When is the level of oxygen content lowest? When organic waste is dumped into a

Question

When is the level of oxygen content lowest?

When organic waste is dumped into a pond, the oxidation process that takes place reduces the pond's oxygen content. However, given time, nature will restore the oxygen content to its natural level. Suppose the oxygen content, as a percent of its normal level, t days after organic waste has been dumped into the pond is given by: f(t) = 100 [t^2 - 4t + 4/t^2 + 4], 0 Lessthanorequalto t Lessthanorequalto infinity When is the level of oxygen content lowest?

Explanation / Answer

The given f(t)=100*{(t^2-4t+4)/(t^2+4)},0< or equal) t < or equal infinite

differentiating with respect to t

f'(t)=100*{(t^2+4)(2t-4)-(t^2-4t+4)*(2t)/(t^2+4)^2}

f'(t)={(400(t^2-4))/(t^2+4)^2}

We notic that this derivative exists for all of our values of t.

derivative set to zero

f'(t)={(400(t^2-4))/(t^2+4)^2}=0

by solving the above equation we get t=(+ or -) 2

when t=-2 this is the out of the bounds our situation.so we eliminate t=-2

Throw in our endpoint as a possible location for an extreme value and we get that

f(0)=100*{(0^2-4(0)+4)/(0^2+4)}=(4*100)/4=100

Oxygen is at 100 percent of its normal level at t=0 ,When the waste is dumped

f(2)=100*{(2^2-4(2)+4)/(2^2+4)}=100{0/8}=0

There is no absolutely no oxygen two days after the dumping and can't go any lower than that.

when t=2, f(2)=0 this is the level of oxygen content lowest

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