Weather: Temperature. The average temperature for Boston in...

Weather: Temperature. The average temperature for Boston in January 2005 was $43^{\circ} \mathrm{F}$. In 2007 the average January temperature was $44.5^{\circ} \mathrm{F}$. What is the rate of change of the temperature per year? If this trend continues, what is the expected average temperature in January $2016 ?$

Key Concepts

Rate of Change

The rate of change measures how a particular quantity, such as temperature, increases or decreases over a specific period. It is calculated by dividing the change in the value of the quantity by the change in time, providing insight into the trend's intensity and direction.

Linear Function

A linear function is a mathematical model that describes a constant rate of change through a straight-line equation, typically expressed in the form y = mx + b. This model is used to represent relationships where increases or decreases occur consistently over time.

Extrapolation

Extrapolation is the method of estimating a value beyond the range of observed data by extending a known trend. It relies on the assumption that the established pattern continues into the future, making it a useful technique for forecasting outcomes based on historical data.

Transcript

00:01 In this equation, we are going to use our slope formula to find rate of change, and we're going to use our equation of a line of a linear equation, the point slope formula to write a prediction equation.

00:17 So the information that we have is that in 2005, the average temperature in january was 43 degrees.

00:24 In 2007, the average temperature in january was 44 .5.

00:31 And we're going to try to figure out, or we are going to make a prediction for the average temperature in 2016.

00:40 So what we have right now is two ordered pairs.

00:45 We're going to let x be the year and y be the average temperature in january.

00:52 So we have two ordered pairs, x, y.

00:56 And we're going to use our slope formula to compute the rate of change.

01:01 So we're going to do y2 minus y1.

01:05 We have 44 .5 minus 43 over x2 minus x1 is 2007 minus 2005.

01:19 So we have 1 .5 over 2, which is equal to.

01:26 We got to clean that up.

01:28 We have a decimal right now.

01:29 Let's let that decimal write that decimal as a fraction.

01:33 1 .5 is 3 halves.

01:37 And then the denominator 2, we can write us 2 over 1.

01:40 Now we can easily get those computed.

01:43 When we divide fractions, we're going to write the numerator times the reciprocal of the denominator.

01:51 Multiplying straight across, we have 3 .4s.

01:56 So the slope of the line that would go through those two points or the rate of change is 3 .4s.

02:02 Now we're going to use that in one of our points.

02:05 So write a prediction equation.

02:07 So we have y minus our y value from one of the points equals the slope times x minus the corresponding x value.

02:16 It doesn't matter which one we pick.

02:18 We're going to use the first one.

02:20 We would get the same answer regardless...

Please give Ace some feedback