The number of streetlights in a town is growing linearly. Four months ago (n = 0)
there were 130 lights. Now (n = 4) there are 146 lights. If this trend continues,
a. Find an explicit formula for the number of lights in month n
b. How many months will it take to reach 200 lights?
z = 130+4n
z=200 when n=17.5
a. Well, it seems like the number of lights is creeping up at a constant rate. Let's say that the number of lights in month n is L. We know that L = 130 when n = 0, and L = 146 when n = 4.
To find our explicit formula, let's first determine the rate at which the number of lights is growing. The difference between L at n = 4 and n = 0 is 146 - 130 = 16. So, the number of lights increases by 16 units in 4 months.
Now, let's figure out how many lights are added each month. 16 divided by 4 gives us 4, so we have an increase of 4 lights every month.
Therefore, our explicit formula for the number of lights in month n would be L = 4n + 130.
b. We need to find the value of n when L = 200. Substituting L = 200 into our formula, we get 200 = 4n + 130.
Solving for n, we subtract 130 from both sides: 200 - 130 = 4n, which gives us 70 = 4n.
Dividing both sides by 4 gives us n = 17.5.
Wait a minute, 17.5 months? That's a strange answer! We can't have half a month, can we?
So, using logic rather than math, if it took 4 months to go from 130 to 146 lights and the growth rate is constant, we can assume it will take approximately 4 more months to reach 200 lights.
Therefore, it will take about 8 months to reach 200 lights.
To find an explicit formula for the number of lights in month n, we can use the given information.
We are given that 4 months ago (n = 0), there were 130 lights, and now (n = 4), there are 146 lights.
Using these two data points, we can find the slope of the line (rate of increase) by using the formula:
slope = (change in y) / (change in x)
In this case, the change in y is 146 - 130 = 16, and the change in x is 4 - 0 = 4.
So, the slope of the line is 16 / 4 = 4.
Now, we can use the point-slope form of a linear equation to find the explicit formula. The formula is:
y - y1 = m(x - x1)
Where:
y1 is the y-coordinate of one of the points (130 lights)
x1 is the x-coordinate of the same point (n = 0)
m is the slope of the line (4)
Using (n = 0, y = 130), we can substitute these values into the formula:
y - 130 = 4(x - 0)
Simplifying this equation, we get:
y - 130 = 4x
Adding 130 to both sides, we get:
y = 4x + 130
So, the explicit formula to find the number of lights in month n is y = 4n + 130.
Now, to find how many months it will take to reach 200 lights, we can substitute y = 200 into the formula and solve for n:
200 = 4n + 130
Subtracting 130 from both sides, we get:
70 = 4n
Dividing both sides by 4, we get:
n = 17.5
Since the number of months cannot be a decimal, we can round up to the nearest whole number:
n = 18
Therefore, it will take 18 months to reach 200 lights.
To find the explicit formula for the number of lights in month n, we can use the formula for a linear function, which is given by:
y = mx + b
Where:
y represents the number of lights in month n
m represents the rate of change (slope) of the function
x represents the number of months since the starting point
b represents the initial number of lights at the starting point
In this case, we are given two data points:
When n = 0, y = 130
When n = 4, y = 146
We can use these data points to find the values of m and b. Let's start by finding the slope (m):
m = (y2 - y1) / (x2 - x1)
m = (146 - 130) / (4 - 0)
m = 16 / 4
m = 4
Now that we have the slope, we can substitute it into the formula along with one of the data points (let's use n = 0, y = 130) to find the value of b:
130 = 4(0) + b
130 = 0 + b
b = 130
Now that we have the values of m and b, we can substitute them into the formula to find the explicit formula for the number of lights in month n:
y = 4n + 130
Therefore, the explicit formula for the number of lights in month n is y = 4n + 130.
To find out how many months it will take to reach 200 lights, we can plug in the value of y into the formula and solve for n:
200 = 4n + 130
Subtracting 130 from both sides:
70 = 4n
Dividing both sides by 4:
n = 70 / 4
n = 17.5
Since we cannot have a fraction of a month, we can round up to the next whole number. Therefore, it will take approximately 18 months to reach 200 lights.