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)`

Accepted Solution

Answer:The change of area A(t) = t^4 - 3t^(5/2) + 2t ⇒ answer CStep-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)∴ [tex]A(t)=(t^{2}-t^{\frac{1}{2}})(t^{2}-2t^{\frac{1}{2}})[/tex]⇒use the foil method∴ [tex]A(t)=(t^{2})(t^{2})+(t^{2})(-2t^{\frac{1}{2}})+(-t^{\frac{1}{2}})(t^{2})+(-t^{\frac{1}{2}})(-2t^{\frac{1}{2}})[/tex]- If we multiply two same numbers have exponents, then we add  the power of them∴ [tex]A(t)=(t^{2+2})-2t^{2+\frac{1}{2}}-t^{\frac{1}{2}+2}+2t^{\frac{1}{2}+\frac{1}{2}}[/tex]∴ [tex]A(t)=t^{4}-2t^{\frac{5}{2}}-t^{\frac{5}{2}}+2t[/tex]* Now lets add the like terms∴ [tex]A(t)=t^{4}-3t^{\frac{5}{2}}+2t[/tex]* The change of area A(t) = t^4 - 3t^(5/2) + 2t