Question

The speed of sound in air is a linear function of the air temperature. When the air temperature is 10°C, the speed of sound is 337 m/s. When the air temperature is 30°C, the speed of sound is 349 m/s.
Can you please explain the process of writing this function as a linear equation?

The answer is s-337=0.6(t-10), but I don't know how the textbook got this answer.​

Answer 1

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Answer:

  process: substitute the given point values into the 2-point form of the equation for a line

Step-by-step explanation:

There are more than a half-dozen different forms of the equation for a line. They are useful for different purposes. One of them is the "two-point form".

Using x as the independent variable, and y as the dependent variable, the equation can be written as ...

  y -y1 = (y2 -y1)/(x2 -x1)/(x -x1)

where (x1, y1) and (x2, y2) are the two points.

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Here, your two points are ...

  (t, s) = (10, 337) and (30, 349)

Using s in place of y, and t in place of x, these two points go into the formula like this:

  s -337 = (349 -337)/(30 -10)(t -10)

Simplifying the fraction, this is ...

  s -337 = (12/20)(t -10)

And writing it as a decimal, we get ...

  s -337 = 0.6(t -10)

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Additional comments

Adding y1 to both sides of the above form gives you ...

  y = (y2 -y1)/(x2 -x1)/(x -x1) +y1

This is the form I usually prefer to use, because it can lead directly to slope-intercept form. For this problem, the form shown above gets you to the answer you're looking for.

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This "two-point form" is an expansion of the "point-slope form", which is ...

  y - k = m(x -h) . . . . . . . line with slope m through point (h, k)

where the equation for slope is ...

  m = (y2 -y1)/(x2 -x1)

and (x1, y1) is used instead of (h, k).

Answer 2

Answer:

The actual linear equation is s= 0.6t + 331 which can be rearranged to make s-337 = 0.6( t-10 ).

Step-by-step explanation:

Any linear equation has the formula y=Mx + c , where x is the independent variable , y is the dependent variable , m is the slope of the line and c is the y-intercept .

here y is replaced by “s” which stands for speed of sound and x is replaced by “t” which stands for temperature.

To find the slope , we divide gain  of “s” by gain of “t “

i.e. $m= \frac{349-337}{30-10}= \frac{12}{20} = 3/5$

So the slope of the equation is 3/5 Or 0.6

lets plug s=337 and t= 10 into the equation s= 0.6t+ c

$337= \frac{3}{5} (10) + c\\337 = 6 + c\\C= 331\\Hence the equation becomes s= 0.6t+ 331 \\Rearranging the equation , s-331 = 0.6t \\ subtracting 6 from both sides, s-331-6 = 0.6t-6\\S-337 = 0.6 ( t- 10 ) , which is given in your book.$

The last line is s-337 = 0.6t - 6 = 0.6(t-10 ) , which is given in your book.

Hope it helps