We are told that it takes 7 joules to raise the gold bar's temperature one degree celsius' We are also told that the initail temp of the bar is 25 degrees.
And we are asked to write an equation that gives heat in Joules required to heat the bar to a tempurature "t".
We notice that we need to start with a temperature of 25 degrees. And that is without giving any energy(in joules) to the bar.
So if we are plotting the energy dlivered (Joules) on the y-axis, and temperature on degree Celsius on the x axis, we are going to have the equation of a line representing this relationship. The slope of the line is going to be the rate 7 joules / 1 degree C.
and we know that the line should go through the point (25, 0) (initial temperature of the bar (25 degrees) when no addition of Joules have been made (zero joules)
Then the equation becomes:
h(t) = 7 * t + b
uisng the point (25, 0) to find the y-intercept "b"
0 = 7 (25) + b
b = - 175 joules
The final expression becomes:
h(t) = 7 * t - 175
where "t" must be in degrees celsius, to render the heat in joules.
The rate of change of the function is that slope: 7 joules per degree celsius.
It is what change in the y-variable yo get per unit of change in the horizontal variable (temperature)
Since it is a line, the rate of change is constant and equal to 7 joules per degree C.