A high school had 1200 students enrolled in 2003 and 1500 students in 2006. If the student population P; grows a linear function of time t, where t is the number of years after 2003.

Question

A high school had 1200 students enrolled in 2003 and 1500 students in 2006. If the student population P; grows a linear function of time t, where t is the number of years after 2003.

a. How many students will be enrolled in the school in 2010
b. Find a linear function that relates the student population to the time t.

Answer 1

This is just a linear function of the type y = mx + b

let's assume that 2000 corresponds with 0, so we have the two points:
(3,1200) and (6,1500). You want (10, ?)

slope = (1500-1200)/(6-3) = 300/3 = 100

so P = 100t + b
use the first point
1200 = 100(3) + b
b = 900

P = 100t + 900 , where P is the population and t is the year after 2000

so plug in t=10 for the other part of the question

Answer 2

To find the linear function that relates the student population to the time t, we can use the formula for the equation of a line: y = mx + b, where y is the student population, x is the time (t), m is the slope, and b is the y-intercept.

Step 1: Determine the slope (m)
We can calculate the slope using the given data points:
m = (change in y) / (change in x) = (1500 - 1200) / (2006 - 2003)
m = 300 / 3
m = 100

Step 2: Determine the y-intercept (b)
We know that in 2003 (when t = 0), the student population was 1200. So, b = 1200.

The linear function that relates the student population (P) to the time (t) is:
P = 100t + 1200

a. To determine the number of students enrolled in the school in 2010 (t = 2010 - 2003 = 7), substitute t = 7 into the linear function:
P = 100(7) + 1200
P = 700 + 1200
P = 1900

Therefore, there will be 1900 students enrolled in the school in 2010.

Answer 3

To answer these questions, we will use the given information about the student population growth as a linear function of time. Let's begin:

a. To determine the number of students enrolled in the school in 2010, we need to find the value of P for t = 2010 - 2003 = 7 years.

To find the student population for a given year, we can use the equation of a straight line, which can be written as:
P = mt + b

Now, let's use the information we have:

Year 2003: P = 1200 students
This corresponds to t = 0 years. So, we can write:
1200 = m(0) + b ---(1)

Year 2006: P = 1500 students
This corresponds to t = 3 years. So, we can write:
1500 = m(3) + b ---(2)

Now, we have a system of two linear equations with two unknowns (m and b). We can solve these equations simultaneously to find their values.

Subtract equation (1) from equation (2) to eliminate b:
1500 - 1200 = 3m - 0m + b - b
300 = 3m

Simplifying, we find:
m = 100

Now, substitute the value of m into equation (1):
1200 = 0(100) + b
1200 = b

Therefore, we have the equation for the linear relationship between student population P and time t:
P = 100t + 1200

To find the number of students enrolled in the school in 2010 (t = 7 years), substitute t = 7 into the equation:
P = 100(7) + 1200
P = 700 + 1200
P = 1900

So, there will be 1900 students enrolled in the school in 2010.

b. The linear function that relates the student population P to the time t is:
P = 100t + 1200.