# Math Storyboard

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#### Storyboard Description

Thermal Expansion Problem

#### Storyboard Text

• Why is there a gap on the roadway on that bridge?
• The gaps are put there so the bridge will have room to expand when the weather gets hot. So when the temperature is 22 degrees Celsius the gap is 1.3 cm. When it's 30 degrees Celsius the gap narrows to 0.9 cm.
• How wide would the gap be if it was 35 degrees Celsius?
• Well, you first have to make an equation using the slope intercept format which is y=mx+b. Where m is the slope or rate of change and b is the y intercept or starting point. In this equation w is the width and t is the temperature.
• To find slope use the formula: w2-w1/t2-t1. X is the independent variable which is the temperature. The y value is the dependent variable which is the width of the gap.
• Yes, then you solve it. 0.9-1.3= -0.4 and 30-22=8. Then divide. -0.4/8= -0.05. m= -0.05.
• So would the equation to find the slope look like this?: 0.9-1.3/30-22.
• So how would you find the y-intercept in the equation?
• And can you use this equation to find any temperature?
• To find the y-intercept, I used the point slope equation which is w-w1=m(t-t1) Then you plug in the values to solve the equation. w-1.3=-0.05(t-22). Then you use distributive property to simplify the equation. w-1.3=-0.05t+1.1, w-1.3+1.3=-0.05t+1.1+1.3, the equation is w=-0.05t+2.4
• Yes, you just plug in any temperature for t in y=-0.05t+2.4
• So if the temperature is 35 degrees, then the gap would be 0.65 cm. because you plug in 35 for t. w=-0.05(35)+2.4, w=-1.75+2.4, w=0.65 cm.
• If you wanted to figure out what temperature it would be if the gap was completely closed, you have to substitute 0 for w in the equation to solve for t. 0=-0.05t+2.4, 0-2.4=-0.05t+2.4-2.4, -2.4=-0.05t, -2.4/-0.05=-0.05t/-0.05, t=48, this is the x-intercept.
• Yes, you would do the same thing if the temperature was -10 degrees. w=-0.05(-10)+2.4, w=0.5+2.4, w=2.9 cm.
• Let's review. The width intercept is 2.4 cm. which tells you in the real world, that the temperature would be 0 degrees celsius when the gap is 2.4 cm. The slope is -0.05 which is the rate of change. And a linear equation is in the form of y=mx+b
• This is so easy now! It seemed hard at first but now I know how to do it.
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