i am having trouble with a graphing calculator please help with these two questions

1. a census was taken from 2012 to 2020. the data collected is given in the following table. 2012-2020
L1 l2
0 6000
1 6300
2 6615
3 7012
4 7362
5 7878
6 8429
7 9272
8 10199

a) what is the exponentail equation for the data (is it P=pe(rt)

B) Using your calculator determine the predidcted annual population to the nearest whole number in the year 2030.

2. A student fires a model rocket into the air. The rocket reaches a maximum height of 24.8 meters and lands 55 meters from the student. (assume parabola)
a) list thress points on the trajectory
b) determine the regression equation that best fits the trajectory using your graphing calculator.

I have spent the last two hours on this please help me walk through this as i really just want to know how to do this

Yes, the general equation is P=pe^(rt), or P=p*k^t

If that's the equation, then the terms form a geometric sequence, since each is k times the one before. Let's examine the ratios:

6300/6000 = 1.05
6615/6300 = 1.05
...

Looks like P = 6000 * 1.05^t

If ou want it in terms of e^x, then since 1.05 = e^(ln 1.05)

P = 6000 * e^(ln 1.05 * t) = 6000*e^(.049t)

where t is years since 2012

so i came up with 14440 people rounded to the nearest whole number.

is there a way to check the formula by using the previous information?

Sure, I can help you with both of these questions. Let's start with the first one.

1. To find the exponential equation for the data, we need to determine if the given data follows an exponential growth pattern of the form P = P₀e^(rt), where P is the population, P₀ is the initial population, r is the growth rate, and t is the time.

a) To check if the data follows an exponential growth pattern, we can plot the data points on a graphing calculator and see if they form a curve that resembles exponential growth. Here's how you can do it:

- Enter the data into two lists on your calculator. Let's say the years are in L₁ and the population values are in L₂.
- Go to the graphing feature on your calculator and plot the points (L₁, L₂). Make sure to select the appropriate graphing mode (scatter plot or xy-line plot) to view the individual data points.
- If the plotted points form an exponential curve, then an exponential equation may be a good fit for the data.

b) To predict the annual population in the year 2030, you can use the exponential equation obtained in part a) by substituting the value of t (time) as 2030 - 2012 = 18 (since the given data ranges from 2012 to 2020).

Since we don't have the values of P₀ and r yet, we can't calculate the exact population. However, you can use your calculator's exponential regression function to find the values of P₀ and r that best fit the data. Once you have these values, you can substitute them into the exponential equation to find the predicted population for the year 2030.

Now, let's move on to the second question.

2. To find the regression equation for the rocket trajectory, we need to determine if the trajectory follows a parabolic path. Here's how you can do it:

a) To list three points on the trajectory, you can choose any three data points from the given information. Let's select the maximum height point and two other points on the descending part of the trajectory. For example:
- Point 1: (-55, 0) - representing the starting point
- Point 2: (0, 24.8) - representing the maximum height
- Point 3: (55, 0) - representing the landing point

b) To determine the regression equation that best fits the trajectory, you can use your calculator's quadratic regression function. Follow these steps:

- Enter the three selected points into two lists on your calculator, where the x-coordinates are in L₁ and the y-coordinates are in L₂.
- Go to the regression feature on your calculator and choose the quadratic regression option.
- Select the appropriate lists (L₁, L₂) and calculate the regression equation.
- Once you have the quadratic regression equation, it will represent the best fit of the trajectory based on the given data points.

I hope these explanations help you understand the approach to solve these questions. If you have any further questions, feel free to ask!