If the price of a commodity is directly proportional to m^3

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Hope it helps.

Let price of the commodity be P = \(k *\)\(m^3\) /\(Q^2\)

A. P = \(k * 27/4\) = 6.75K
B. P = \(K * 12^3/12^2\) = 12k
C. P = \(K * 20^3/20^2\) = 20k
D. P = \(K * 30^3/36^2\)= \(k*5*5*30/6*6\) = \(k * (125/6)\) = 21K (approx)
E. P = \(k * 36^3/72^2\) = \(K*(36/2*2)\) = 9k

Ans : Option D.

Could you please explain how you get \(\frac{m^3}{q^2}\) or \(\frac{Km^3}{q^2}\)? Where am I wrong expressing this thing firstly as \(p=m^3k\) and \(p=\frac{k}{q^2}\)?

"directly proportional to m^3 and inversely proportional to q^2" is what the text says. So if m increases the price increases, if q increases the price decreases. The right formula here is \(\frac{m^3}{q^2}\) ( or with k, it doesn`t change anything), if the numerator grows, the price does the same; and as in every fraction, if the denominator grows, the price drops.
Your formulas are right but the price depends on both m and q, you have to include them in one equation.

Let me know if it is clear

I see that makes sense. I guess I am making it more complicated than it really is. If you say that my formulas are correct, then \(\frac{m^3}{q^2}\) should somehow be extracted from \(p=m^3k\) and \(p=\frac{k}{q^2}\), no? Technically speaking, I thought there was a way to combine them into \(\frac{m^3}{q^2}\), just from pure algebraic manipulation (which I am very interested in here)... Or is it simply about common sense or the way you explained it?

P.S. Like your "kudos" tagline man!

To create the formula we should refer to the text:
"If the price of a commodity is directly proportional to m^3 [and at this point we write down Price= \(m^3\)] and inversely proportional to q^2 [and at this point we complete the foumula adding the Denominator so Price=\(\frac{m^3}{q^2}\)], which of the following values of m and q will result in the highest price for the commodity?"
\(\frac{m^3}{q^2}\) is not extracted from \(p=m^3k\) and \(p=\frac{k}{q^2}\), and you cannot obtain it from pure algebraic manipulation.
The idea behind your formulas is correct p=m^3 expresses the direct correlation between p and m; and also p=1/q^2 expresses the inverse correlation between p and q. But the text uses "and" so those ideas must be expressed in one formula, so to obtain this final formula you "complete" one with the other =>m^3/p^2

Got it! Thanks much for help.

Please check my signature for the link to the relevant blog posts.

not sure why i dont understand this, but why does

30^3 / 36^2 = (30)(30)(30) / (36)(36) = (5)(5)(5) / 6

?

\(\frac{30^3}{36^2} = \frac{30*30*30}{36*36} =\frac{(5*6)*(5*6)*(5*6)}{(6*6)*(6*6)}=\frac{5*5*5}{6}\)

Hope it helps.

In the joint variation blog post (#3), there is a sample question



Could you please explain how the solution is derived?

Solution:

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