apr 28 '19 at 21:58
  • @UbiquitousStudent do you want to change both of the first two parameters? or just one of them? – Saketh Malyala Apr 28 '19 at 22:04
  • you can multiply 3.833 by 76.04/76.02 instead – Saketh Malyala Apr 28 '19 at 22:05
  • You can take any two positive numbers $a$ and $b$, let $c = (76.04/76.02)/ab$ and then multiply each of the factors by $a$, $b$ and $c$. So you can make any two factors anything you like and adjust the third accordingly. If you want to keep the third factor the same and keep the first two equal, let $a = b =$ the square root of $76.04/76.02$. – Ethan Bolker Apr 28 '19 at 22:05
  • Thank you Ethan. I'll try to find in the way that you mentioned. @SakethMalyala I'd like to change both of them equally. – Arithmetic-Enthusiast Apr 28 '19 at 22:33
  • Okay. Try multiplying 3.833 x sqrt(76.04/76.02) for both of them – Saketh Malyala Apr 29 '19 at 04:14
2

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 $0.0242\ \text{cm}^3$. How thick does it have to be to equal that volume?

Hence, $\frac{0.0242}{3.833^2} = .00165$

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

$3.833 \cdot 3.833 \cdot 5.17565 = 76.040$

Phil H
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