a. To write an explicit equation for the number of lights in month n, we need to find the equation of the linear relationship between the number of lights and the number of months.
We are given that the number of streetlights is growing linearly, and that 5 months ago (n = 0) there were 108 lights, and now (n = 5) there are 138 lights.
To find the equation, we can use the slope-intercept form of a linear equation, y = mx + b, where y is the dependent variable (the number of lights), x is the independent variable (the number of months), m is the slope, and b is the y-intercept.
We can find the slope, m, by using the formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) is the point (0, 108) and (x2, y2) is the point (5, 138).
Plugging in the values, we get: m = (138 - 108) / (5 - 0) = 30 / 5 = 6.
Now that we have the slope, we can find the y-intercept, b, by plugging in the values of one of the points into the equation: 108 = 6(0) + b.
Simplifying the equation, we find that b = 108.
Therefore, the explicit equation for the number of lights in month n is: Pn = 6n + 108.
b. To find how many months it will take to reach 216 lights, we can plug in the value of Pn into the equation and solve for n: 216 = 6n + 108.
Subtracting 108 from both sides of the equation, we get: 216 - 108 = 6n.
Simplifying the equation, we find that: 108 = 6n.
Dividing both sides of the equation by 6, we get: n = 108 / 6 = 18.
Therefore, it will take 18 months to reach 216 lights.