Answer:
[tex]\displaystyle{\int^{\infty}_0 {50e^{-0.04t}}\, dt}=1250[/tex]
Step-by-step explanation:
Given:
[tex]T =\displaystyle{\int^{\infty}_0 {Pe^{-kt}} \, dt}[/tex]
where,
T = total amount of waste
P = 50, the initial rate
k = 0.04
t = time
[tex]T =\displaystyle{\int^{\infty}_0 {50e^{-0.04t}} \, dt}[/tex]
now we need to solve this integral!
[tex]T =\displaystyle{50\int^{\infty}_0 {e^{-0.04t}} \, dt}[/tex]
[tex]T = \left|50\left(\dfrac{e^{-0.04t}}{-0.04}\right)\right|^{\infty}_0[/tex]
[tex]T = \left|-1250e^{-0.04t}\right|^{\infty}_0[/tex]
[tex]T = (-1250e^{-0.04(\infty)})-(-1250e^{-0.04(0)})[/tex]
when any number has a power of negative infinity it is 0. because: [tex]a^-{\infty} = \dfrac{1}{a^{\infty}} = \dfrac{1}{\infty} = 0[/tex], like something being divided by a very large number!
[tex]T = (-1250(0))-(-1250e^0)[/tex]
[tex]T = 1250[/tex]
this is the total amount of waste
