# Express temperature as a function of height, using linear model?The ground temperature is 20 degrees C. The temperature of 10 degrees C is at a height of 1Km.

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### 3 Answers

Given:

Temperature at ground level, that is when height is 0 (h0) meters

= t0 = 20 degrees centigrade

Temperature at height 1 km (h) meters

= t = 10 degrees centigrade

It is assumed that the temperature drops in proportion to increase in height.

This can be expressed in the form of equation as follows:

t = t0 - C*(h - h0) ... (1)

Where C is a constant.

Substituting values of t0, t h0 and h in equation (1) we get:

10 = 20 - C*(1 - 0) = 20 - C

Therefore:

C = 20 - 10 = 10

Substituting value of C, t0, and h0 in equation (1) we get:

t = 20 - 10*(h - 0) = 20 - 10h

Therefore temperature can be expressed as a function of height by the equation:

Temperature in degrees centigrade = 20 - 10*(Height in km)

Let h denote the height from the ground level.So

(h,t) = (0,20degree.) at ground level and (1km, 10degree)

We know that the linear equation for joining (x1,y1) to (x2,y2) is

(y-y1) = (y2-y1)/(x2-x1)(x-x1).

Therfore the linear model for (0,20) to (1,10) is

(t-20) = (10-20)/(1-0) (h-0). Or

t-20 = -2(*h-0). Or

t-20 = -2h . Or

t = 20-2h which is temperature as function of height. Or

2h+t-20 = 0, is the linear model between height h and temperature t.

From enunciation we find out that the temperature has to be expressed, using the model of a linear function, depending on the height.

T(h)=m*h+b, where m is the slope and b is the y-intercept.

It is given, from enunciation, that when h=0, T=20, so, we'll substitute the values into the expression of the function T(h).

20=m*0+b

b=20

So, the y-intercept is b=20

Also, from enunciation, we have that at a height h=1Km, T=10.

10=m*1+20

m+20=10

m=10-20

m=-10

So, the expression of the linear function T(h) is:

**T(h)=-10*h+20**