The restorative power of sleep is graphically approximated by the function −x^2 + 16x + 36, where the x-axis measures sleeping hours and the y-axis measures the restoration value. After how many hours does sleep no longer perform restorative duties according to the function?
A. −2
B. 4
C. 8
D. 13
E. 18
OA E
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The restorative power of sleep is graphically approximated b
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BTGmoderatorDC
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Multiplying the equation by -1, we get:BTGmoderatorDC wrote:The restorative power of sleep is graphically approximated by the function −x^2 + 16x + 36, where the x-axis measures sleeping hours and the y-axis measures the restoration value. After how many hours does sleep no longer perform restorative duties according to the function?
A. −2
B. 4
C. 8
D. 13
E. 18
y = x² - 16x - 36
y = (x-18)(x+2)
The restorative power -- the value of y -- is equal to 0 when x=18 or x=-2.
Since the time cannot be a negative value, only x = 18 hours is viable.
The correct answer is E.
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The restorative power is given by the equation \(-x^{2}+16x+36\), which is equal to \(x^2-16x-36\). We are told that the restoration value is in the y-axis, and we want to find out after how much sleep we would not get any restoration for our sleep.
So if y becomes 0, we have \(x^2-16x-36=0\).
or
\((x-18)(x+2)=0\)
Which means that \(x=18\) or \(x=-2\)
Since we are counting hours of sleep, the answer cannot be negative, thus \(x=18\) and the answer is __E__.
So if y becomes 0, we have \(x^2-16x-36=0\).
or
\((x-18)(x+2)=0\)
Which means that \(x=18\) or \(x=-2\)
Since we are counting hours of sleep, the answer cannot be negative, thus \(x=18\) and the answer is __E__.
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Sleep no longer performs any restorative duties if the value of the function is below 0. So let's set the function equal to 0 and solve for x:BTGmoderatorDC wrote:The restorative power of sleep is graphically approximated by the function −x^2 + 16x + 36, where the x-axis measures sleeping hours and the y-axis measures the restoration value. After how many hours does sleep no longer perform restorative duties according to the function?
A. −2
B. 4
C. 8
D. 13
E. 18
OA E
Source: Veritas Prep
-x^2 + 16x + 36 = 0
x^2 - 16x - 36 = 0
(x - 18)(x + 2) = 0
x = 18 or x = -2
We see that sleep performs a restorative duty when x is between -2 and 18 (just substitute any value between -2 and 18 into the function, and the function value will be positive). Therefore, sleep will not perform a restorative duty after 18 hours (that is, substitute any value greater than 18 into the function, and the function value will be negative).
Answer: E
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