fish population: the fish population in a certain lake rises and falls according to the formula F=1000(30+17t-t^2) , Here F is the number of fish at time t, where t is measured in years since January 1, 2002, when the fish population was first estimated. a) On what date will the fish population again be the same as it was on January 1, 2002? b) By what date will all the fish in the lake have died?

i badly needed an answer for this...

To find the date when the fish population will again be the same as it was on January 1, 2002, we need to set the equation equal to the initial fish population.

a) On what date will the fish population again be the same as it was on January 1, 2002?

Step 1: Set F = 0 (the initial fish population on January 1, 2002).

0 = 1000(30 + 17t - t^2)

Step 2: Solve for t by factoring or using the quadratic formula.

1000(30 + 17t - t^2) = 0

Now, either the first factor (1000) can be zero, which means F would be zero before January 1, 2002, which would not make sense in this context. So, we consider the second factor.

30 + 17t - t^2 = 0

Step 3: Solve the quadratic equation.

You can either factor it or use the quadratic formula.

Factoring:
(30 - t)(t + 1) = 0

This gives two possible values for t: t = 30 and t = -1.

But since t is measured in years since January 1, 2002, a negative value for t is not applicable in this case.

Therefore, t = 30 is the correct value.

Step 4: Find the date.
Since t is measured in years, t = 30 represents 30 years after January 1, 2002.

Hence, on January 1, 2002 + 30 years, the fish population will again be the same as it was on January 1, 2002.

The date will be January 1, 2002 + 30 years = January 1, 2032.

b) By what date will all the fish in the lake have died?

To find the date when all the fish in the lake have died, we need to find when the fish population, F, reaches zero.

Step 1: Set F = 0.

0 = 1000(30 + 17t - t^2)

Step 2: Solve for t by factoring or using the quadratic formula.

30 + 17t - t^2 = 0

Step 3: Solve the quadratic equation.

You can either factor it or use the quadratic formula.

Factoring:
(30 - t)(t + 1) = 0

This gives two possible values for t: t = 30 and t = -1.

Since we are looking for the time when the fish have died, we consider t = 30, as negative values for t are not applicable in this case.

Step 4: Find the date.
Since t is measured in years, t = 30 represents 30 years after January 1, 2002.

Hence, the fish population will reach zero on January 1, 2002 + 30 years.

The date will be January 1, 2002 + 30 years = January 1, 2032.

Therefore, all the fish in the lake will have died by January 1, 2032.