The restorative power of sleep is graphically approximated by the function andminus;x2 + 16x + 36, where the x-axis measures sleeping hours and the y-axis measure the restoration value. After how many hours does sleep no longer perform restorative duties according to the function?a)andminus;2b)4c)8d)13e)18Correct answer is option 'E'. Can you explain this answer?

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The restorative power of sleep is graphically approximated by the function −x² + 16x + 36, where the x-axis measures sleeping hours and the y-axis measure 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

Correct answer is option 'E'. Can you explain this answer?

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The restorative power of sleep is graphically approximated by the func...

To determine when sleep no longer performs restorative duties according to the function −x² + 16x + 36, we need to find the x-coordinate of the vertex.

The x-coordinate of the vertex of a quadratic function in the form ax² + bx + c can be found using the formula:

x = -b / (2a)

In this case, the function is −x² + 16x + 36, so a = -1, b = 16, and c = 36.

Using the formula, we have:

x = -16 / (2*(-1))

x = -16 / (-2)

x = 8

Therefore, the vertex of the function occurs at x = 8. However, we need to determine after how many hours sleep no longer performs restorative duties. Since the function is a downward-facing parabola, the restoration value decreases as x increases beyond the vertex.

As x increases beyond 8, the restoration value will continue to decrease. We can observe this by calculating the restoration value at x = 8 and x = 9:

At x = 8:
Restoration value = -8² + 16(8) + 36 = -64 + 128 + 36 = 100

At x = 9:
Restoration value = -9² + 16(9) + 36 = -81 + 144 + 36 = 99

As x increases further, the restoration value will continue to decrease. Therefore, after 8 hours of sleep, according to the function, sleep no longer performs restorative duties.

Hence, the correct answer is E: 18.

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The restorative power of sleep is graphically approximated by the func...

To determine when sleep no longer performs restorative duties according to the function −x² + 16x + 36, we need to find the x-coordinate of the vertex.

The x-coordinate of the vertex of a quadratic function in the form ax² + bx + c can be found using the formula:

x = -b / (2a)

In this case, the function is −x² + 16x + 36, so a = -1, b = 16, and c = 36.

Using the formula, we have:

x = -16 / (2*(-1))

x = -16 / (-2)

x = 8

As x increases beyond 8, the restoration value will continue to decrease. We can observe this by calculating the restoration value at x = 8 and x = 9:

At x = 8:
Restoration value = -8² + 16(8) + 36 = -64 + 128 + 36 = 100

At x = 9:
Restoration value = -9² + 16(9) + 36 = -81 + 144 + 36 = 99

As x increases further, the restoration value will continue to decrease. Therefore, after 8 hours of sleep, according to the function, sleep no longer performs restorative duties.

Hence, the correct answer is E: 18.

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The restorative power of sleep is graphically approximated by the function −x2 + 16x + 36, where the x-axis measures sleeping hours and the y-axis measure the restoration value. After how many hours does sleep no longer perform restorative duties according to the function?a)−2b)4c)8d)13e)18Correct answer is option 'E'. Can you explain this answer?

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