Answer:
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)
∴ [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
