Math-Precalculus

Figure SEQ Figure \ ARABIC 1

1

A . Figure 1 gives the plot for the temperature versus distance east Based on the appearance of the graph , it is a good linear model for predicting temperature as a function of the distance east . To do it more quantitatively , the slope of the graph at different points can be calculated . If they are all equal with one another , then the graph can be said to be linear . Based on the data , there are three points given (0 , 71 .6 (10 , 69 .8 , and (20 , 68

.0

Determine the slope between the first two points . The equation is simply

m1 (y2 - y1 (x2 - x1

m1 (69 .8 - 71 .6 (10 - 0 The slope for the second pair of points is

m2 (y3 - y2 (x3 - x2

m2 (68 - 69 .8 (20 - 10

Since m1 and m2 are equal , this indicates that the graph is indeed linear . The slope of the whole line is given by -0 .18

In for this plot to be of use , the equation governing this linear model has to be derived . In this case , the y-intercept of the graph is already known and is equal to 71 .60F . Combining this with the given slope , the linear equation is (Equation 1

In this case , y stands for the temperature as a function of x , the distance east . Furthermore , the linear equation , which has a negative slope , tells us that the farther away we go from the starting point , the lower the temperature goes

B . We can use Equation 1 to determine the expected temperature . In this case , let x be the distance east , which is equal to 50 miles . Hence

y -0 .18x 71 .6 , where x 50 miles

y -0 .18 (50 71 .6

y -0 .18x 71 .6 At a distance of 50 miles east , we can expect the temperature to be 62 .6 0F

C . Also , we can use Equation 1 to determine the distance east given any temperature . We only need to rearrange to equation to yield x instead of y . In this case , isolate all x 's in one side of the equation

-0 .18x y - 71 .6

Divide both sides by the coefficient of x , which is -0 .18 to yield x

x (y /-0 .18 397 .78

Plugging in the temperature (represented by y , which is equal to 65 0F , we get

x (65 /-0 .18 397 .78

x -361 .11 397 .78 So , at a temperature of 65 0F , the distance east is 36 .67 miles

D . Using the plot and the equation obtained from the given data , the following questions can also be answered : What is the distance east such that the temperature reaches freezing point (Answer : 220 miles ) Suppose instead of going east , the person goes west . How will the temperature be affected in this case (Note that east and west are opposite to each other , and going west will imply a negative direction or a negative value for x...