more math, thanks for all your help?

Question

Suppose that the population of a town is described by P=0.16t² +7.2t +100, where P is the population in thousands and t is the time in years, with t=0 representing the year 2000.

1.what will the population be in 2010
2.what was the population in 1995
3.when will the population reach 52,000
4.will the population ever reach zero under this model, explain

no sarcatic answers please. we don't all get it...

Answer 1

Given: P = 0∙16t² + 7∙2t + 100

(1) t = 0 represents the year 2000.
So t = 10 in the year 2010.
Substitute the t value into the equation.
P = 0∙16t² + 7∙2t + 100
P = 0∙16(10)² + 7∙2(10) + 100
P = 16 + 7∙2 + 100
P = 188 thousand.


(2) t = 0 represents the year 2000.
So t = -5 in the year 1995.
Substitute the t value into the equation.
P = 0∙16t² + 7∙2t + 100
P = 0∙16(-5)² + 7∙2(-5) + 100
P = 4 - 36 + 100
P = 68 thousand.


(3) Since P is in thousands, P = 52.
Substitute the t value into the equation.
P = 0∙16t² + 7∙2t + 100
52 = 0∙16t² + 7∙2t + 100
0∙16t² + 7∙2t + 100 - 52 = 0
0∙16t² + 7∙2t + 48 = 0

Use quadric equation: [ -b ± √(b)² - 4ac ] / 2c
[ -(7∙2) ± √(7∙2)² - 4(0∙16)(48) ] / 2(0∙16)
t = -8∙13589.. (or) t = -36∙8614...

So the year 2000 - 8∙13589.. = 1991∙8641....yr.
and 2000 - 36∙8614... = 1963∙138....yr.

(4)
Once you pass the year 2000, the numbers will all be positive, so the population will always grow.
If the population ever reaches zero, that's the end of the population, it can't increase.

Let the population be equal to zero. The equation becomes:
P = 0∙16t² + 7∙2t + 100
0 = 0∙16t² + 7∙2t + 100
Solve for t with the formula [ -b ± √(b)² - 4ac ] / 2c.
The square root part of the formula is minus. That is, imaginary roots. This suggest that the population will never reach zero.

Answer 2

1. Pop in 2010. Put 10 in for t

.16 * 10^2 + 7.2 * 10 + 100
16+72+100 = 188 (in thousands)

2. t = -5

.16 (-5)^2 + 7.2(-5) + 100
4+(-36)+100 = 68

3. Put in 52 for P, solve for t

52 = ...

Subtract 52 from both sides

0 = .16t^2 + 7.2t + 48
Too lazy to figure if it factors. Use the quadratic equation.

4. No. from year 0 (2000) the pop is at least 48,000, from then it goes up.

Answer 3

Questions 1&2, substitute t=2010 and t=1995 in your equation and then you get the answer.

Question 3, set P=52,000. This is a quadratic equation. Not hard to solve. Remeber you are solving for t!

Question 4, is a litlle trickier...but I'll give you a hint, t will never be less than zero. If you substitue T=0 into the equation, you will see that the P is never less than zero as well.

Give me your answers and I will tell you if you are right!