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
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vivecan2005
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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.
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.
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vivecan2005 wrote:The longevity of a certain metal construction is determined by the following formula:
L = (7.5 - x)� + 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
Plug in c = 1.05 to get: L = (7.5 - x)� + 8.97(1.05)
Simplify: L = (7.5 - x)� + some positive number
Our goal is to MINIMIZE the value of L
To do this, we must MINIMIZE the value of (7.5 - x)�
Since we exponent is EVEN, we know that (7.5 - x)� will be greater than or equal to zero for all values of x.
So, the MINIMUM value of (7.5 - x)� is ZERO
This occurs when x = 7.5
Answer: C
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