A heated piece of metal cools according to the function c(x) = (.5)x − 11, where x is measured in hours. A device is added that aids in cooling according to the function h(x) = −x − 3. What will be the temperature of the metal after five hours?
To find the temperature of the metal after five hours, we need to evaluate the combined cooling function c(x) + h(x) at x = 5.
First, let's start with the original cooling function c(x) = (.5)x - 11:
c(x) = (.5)x - 11
Next, let's consider the additional cooling function h(x) = -x - 3:
h(x) = -x - 3
To find the combined cooling function, we add the two functions together:
c(x) + h(x) = (.5)x - 11 + (-x - 3)
= .5x - x - 11 - 3
= -.5x - 14
Now, let's substitute x = 5 into the combined cooling function:
c(5) + h(5) = -(.5)(5) - 14
= -2.5 - 14
= -16.5
Therefore, the temperature of the metal after five hours will be -16.5 degrees.
To find the temperature of the metal after five hours, we need to calculate the sum of the cooling effect from the heated piece of metal (c(x)) and the cooling effect from the added device (h(x)) after five hours.
First, let's evaluate c(x) = (.5)x − 11 for x = 5 hours:
c(5) = (.5)(5) − 11
c(5) = 2.5 - 11
c(5) = -8.5
Next, let's evaluate h(x) = −x − 3 for x = 5 hours:
h(5) = -(5) - 3
h(5) = -5 - 3
h(5) = -8
Now, let's find the total cooling effect after five hours by adding c(x) and h(x):
Total cooling effect = c(5) + h(5)
Total cooling effect = -8.5 + (-8)
Total cooling effect = -16.5
Therefore, the temperature of the metal after five hours will be -16.5.
I don't believe that c(x) is correct as stated. It means that c(x) increases with time, not likely for a cooling piece of metal.
So, either c(x) does not represent the metal's temperature after x hours, or it indicates how many degrees it has cooled. In either case, c(x) and h(x) appear to oppose each other.