Question

scientists are measuring a distant planet’s temperature. the graph shows the temperature (in $^circ$c) versus the height (in kilometers) above the planet’s surface.
(a) at what height above the planet’s surface is the temperature $0^circ$c?
\boxed{} kilometers
(b) choose the statement that best describes how the height and temperature are related. then fill in the blank.
\bigcirc as the height increases, the temperature decreases.
at what rate is the temperature decreasing?
\boxed{} $^circ$c per kilometer
\bigcirc as the height increases, the temperature increases.
at what rate is the temperature increasing?
\boxed{} $^circ$c per kilometer

Explanation:

Step1: Analyze the graph for part (a)

We need to find the height when temperature is \(0^\circ\text{C}\). From the graph, we can identify two points. Let's assume the line passes through \((0, 40)\) (height = 0, temperature = \(40^\circ\text{C}\)) and \((3, 0)\) (height = 3 km, temperature = \(0^\circ\text{C}\))? Wait, no, looking at the axes: the y - axis is temperature (°C) and x - axis is height (km)? Wait, the labels: "Height (kilometers)" is on the x - axis (horizontal) and "Temperature (°C)" is on the y - axis (vertical). Wait, the line goes from (0, 40) to (3, 0)? Wait, no, the graph: when height (x) is 0, temperature (y) is 40? Wait, no, the user's graph: the x - axis is height (km) with values 0,1,2,3,... and y - axis is temperature (°C) with values 0,5,10,15,20,25,30,35,40,45. The blue line starts at (3, 0) and goes to (0, 40)? Wait, no, the coordinates: let's check the intercepts. To find when temperature \(T = 0\), we look for the x - value (height) where \(y = 0\). From the graph, when \(T = 0\), the height \(h\) is 3 kilometers? Wait, no, maybe I got the axes reversed. Wait, the problem says "the graph shows the temperature (in °C) versus the height (in kilometers) above the planet’s surface". So temperature is the dependent variable (y - axis), height is independent (x - axis). So the line: let's find two points. Let's say when height \(h = 0\) (surface), temperature \(T = 40^\circ\text{C}\), and when height \(h = 3\) km, temperature \(T = 0^\circ\text{C}\). So for part (a), we need to find the height when temperature is \(0^\circ\text{C}\), which is 3 kilometers? Wait, no, wait: if the line is from (0, 40) to (3, 0), then when \(T = 0\), \(h = 3\) km. So part (a) answer is 3 kilometers.

Step2: Analyze the relationship for part (b)

As height (x) increases, temperature (y) decreases (since when x goes from 0 to 3, y goes from 40 to 0). So the statement is "As the height increases, the temperature decreases". Then we need to find the rate of temperature decrease per kilometer. The slope of the line is \(\frac{\Delta T}{\Delta h}=\frac{0 - 40}{3 - 0}=\frac{- 40}{3}\approx - 13.33^\circ\text{C}\) per km? Wait, no, wait: if height increases by 3 km, temperature decreases by 40 °C. So the rate is \(\frac{40}{3}\approx13.33^\circ\text{C}\) per km decrease. Wait, the formula for slope (rate of change) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \(x_1 = 0\), \(y_1 = 40\); \(x_2 = 3\), \(y_2 = 0\). Then \(m=\frac{0 - 40}{3 - 0}=\frac{- 40}{3}\approx - 13.33\). The negative sign indicates that as height (x) increases, temperature (y) decreases. So the rate is \(\frac{40}{3}\approx13.33^\circ\text{C}\) per kilometer decrease, or \(-\frac{40}{3}^\circ\text{C}\) per kilometer. But the options for part (b) are "As the height increases, the temperature decreases" and the rate is \(\frac{40}{3}\approx13.33^\circ\text{C}\) per kilometer (but since it's decreasing, the rate is negative, but the question says "at what rate is the temperature decreasing?" So the rate is \(\frac{40}{3}\approx13.33^\circ\text{C}\) per kilometer. Wait, let's recast: the change in temperature \(\Delta T=0 - 40=- 40^\circ\text{C}\) when change in height \(\Delta h = 3 - 0 = 3\) km. So the rate of change of temperature with respect to height is \(\frac{\Delta T}{\Delta h}=\frac{- 40}{3}\approx - 13.33^\circ\text{C}\) per km. This means that for each kilometer increase in height, the temperature decreases by \(\frac{40}{3}\approx13.33^\circ\text{C}\). So the statement is "As the height increases, the temperature decreases" and the rate is \(\frac{40}{3}\approx13.33^\circ\text{C}\) per kilometer (or \(\frac{40}{3}\) which is approximately 13.33).

Step3: Answering part (a)

To find the height when temperature is \(0^\circ\text{C}\), we look at the graph. The line intersects the temperature - 0 line (y = 0) at x = 3 (height = 3 kilometers). So the height is 3 kilometers.

Step4: Answering part (b)

First, the relationship: as height (x) increases, temperature (y) decreases (since the line has a negative slope). Then, to find the rate, we calculate the slope. Using two points \((h_1,T_1)=(0, 40)\) and \((h_2,T_2)=(3, 0)\), the slope \(m=\frac{T_2 - T_1}{h_2 - h_1}=\frac{0 - 40}{3 - 0}=-\frac{40}{3}\approx - 13.33^\circ\text{C}\) per kilometer. This means the temperature decreases at a rate of \(\frac{40}{3}\approx13.33^\circ\text{C}\) per kilometer as height increases.

Answer:

(a) 3 kilometers
(b) Statement: "As the height increases, the temperature decreases."; Rate: \(\frac{40}{3}\approx13.33^\circ\text{C}\) per kilometer (or \(\frac{40}{3}\) °C per kilometer)