If the price of a commodity is directly proportional to m^3 and inversely proportional to q^2, which of the following values of m and q will result in the highest price for the commodity?
A. m=3, q=2
B. m=12, q=12
C. m=20, q=20
D. m=30, q=36
E. m=36, q=72

Let the price is P & constant is K
So P = \(\frac{K m^3}{q^2}\)

Now Just use the values in the options and put in the before mentioned equation.
A) P = \(\frac{K m^3}{q^2}\) = \(\frac{K 3^3}{2^2}\) = \(\frac{K 27}{4}\) = 6.75 K
B) P = \(\frac{K m^3}{q^2}\) = \(\frac{K 12^3}{12^2}\) = 12 K
C) P = \(\frac{K m^3}{q^2}\) = \(\frac{K 20^3}{20^2}\) = 20 K
D) P = \(\frac{K m^3}{q^2}\) = \(\frac{K 30^3}{36^2}\) = 125/6 = >20K
E) P = \(\frac{K m^3}{q^2}\) = \(\frac{K 36^3}{72^2}\) = 9 K

So the answer is D. This is probably 600-700 level question.

Hope the explanation helps.

Function = \(\frac{m^3}{q^2}\)

A=\(\frac{3*3*3}{2*2}=\frac{27}{4}=7\) almost
B=\(\frac{12*12*12}{12*12}=12\)
C=\(\frac{20*20*20}{20*20}=20\)
D=\(\frac{30*30*30}{36*36}=125/6\)
E=\(\frac{36*36*36}{72*72}=9\)

It's down to C or D, because \(\frac{120}{6}=20\), \(\frac{125}{6}>20\)
D

"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

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

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!

Similar questions to practice:

DS:
the-amount-of-coal-a-train-burns-each-mile-is-directly-93667.html
in-a-certain-business-production-index-p-is-directly-63570.html

PS:
a-is-directly-proportional-to-b-when-a-8-b-88971.html
in-a-certain-formula-p-is-directly-proportional-to-s-and-80941.html
the-rate-of-a-chemical-reaction-is-directly-proportional-to-76921.html

Hope it helps.
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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 \(p=\frac{k}{q^2}\)?

For more on direct, inverse and joint variation, check out these posts:

http://www.veritasprep.com/blog/2013/01 ... -directly/
http://www.veritasprep.com/blog/2013/02 ... inversely/
http://www.veritasprep.com/blog/2013/02 ... g-jointly/
_________________

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.

Got it! Thanks much for help.

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



Could you please explain how the solution is derived?

Solution: