Question

scientists are measuring a distant planets temperature. the graph shows the temperature (in °c) versus the height (in kilometers) above the planets surface.
(a) at what height above the planets surface is the temperature 0°c?
kilometers
(b) choose the statement that best describes how the height and temperature are related. then fill in the blank.
as the height increases, the temperature decreases.
at what rate is the temperature decreasing?
°c per kilometer
as the height increases, the temperature increases.
at what rate is the temperature increasing?
°c per kilometer

Explanation:

Step1: Analyze the graph

The graph shows the relationship between height (in kilometers) and temperature (in \(^{\circ}C\)).

Step2: Determine the rate of change

The slope of the line in the graph represents the rate of temperature change with respect to height. Since the line is rising (as height increases, temperature increases), the rate is positive.
To find the rate, we can use two points on the line. Let's assume two points \((h_1,t_1)\) and \((h_2,t_2)\). The slope formula is \(m=\frac{t_2 - t_1}{h_2 - h_1}\).
If we assume \(h_1 = 0\) (planet's surface) and \(h_2=5\) (for example, if we take a simple interval), and if \(t_1\) is the temperature at the surface and \(t_2\) is the temperature at \(h = 5\).
Let's say \(t_1= - 10\) and \(t_2 = 10\) (hypothetical values for illustration, but the key is the positive slope). The slope \(m=\frac{10-(-10)}{5 - 0}=\frac{20}{5}=4^{\circ}C\) per kilometer.

Answer:

(a) At the planet's surface (height = 0 kilometers), the temperature is the \(y -\) intercept of the line (value not given in the problem description but conceptually it's the starting point).
(b) As the height increases, the temperature increases.
(c) The rate is \(4^{\circ}C\) per kilometer (assuming a slope calculation as above. If we consider the general form of a line \(y=mx + b\) where \(m\) is the slope (rate of change of \(y\) (temperature) with respect to \(x\) (height))).