Question

A machine 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)`

Answer 1

the answer is C

Explanation:

Answer 2

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

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)

∴
`A(t)=(t^(2)-t^{(1)/(2)})(t^(2)-2t^{(1)/(2)})`⇒use the foil method

∴
`A(t)=(t^(2))(t^(2))+(t^(2))(-2t^{(1)/(2)})+(-t^{(1)/(2)})(t^(2))+(-t^{(1)/(2)})(-2t^{(1)/(2)})`

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

the power of them

∴
`A(t)=(t^(2+2))-2t^{2+(1)/(2)}-t^{(1)/(2)+2}+2t^{(1)/(2)+(1)/(2)}`

∴
`A(t)=t^(4)-2t^{(5)/(2)}-t^{(5)/(2)}+2t`

* Now lets add the like terms

∴
`A(t)=t^(4)-3t^{(5)/(2)}+2t`

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