equation word problem

Question

The temperature in a city began to decrease at a constant rate once a cold front began to move in. The temperature was 49° F after 2 hours and 39° after 6 hours. Suppose x is the number of hours since the cold front began to move in, and y is the temperature in °F

 

The answer is y=-2.5+54 this is My guess but I have no idea why, the others just look wrong.  Is this correct and can you explain please in very simple terms? Thanks.

 

a) y=-2.5+54

b) y=-1.25+54

c) y=-2.5x+56

d) y=-1.25+56

 

Answer 1

If x is the number of hours and y is the temperature, then we can make two points from the data:

(2, 49) and (6, 39)

 

We can find the slope of a line with the formula:

(change in y) / (change in x) = (y₂ - y₁) / (x₂ - x₁)

⇒ (39 - 49) / (6 - 2) = -10 / 4 = -2.5

 

The equation of a line is:

y = mx + b, where m = slope and b = y-intercept.

 

We just solved for m, so now all we need is b. To get b, substitute the values of one of the points into the equation.

 

y = -2.5x + b

⇒ (49) = -2.5(2) + b, I chose point (2, 49)

⇒ 49 = -5 + b

⇒ 54 = b

 

Our final equation is:

y = -2.5x + 54

Answer 2

Andrea;

 

First off you must determine the amount of change per hour by degrees Farehneit (*F)

 

The initial temperature (2 hours into the cold front) was 49*F

4 hours later (6 hours into the cold front) the temperature was 39*F

Thus, the change in temperature (changeT) was -10*F in a 4 hour period....

Given that they stated that the change in temperature was constant all you have

to do to find that changeT per hour is take the (ChangeTemp)/hours

 

The constant for changeT/hour is (-10*F)/4hr  = -2.5*F/hr

 

Next one must find the actual start point.  2 hours into the cold front we were at

49*F but that would be 5*F less than the actual start point because we have lost

2.5*F/hr for 2 hours....    Thus the actual beginning temperature was 49*F + 5*F = 54*F

 

Take your start point (54*F) and apply to it the constant of change (-2.5*F/hr)(hours)

to set up the equation.  Letting x = #hours into the cold front and y be the current

temperature your equation is:

 

y = -2.5x + 54

 

Note:  the correct answer is not actually on the choice list because whoever typed this out

left the variable "x" out of the answer.