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In a certain industry, production index x is directly

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In a certain industry, production index x is directly

by VJesus12 » Wed Jul 04, 2018 2:54 am

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In a certain industry, production index x is directly proportional to the square of efficiency index y, and indirectly proportional to investment index z. If a business in this industry halves its investment index, which of the following is closest to the percent change in the business's efficiency index required to keep the production index the same?

(A) 100% increase
(B) 50% increase
(C) 30% increase
(D) 30% decrease
(E) 50% decrease

The OA is the option D.

What? I didn't understand the question. Please, help. <i class="em em-sob"></i>
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by Shahrukh_mbabreakspace » Wed Jul 04, 2018 3:11 am
Since x is directly proportional to y^2 and inversely proportional to z

x=(ky^2)/z, where k is a constant

Now, if z becomes half, then to keep x unchanged, y^2 should also become 1/2
This means y will become 1/sqrt(2) times of y

Now, % change in y= (y-(1/sqrt(2)*y))/y * 100
Which approximately is 30%


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by Scott@TargetTestPrep » Mon Jul 09, 2018 7:01 pm
VJesus12 wrote:In a certain industry, production index x is directly proportional to the square of efficiency index y, and indirectly proportional to investment index z. If a business in this industry halves its investment index, which of the following is closest to the percent change in the business's efficiency index required to keep the production index the same?

(A) 100% increase
(B) 50% increase
(C) 30% increase
(D) 30% decrease
(E) 50% decrease
"Production index x is directly proportional to the square of efficiency index y, and indirectly proportional to investment index z" means x = k * y^2/z for some positive value k.

Now we are given that z becomes z/2 and we can let y become ty for some positive value t. So we have x = k * (ty)^2/(z/2). Since we are keeping x the same and recall that x = k * y^2/z, so we have:

k * y^2/z = k * (ty)^2/(z/2)

y^2/z = t^2 * y^2 * 2/z

1 = t^2 * 2

1/2 = t^2

t = √(1/2) ≈ 0.7

Thus we see that the efficiency index y has to become 0.7y, or a 30% decrease, in order to keep the production index x the same.

Answer: D

Scott Woodbury-Stewart
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by swerve » Fri Jul 13, 2018 9:21 am
The entire expression can be written in the form x=k*y^2/z. When z is halved, the expression becomes x = 2k*y^2/z, in order to make this expression equal to its initial form we need to divide this expression by 2. This can be done by reducing y to y/sqrt{2}.

Therefore percent change = y−y/1.41 4 --> 0.414y/1.414. This is almost equal to 30%.

Regards!
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