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

Question

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

emmak · Accepted Answer

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. :arrow:

Hope the explanation helps.

Zarrolou · Answer

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

Bunuel · Answer

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.

subhendu009 · Answer

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.

obs23 · Answer

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} ?

Zarrolou · Answer

"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

obs23 · Answer

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? 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 tagline man!

Zarrolou · Answer

To create the formula we should refer to the text:
"If the price of a commodity is directly proportional to m^3  m^3 ]   and   inversely proportional to q^2  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

obs23 · Answer

Got it! Thanks much for help.

KarishmaB · Answer

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

krebsbr · Answer

not sure why i dont understand this, but why does

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

?

Bunuel · Answer

\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.

mrauj · Answer

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

Could you please explain how the solution is derived?

Solution:
 Joint variation: (x*p^6)/(y^2*z^2) = k

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

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