Question

Scientists plan to release a space probe that will enter the atmosphere of a gaseous planet. The temperature of the gaseous planet varies linearly with the height of the atmosphere as measured from the top of a visible boundary layer. The instruments on board can withstand a temperature of 601 K. At what altitude will the probe's instruments fail?

Answer 1

We need to extrapolate the altitude from the graph.  To do this we can write an equation that describes the line, then any point not shown on the graph can be found.  The linear graph is in the form of y=mx+b.  The slope, m, (rise/run) can be found from the two points as
m= (567K - 147.54K)/(78.11km - 18.4km) = 7.02495.
Now to get the y intercept, b, we can use this slope and the x and y values from one of the data points given into the y=mx+b equation:
147.54 = 7.02495(18.4) + b
and then solve for b with:
147.54 = 129.259 + b
18.2808 = b
Now our equation is 
y = 7.02495x + 18.2808
y is the temperature, x is the depth.  Now solve the original question:
601 = 7.02495x + 18.2808
582.719 = 7.02495x
582.719/7.02495 = x
x = 82.95km