Amachine is set up to cut metal strips of varying lengths and widths based on the time (t) in minutes. the change in length is given by the function `l(t) = t^2 - sqrt(t)`, and the change in width is given by `w(t) = t^2 - 2t^(1/2)`. which function gives the change in area of the metal strips? a. `a(t) = t^4 + 2t` b. `a(t) = t^4 + 2t + 3t^(5/2)` c. `a(t) = t^4 - 3t^(5/2) + 2t` d. `a(t) = t^4 + 2t - 2t^(1/2) + sqrt(t)`

The change of area A(t) = t^4 - 3t^(5/2) + 2t ⇒ answer C

Step-by-step explanation:

* Lets study the problem

- The metal strip is in a shape of rectangle

- The change in length l(t) = t² - √t

- The change is the width w(t) = t² - 2t^1/2

* We must find function gives the change of area

∵ The area of the rectangle = length × width

∴ The change of rate of area A(t) = l(t) × w(t)

- We can write the √t in exponential form t^1/2

∴ l(t) = t² - t^1/2

∵ w(t) = t² - 2t^1/2

∵ A = l × w

∴ A(t) = l(t) × w(t)

∴ ⇒use the foil method

∴

- If we multiply two same numbers have exponents, then we add

 the power of them

∴

∴

* Now lets add the like terms

∴

* The change of area A(t) = t^4 - 3t^(5/2) + 2t

the answers are in the images, i received 100% on this assignment. hope this !