Problem Solving

The longevity of a certain metal construction is determined by the following formula:
l = (7.5 - x)^4 + 8.97c, where l is the longevity of the construction, in years, x is the density of the underlying material, in g/cm3, and c is a positive constant equal to 1.05 for this type of metal constructions. For what value of density, x, expressed in g/cm3, will the metal construction have minimal longevity?

A. -7.5

B. 0

C. 7.5

D.15


E. 75

Source : MGMAT CAT

Need different approaches

l =(7.5 - x)^4 + 8.97c
(7.5 - x)^4 will always be +ve so the least value is 0
IMO C

OA C

bhumika.k.shah wrote:
Source : MGMAT CAT

Which MGMAT guide is this from : Word Translations or Number Properties ? Please also mention the Edition .

Its from MGMAT CAT and not QB

Hence NO edition as such...

CAT = Computer Adaptive Test.

Haldiram Bhujiawala wrote:

bhumika.k.shah wrote:
Source : MGMAT CAT

Which MGMAT guide is this from : Word Translations or Number Properties ? Please also mention the Edition .

Such probs can be solved using derivative,Which I think the GMAt doesn't test.

I = (7.5 - x)^4 + 8.97c Taking derivative on both side I get

dI/dx = 4*(7.5 - x)^3

Now for least value of L means dI/dx = 0

means x = 7.5

If there is any other solution please let us know .

gauravgundal wrote:Such probs can be solved using derivative,Which I think the GMAt doesn't test.

I = (7.5 - x)^4 + 8.97c Taking derivative on both side I get

dI/dx = 4*(7.5 - x)^3

Now for least value of L means dI/dx = 0

means x = 7.5

If there is any other solution please let us know .

kstv's solution is the fastest way to solve. Common sense/logic goes a long way to saving time on the GMAT.

Whenever a question asks us to minimize or maximize something, think through the task: what other variable should I minimize or maximize to reach my goal?

As an aside, you definitely don't need to know any calculus or high end functions (e.g. limits) for the GMAT.

this should be solved simply by inspection. Any number greater than or less than 7.5 is additive to the result, since the result is raised to an even power

l = (7.5-x)^4 + 8.97^c , where c is a constant
dl/dx = 0 to minimze l

dl/dx = 4(7.5-x)^3 + 0 = 0
dl/dx = 4(7.5-x)^3, x = 7.5 for dl/dx = 0 and l to be minimized

The longevity of a certain metal construction is determined by the following formula:
l = (7.5 - x)^4 + 8.97c, where l is the longevity of the construction, in years, x is the density of the underlying material, in g/cm3, and c is a positive constant equal to 1.05 for this type of metal constructions. For what value of density, x, expressed in g/cm3, will the metal construction have minimal longevity?

A. -7.5

B. 0

C. 7.5

D.15


E. 75

l is a function of x and c but since c is constant therefore the entire focus is x

l will be minimum when (7.5 - x)^4 is minimum but since it's a perfect squant and the minimum value of a perfect square is "0" therefore (7.5 - x)^4 must be "0" for smallest value of l

(7.5 - x)^4 = 0 @x=7.5 ANSWER Option: C

One request and suggestion to GMAT aspirants

"Refrain from higher maths like Differentiation, Calculus, Trigonometry etc. for keeping the right orientation and limited required concepts to Beat the GMAT"

All the best!!!

To answer this question, we need to minimize the value of l = (7.5 - x)^4 + (8.97)^1.05. Since we do not need to determine the actual minimum longevity, we do not need to find the value of the second component in our formula, (8.97)^1.05, which will remain constant for any level of x. Therefore, to minimize longevity, we need to minimize the value of the first component in our formula,
i.e. (7.5 - x)^4. Since we are raising the expression (7.5 - x) to an even exponent, 4, the value of
(7.5 - x)4 will always be non-negative, i.e. positive or zero. Thus, to minimize this outcome, we need to find the value of x for which (7.5 - x)^4 = 0.
(7.5 - x)^4 = 0
7.5 - x = 0
x = 7.5

Therefore, the metal construction will have minimal longevity for the value of x = 7.5, i.e. when the density of the underlying material will be equal to 7.5 g/cm3.

The correct answer is C.

Also, since the OP asked for multiple approaches, let me give you an easy (= mindless) one: try the answers.

Since l = (7.5 - x)� + 8.97 * 1.05, the only part we need to minimize is the unknown, (7.5 - x)�.

Plugging in the answers, we get

A:: 15�
B:: 7.5�
C:: 0�
D:: (-7.5)�
E:: (-67.5)�

C is clearly smallest, so we're set.