Answer to Question #142934 in Algebra for ben

Question

the temperature, C, in degrees Celsius recorded by a city park's weather station between midnight and 7:00 a.m. could be represented as a linear function of the number of hours after midnight, t. The temperature at 1:30 a.m. was −4.2°C and was −8.6°C at 7:00 a.m. Which equation could be used to represent this function?

Answer 1

Start with the "point-slope" form of the equation of the straight line

$T - T_1 = m (t - t_1)$

where

T - temperature,

t - time from midnight

$m = \cfrac{\Delta T}{\Delta t} = \cfrac{T_2-T_1}{t_2-t_1}$ - slope of the line

$(t_1, T_1) - \text{point representing temperature at the time } t_1 \\ (t_2, T_2) - \text{point representing temperature at the time } t_2$

Put in these values:

$(t_1,T_1) = (1.5, -4.2) \\ (t_2,T_2) = (7, -8.6) \\ m = \cfrac{-8.6 - (-4.2)}{7 - 1.5} = - \cfrac{4.4}{5.5} = - 0.8$

And we get

T - (-4.2) = -0.8(t - 1.5)

Simplify to "slope-intercept" form

T + 4.2 = - 0.8t + 1.2

T = - 0.8t - 3