An insect population P in a certain area flunctuates with the seasons.?

Question

It is estimated that P=17,000+4,500sin pi t /52, where t is given in weeks. Determine the number of weeks it would take for the population to initially reach 20,000.

a. 12 weeks
b. 692 weeks
c. 42 weeks

Answer 1

We know that at time t=0, the population is P = 17,000 + 4,500sin pi t /52 = 17,000 + sin 0 = 17,000. We also know that sin x increases from 0 at x=0 to a local maximum of 1 at x=pi/2. So when t=26, we have P = 17,000 + 4,500sin pi t /52 = 17,000 + 4,500sin pi 26/52 = = 17,000 + 4,500sin pi/2 = 17,000 + 4,500 = 21,500. And since P is a composition of continuous functions, it is itself continuous, so by the intermediate value theorem, there must be some t between 0 and 26 for which P=20,000. (Don't worry about that part if you've never had calculus; the idea is that since P goes from 17,000 to 21,500 as t goes from 0 to 26, P must be cross 20,000 at some point between t=0 and t=26.)

So we know that P first reaches 20,000 sometime in the first 26 weeks, so the answer must be a: 12 weeks.

Answer 2

For population, P to reach 20000,
20000 = 17000+4500sin pi t/52
3000 = 4500sin pi t/52
3000/4500 = sin pi t/52
2/3 = sin pi t/52
After it is simplified, just substiute 12, 694 and 42 into the equation.
If 12 is used, sin pi 12/52 will result in 2/3. Therefore that will be your answer.

Answer 3

P=17,000+4,500sin( pi t /52)
plug in 20000 for p
20000 = 17,000+4,500sin( pi t /52)
So now you need to solve for t. Isolate the t by getting sin( pi t /52) by itself
Subtract 17000 from both sides
3000 = 4,500sin( pi t /52)
Divide both sides by 4500
2/3 = sin( pi t /52)
Take the inverse sin of both sides sin-1 on your calculator
0.7297 = pi t /52
Multiply both sides by 52
37.95 = pi t
Divide both sides by pi
t = 12.07
so the answer is 12

Answer 4

P = 17,000 + 4,500 sin(pi t/52) from the question
We have P = 20,000 and need to determine t. Then,
20,000 = 17,000 + 4,500 sin (pi t/52)
We got an equation with one unknown (t). Subtract 17,000 from both sides:
3,000 = 4,500 sin (pi t/52)
devide both sides over 4,500 gives:
0.6667 = sin (pi t/52)
We use the sine inverse to know the angel whose sin is .6667
--> pi t/52 = arcsin (.6667) = 41.82 degrees
we need to change the angle to radians
pi t/52 = 41.82* pi / 180 rads
t = 41.82 * 52 / 180 = 12.083 weeks or (a).

Note: if the angel is inside a trigonometric function (sin, cos) we can put it in degrees. If it's outside, then we have to use radians.

Answer 5

Hi,

If you graph Y1 = 17000 + 4500 sin(Pi(x/52)) and Y2 = 20000, you can look for their first intersection on your calculator. I used a window of xmin=0,xmax=104, ymin=13000, ymax=23000 so I can see their intersection. It happens when x = 12.0785, so answer a is correct.

I hope that helps!! :-)

Answer 6

20000=17000+4500sin(πt/52)
3000=4500sin(πt/52)
3000/4500=sin(πt/52)
arcsin(2/3)=.72972Rad=πt/52
52*.7292=πt
37.9184/π=12.07=t
I'm guessing that the unit of t is weeks ☺

HTH

Doug