Question 714739: Highway bridges often have expansion joints- small gaps between the bridge sections-that allow for the expansion of the bridge when the weather gets hot. Suppose that a bridge's expansion joint has a gap of 1/2 inches when the temperature if 75°F and that gap increases to 1 inch when the temperature is 50°F . Assume that there is a linear relationship between the width of the gap and the temperature.
1. What would you expect the width of the gap to be when the temperature is 80 F
Answer by DrBeeee(684)   (Show Source):
You can put this solution on YOUR website! Let g = width of gap in inches
Let t = temperature in degrees fahrenheit
Since we assume a linear relationship between the gap and temperature, we can use
(1) g = m*t + b
The two data point of gap vs temperature enable us to find m and b.
Using (75, 1/2) in (1) we get
(2) 1/2 = m*75 + b
And (50,1) in (1) we get
(3) 1 = m*50 + b
Now we solve (2) and (3) for m and b. Subtract (3) from (2) yields
(4) -1/2 = 25*m or
(5) m = -1/50
Now put (5) into (3) and get
(6) 1 = (-1/50)*50 + b or
(7) 1 = -1 + b or
(8) b = 2
The linear relationship between g and t is
(9) g = -(1/50)*t + 2
Let's check this with (2).
Is (1/2 = -(1/50)*75 + 2)?
Is (1/2 = -(3/2) + 2)?
Is (1/2 = 1/2)? Yes, therefore (9) is correct.
At t = 80 using (9), the gap is
(10) g = -(1/50)*80 + 2 or
(11) g = 2 - 8/5 or
(12) g = 2/5
Answer: We would expect the gap to be 2/5 in. when the temperature is 80 F.
|
|
|
