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I will answer part (a) for 15 days and part (b).

You can solve part (a) for 25 days following my steps.

Ready?

We are given the function:

p(t) = 100[3^(t/10)]

We let t = 15 days and simplify.

P = population.

Then:

p(15 days) = 100[3^(15/10)]

p(15 days) = 100[5.196152423]

p(15 days) = 519.6152423, which can rounded off to the nearest hundredths and becomes 519.62. This is the population size in 15 days.

Do the same for 25 days.

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Part (b)

For part (b), we have to solve for t.

Let p(t) = 2500

We then have this equation:

2500 = 100[3^(t/10)]

To solve for t, start by dividing both sides of the equation by 100.

Doing so we get this:

25 = 3^(t/10)

We now take the natural log of both sides of the equation to find t.

Let LN = natural log.

LN(25) = LN3(t/10)

Divide both sides by LN(3) as step one.

LN(25) divided by LN(3) = (t/10)

2.9299470441 = (t/10)

To find t, multiply both sides of the equation by 10.

2.9299470441 times 10 = (t/10) times 10

29.29947041 = t....This can be rounded off to the nearest tens and it becomes about 30 days. When will the population reach 2500? In about 30 days.

1. Since the coefficients of the polynomial are real ..complex roots will exist in conjugate pairs. hence another root is 2+i

Hence the polynomial should be divisible by a = (x-(2-i))*(x-(2+i)) that is it should be divisible by a = x^2 -4x+5

The polynomial can be rewritten as = x^2(x^2 -4x +5) +x^2-4x+5

=( x^2 -4x +5)(x^2 +1)

Hence the other roots are +i and -i

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