Question

A heated piece of metal cools according to the function c(x) = (.5)x − 7, 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 two hours?

-4 celsius
28 celsius
32 celsius
38 celsius

Answer 1

Option B is correct.

28 Celsius

Explanation:

As per the statement:

A heated piece of metal cools according to the function

`c(x) = (0.5)^(x-7)` where x is measured in hours.

It is also given that:

A device is added that aids in cooling according to the function h(x) = −x − 2.

then the resultant metal of function becomes:

`f(x) = c(x)+h(x)`

Substitute the functions:

`f(x) = (0.5)^(x-7) +(-x-2)` ......[1]

We have to find the temperature of the metal after two hours.

⇒x =2 hours

Substitute the value in [1] we have;

`f(2) = (0.5)^(2-7) +(-2-2)`

`f(2) = (0.5)^(-5) +(-4)`

⇒
`f(2) = 32 -4 = 28`

Therefore, the temperature of the metal after two hours will be 28 Celsius

Answer 2

Option (b) is correct.

The temperature of the metal after two hours is 28 Celsius

Explanation:

Given : A heated piece of metal cools according to the function
`c(x)=(0.5)^(x-7)` , where x is measured in hours and then a cooling device id added that aids in cooling according to the function
`h(x)=-x-2`

We have to determine the temperature of the metal after two hours.

Let w(x) denotes the temperature of metal .

Thus, w(x) can be given by function c(x) + h (x) = [c + h](x)

Thus,
`w(x)=[c+h](x)\\\\w(x)=(0.5)^(x-7)+(-x-2)\\\\` , where x is measured in hours.

Thus, after two hours that is when x = 2

the temperature of metal is given by w(2)

`w(x)=(0.5)^(x-7)+(-x-2)\\\\ w(2)=(0.5)^(2-7)+(-2-2)`

Solving , we get,

`w(2)=(0.5)^(2-7)+(-2-2)\\\\ w(2)=(0.5)^(-5)+(-4)\\\\ w(2)=32-4=28`

Thus, the temperature of the metal after two hours is 28 Celsius.

Hence, Option (b) is correct.