A town has population 225 people at year t=0. Write a formula for the population, P, in year t.
a) grows by 10% per year
I don't understand how to do this, and keep getting the wrong answer.
Also, Find a formula for the linear function q(x) whose graph intersects the graph of y=6000e−x/20 at x=20, x=120.
I have no clue to even attempt this. Thank you!
P = 225 * 1.1^t
Explanation: Every year, the population grows by 10%, so you multiply by 110% for the first year, 110%*110% for the second year, etc.
Evaluate y=6000e−x/20 at x = 20 and at x = 120
y(20)=6000e−20/20 = 6000 * e^-1 = 2207
y(120)=6000e−120/20 = 6000 * e^-6 = 14.87
So find a linear equation through the two points (20, 2207) and (120, 14.87)
which is of the form y = mx + b, where m is the slope, and b is the y intercept
The slope is (14.87-2207)/(120-20)
The y intercept is found by plugging one point into the equation and solving for b
2207 = ((14.87-2207)/(120-20))*20 + b
Solve for b and you have found your equation
To find the formula for the population, P, in year t, given that it grows by 10% per year, we can use the formula for compound interest.
The formula for compound interest is:
A = P(1 + r/n)^(nt)
Where:
A is the final amount (population) after time t,
P is the initial amount (population) at t=0,
r is the growth rate expressed as a decimal (e.g., 10% = 0.10),
n is the number of times interest is compounded per time period,
t is the number of time periods.
In this case, the growth rate is 10% per year, expressed as 0.10. Since we're looking for the population at a specific year t, we can set the initial population P to 225.
Therefore, the formula for the population P in year t, growing by 10% per year, is:
P = 225 * (1 + 0.10)^t
As for the second question regarding the linear function q(x) that intersects the graph of y = 6000e^(-x/20) at x = 20 and x = 120, we first need to find the y-values at those points.
1. Substitute x = 20 into the equation:
y(20) = 6000e^(-20/20) = 6000e^(-1) ≈ 6000(0.3679) ≈ 2207.4
2. Substitute x = 120 into the equation:
y(120) = 6000e^(-120/20) = 6000e^(-6) ≈ 6000(0.00248) ≈ 14.9
Now, to find the equation of the linear function q(x), we will use the point-slope form of a linear equation.
The point-slope form is:
y - y1 = m(x - x1)
Where:
(x1, y1) is a given point on the line, and
m is the slope of the line.
Using the coordinates (20, 2207.4), we have:
y - 2207.4 = m(x - 20)
Now, using the coordinates (120, 14.9), we can solve for the slope m:
14.9 - 2207.4 = m(120 - 20)
-2192.5 = m(100)
m = -21.925
Substituting m back into the equation, we have:
y - 2207.4 = -21.925(x - 20)
Simplifying:
y - 2207.4 = -21.925x + 438.5
y = -21.925x + 2645.9
Therefore, the formula for the linear function q(x) is:
q(x) = -21.925x + 2645.9