A heated piece of metal cools according to the function c(x)=(.5)^(x−9), where x is measured in hours. A device is added that aids in cooling according to the function h(x)=−x−2. What will be the temperature of the metal after five hours?

-7 Degrees Celsius
6 Degrees Celsius
9 Degrees Celsius
16 Degrees Celsius

The temp of the metal can be given by the function (c+h)(x).

(c+h)(x) = [(0.5)⁵⁻⁹] + [-(5) - 2]
(c+h)(x) = [16] + [-7]
(c+h)(x) = 9

Therefore, the temperature of the metal will be 9 degrees Celcius

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To find the temperature of the metal after five hours, we need to combine the cooling functions c(x) and h(x).

The function c(x) = (.5)^(x-9) represents the rate at which the metal cools over time. The exponent (x-9) indicates that the cooling starts 9 hours before x. The factor of (.5) represents the amount of cooling that occurs per hour.

The function h(x) = -x-2 represents the aid in cooling provided by the device. The negative sign indicates that it increases the rate of cooling. The exponent (-x-2) represents that the device starts cooling at x = -2 (two hours before the current time).

To combine the functions, we add them together:
c(x) + h(x) = (.5)^(x-9) + (-x-2)

Now, let's calculate the temperature after five hours by plugging in x = 5:
c(5) + h(5) = (.5)^(5-9) + (-5-2)
= (.5)^(-4) + (-7)
= 1/((.5)^4) - 7
= 1/(.0625) - 7
= 16 - 7
= 9

Therefore, the temperature of the metal after five hours is 9 degrees Celsius.