Calculus Help Urgent!?

Question

The population P(t) of a city in 1996 is given by P(t) = 0.00005t^2 + 0.01t + 1, where t denotes the number of year since 1990. 

a) How large is the population in 1996 and how fast is it growing that year? 

b) At what year did the population grow at a rate of 12,000 people per year? 

Answer 1

(b).

The equation is P(t) = 0.00005t² + 0.01t + 1.

Differentiate with respect to t.

P'(t) = 0.0001t + 0.01

1 million = 1,000,000 ⇒ 1 = 1 million/1,000,000 ⇒ 12,000 = 12,000 million/1,000,000 = 0.012.

Therefore, P'(t) = 0.012.

0.0001t + 0.01 = 0.012

0.0001t = 0.002

t = 0.002/0.0001 = 20 years.

The difference between 1996 and 1990 is 6 years.

Therefore, t = 26 tears and the year is 1996 + 26 = 2022.

Answer 2

(b).

The equation is P(t) = 0.00005t² + 0.01t + 1.

Differentiate with respect to t.

P'(t) = 0.0001t + 0.01

1 million = 1,000,000 ⇒ 1 = 1 million/1,000,000 ⇒ 12,000 = 12,000 million/1,000,000 = 0.012.

Therefore, P'(t) = 0.012.

0.0001t + 0.01 = 0.012

0.0001t = 0.002

t = 0.002/0.0001 = 20 years.

The difference between 1996 and 1990 is 6 years.

Therefore, t = 26 tears and the year is 1996 + 26 = 2022.

Answer 3

(a).

The equation is P(t) = 0.00005t² + 0.01t + 1.

The difference between 1996 and 1990 is 6 years.

Substitute t = 6 in the original equation.

P(6) = 0.00005(6)² + 0.01(6) + 1 = 1.0618 millions.

Differentiate with respect to t to the original equation.

P'(t) = 0.0001t + 0.01

Substitute t = 6 in the above equation.

P'(6) = 0.0001(6) + 0.01 = 0.0106 million per year.