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.
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
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).
So, the y-intercept is b=20
Also, from enunciation, we have that at a height h=1Km, T=10.
So, the expression of the linear function T(h) is: