Two separate rabbit populations are observed for 80 weeks, starting at the same time and with the same initial populations. The growth rates of two rabbit populations are modeled as follows, where t =0 corresponds to the beginning of the observation period:
r1(t)=4sin((2pi/72)(t)) +.1t+1, where r(1) is rabbits/week, t is time in weeks
r2(t)=t^(1/2) , where r (2) is rabbits per week, t is time in weeks
a. Using your calculator, find (approximately) the first positive time t for which the rates of
growth for the two populations are the same
b. What’s the physical significance of the area between the two curves from time t = 0 until
the first time where the two rates are the same? What does the area represent?
c. Suppose you want to find the first time (call it T) after the beginning of the observation
period at which the two rabbit populations have identical populations. Write an equation to solve for the unknown variable T.
d.Simplify your equation from part C until you can use your calculator on it. Then use your calculator to solve this equation for T.
a. t=31.718 weeks (where the two lines intersect)
b. this is basically r1's accumulated population-r2's accumulated population, or how many more rabbits r1 has accumulated over the period of time.
a. To find the first positive time when the growth rates are the same, we need to set the two growth rate equations equal to each other and solve for t:
4sin((2pi/72)(t)) + 0.1t + 1 = t^(1/2)
Since this equation involves both a trigonometric function and a square root, it is not possible to find an exact solution using a calculator. However, we can use a numerical method like trial and error or graphing to find an approximate solution.
b. The area between the two curves from time t = 0 until the first time where the growth rates are the same represents the difference in the total number of rabbits between the two populations during that time period. It basically measures the "extra" rabbits that one population has compared to the other during that time interval.
c. To find the first time T when the two rabbit populations have identical populations, we need to set the two population equations equal to each other and solve for T:
4sin((2pi/72)(t)) + 0.1t + 1 = t^(1/2)
d. Simplifying the equation further depends on the specific requirements of "using your calculator" mentioned in part d. The method could vary depending on the calculator's capabilities or the software used.
Once the equation is simplified for calculator use, you can input the equation into your calculator and use numerical methods to solve it. One common method is to use the graphing feature of the calculator to find the intersection point of the two curves.
a. To find the first positive time t for which the growth rates for the two populations are the same, we need to set r1(t) equal to r2(t) and solve for t. So,
4sin((2pi/72)(t)) + 0.1t + 1 = t^(1/2)
Unfortunately, we cannot solve this equation analytically. We will need to use a calculator or computer to find an approximate solution.
b. The physical significance of the area between the two curves from time t = 0 until the first time where the two rates are the same is the difference in the populations of the two rabbit populations over that period of time. It represents the cumulative difference in population growth between the two populations.
c. To find the first time T after the beginning of the observation period at which the two rabbit populations have identical populations, we need to set r1(T) equal to r2(T) and solve for T. So,
4sin((2pi/72)(T)) + 0.1T + 1 = T^(1/2)
d. To simplify the equation from part c, we can square both sides to eliminate the square root:
(4sin((2pi/72)(T)) + 0.1T + 1)^2 = T
Expanding and rearranging the equation:
16sin^2((2pi/72)(T)) + 0.2Tsin((2pi/72)(T)) + 0.01T^2 + 8sin((2pi/72)(T)) + 0.2T + 2 = T
16sin^2((2pi/72)(T)) + 0.2Tsin((2pi/72)(T)) + 0.01T^2 + 8sin((2pi/72)(T)) + 0.2T + 2 - T = 0
Next, we can multiply the entire equation by 100 to eliminate the decimal:
1600sin^2((2pi/72)(T)) + 20Tsin((2pi/72)(T)) + T^2 + 800sin((2pi/72)(T)) + 20T + 200 - 100T = 0
Rearranging and simplifying:
1600sin^2((2pi/72)(T)) + (20Tsin((2pi/72)(T)) + 800sin((2pi/72)(T))) + (T^2 + 20T + 200) - 100T = 0
Finally, we have a simplified equation that we can use a calculator to solve for T.
a. To find the first positive time t for which the rates of growth for the two populations are the same, we need to set the growth equations equal to each other and solve for t.
4sin((2π/72)(t)) + 0.1t + 1 = t^(1/2)
We can solve this equation using a graphing calculator or an online graphing tool that allows us to find the points of intersection between two graphs. Here's how you can use a graphing calculator to solve it:
1. Enter the first equation: Y1 = 4sin((2π/72)(X)) + 0.1X + 1
2. Enter the second equation: Y2 = X^(1/2)
3. Graph the two equations on the same coordinate plane.
4. Use the intersection feature of your graphing calculator to find the x-coordinate of the first positive intersection point.
- Usually, you can find the intersection feature by pressing the "2nd" and then "Trace" or "Calc" button on your calculator.
- In some calculators, you might need to adjust the window settings or zoom in to see the intersection point clearly.
5. The x-coordinate of the intersection point is the first positive time t for which the growth rates of the two populations are the same.
b. The area between the two curves from time t = 0 until the first time where the two rates are the same represents the difference in population growth between the two rabbit populations during that time interval. It signifies the cumulative excess or deficit of one population's growth compared to the other population.
c. To find the first time T after the beginning of the observation period at which the two rabbit populations have identical populations, we need to set the growth equations equal to each other and solve for T.
4sin((2π/72)(T)) + 0.1T + 1 = T^(1/2)
d. To simplify the equation from part C, we can square both sides of the equation to eliminate the square root:
(4sin((2π/72)(T)) + 0.1T + 1)^2 = (T^(1/2))^2
Simplifying further:
(4sin((2π/72)(T)) + 0.1T + 1)^2 = T
Now, you can use a calculator to solve this equation for T. Enter the equation into the calculator and use a numerical solver or the equation-solving feature to find the value of T that makes the equation true. The calculator will provide you with an approximate value for T.