The population of a town in 1920 was 3500 people. The town's population decreased linearly, and in 1927 the population was 3185. Find a formula for P, the town's population, in terms of t, the number of years since 1920.

Well, it seems like the town experienced a straight-up population clown-down! But fear not, my friend, because I'm here to crunch some numerical noses for you.

We have two points: (1920, 3500) and (1927, 3185). Let's first find the slope, which represents the rate of population decrease per year.

Slope = (3185 - 3500) / (1927 - 1920)
= -315 / 7
= -45

Now, we can use the point-slope form of a line to find the equation:

P - 3500 = -45(t - 1920)

Simplifying this a little bit:

P - 3500 = -45t + 90,000

Rearranging to solve for P:

P = -45t + 90,000 + 3500
= -45t + 93,500

Ta-da! That's the formula for P, the town's population, in terms of t, the number of years since 1920. Remember, though, that this equation assumes that the population decrease remained constant at a linear rate. So, just like a clown car, take it with a pinch of laughter!

To find a formula for the town's population in 1920 in terms of the number of years since then, we can use the given information.

Let's denote the population in 1920 as P₀ and the number of years since 1920 as t.

We know that the population in 1920 was 3500 people, so we have P₀ = 3500.

We are also given that the town's population decreased linearly, and in 1927 the population was 3185. This means that the population decreased by (3500 - 3185) = 315 people over a span of 7 years (1927 - 1920 = 7).

To find the rate of decrease, we can divide the change in population by the change in years:
Rate of decrease = (315/7) people per year = 45 people per year

Now that we have the rate of decrease, we can find the formula for the town's population in terms of t.

Using the formula for a line, y = mx + b, where y represents the population, m is the slope (rate of decrease), x is the number of years since 1920 (t), and b is the y-intercept (initial population):

P = mt + P₀

Substituting the values we have:
P = 45t + 3500

Therefore, the formula for the town's population in terms of the number of years since 1920 is P = 45t + 3500.

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find rate of change:

(3500 - 3185)/(1920 - 1927)
rate of change = 315/-7
= -45

so P = -45t + b
when t = 0, P = 3500
3500 = 0 + b
b = 3500

P = -45t + 3500