Question

while on a trip, you notice that the video screen on the airplane, in addition to showing movies and news, records your altitude (in kilometers) above the ground. as the plane starts its descent (at time x = 0), you record the following data: time, x (min.) altitude, y (km) 0 12 2 10 4 8 6 6 8 4 10 2 if you were to graph these points on a graph, you would notice that the line that forms represents a linear function. what is the practical meaning of slope in this situation? a. for every minute that passes, the airplane descends 1 kilometer. b. for every minute that passes, the airplane ascends 1 kilometer. c. for every 2 minutes that passes, the airplane descends 1 kilometer. d. for every minute that passes, the airplane ascends 2 kilometers.

Explanation:

Step1: Calculate the slope

The slope \( m \) of a line is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Let's take two points, say \( (0, 12) \) and \( (2, 10) \). Then \( m=\frac{10 - 12}{2 - 0}=\frac{- 2}{2}=- 1 \)? Wait, no, wait, let's check another pair. Take \( (0,12) \) and \( (2,10) \), change in \( y \) is \( 10 - 12=-2 \), change in \( x \) is \( 2 - 0 = 2 \), so slope is \( \frac{-2}{2}=-1 \)? Wait, but let's check with \( (2,10) \) and \( (4,8) \). Change in \( y \) is \( 8 - 10=-2 \), change in \( x \) is \( 4 - 2 = 2 \), slope is \( \frac{-2}{2}=-1 \)? Wait, no, wait the units. Wait, time is in minutes, altitude in km. Wait, but let's see the rate. Wait, from \( x = 0 \) (time 0 min) to \( x = 2 \) (time 2 min), altitude goes from 12 km to 10 km. So the change in altitude is \( 10 - 12=-2 \) km (descent, so negative) over a change in time of \( 2 - 0 = 2 \) min. So the rate is \( \frac{-2\space km}{2\space min}=-1\space km/min \). But that would mean 1 km per minute descent. But wait, the options: option c says "For every 2 minutes that passes, the airplane descends 1 kilometer". Wait, maybe I made a mistake. Wait, let's recalculate. Wait, take \( (0,12) \) and \( (2,10) \): \( \Delta y=10 - 12=-2 \), \( \Delta x = 2 - 0 = 2 \). So slope is \( \frac{\Delta y}{\Delta x}=\frac{-2}{2}=-1 \) km per minute. But that would mean 1 km descent per minute. But option c says 2 minutes for 1 km descent. Wait, no, wait maybe I messed up the points. Wait, wait the table: at \( x = 0 \), \( y = 12 \); \( x = 2 \), \( y = 10 \); \( x = 4 \), \( y = 8 \); \( x = 6 \), \( y = 6 \); \( x = 8 \), \( y = 4 \); \( x = 10 \), \( y = 2 \). So the change in \( y \) when \( x \) increases by 2 is \( - 2 \) (from 12 to 10, 10 to 8, etc.). So for \( \Delta x = 2 \) minutes, \( \Delta y=-1 \) km? Wait no, 12 to 10 is a change of -2 km over 2 minutes. So per 2 minutes, it descends 2 km? Wait no, 12 to 10 is 2 km less in 2 minutes. So 2 km descent in 2 minutes, which is 1 km per minute. Wait, but the options: option c says "For every 2 minutes that passes, the airplane descends 1 kilometer". Wait, no, 12 to 10 is 2 km descent in 2 minutes, so 1 km per minute. But let's check the options. Option a: 1 km per minute descent. Option c: 1 km descent per 2 minutes. Wait, maybe I miscalculated. Wait, let's take \( x = 0 \) (12 km) and \( x = 2 \) (10 km). So in 2 minutes, it descends 2 km? No, 12 - 10 = 2 km descent in 2 minutes. So per minute, it's 1 km descent. But option a says "For every minute that passes, the airplane descends 1 kilometer". But wait, let's check the slope again. Wait, slope is \( \frac{\Delta y}{\Delta x} \). \( \Delta y = 10 - 12=-2 \), \( \Delta x = 2 - 0 = 2 \), so slope is \( -1 \) km per minute. The negative sign indicates descent. So the slope (rate) is -1 km per minute, meaning for each minute, the altitude decreases by 1 km (descends 1 km per minute). But wait, the options: option c says "For every 2 minutes that passes, the airplane descends 1 kilometer". Wait, that would be a slope of \( \frac{-1\space km}{2\space min}=-0.5\space km/min \). But our calculation shows slope -1 km/min. Wait, maybe I made a mistake in points. Wait, let's take \( (0,12) \) and \( (4,8) \). \( \Delta y = 8 - 12=-4 \), \( \Delta x = 4 - 0 = 4 \), slope is \( \frac{-4}{4}=-1 \) km per minute. So that's -1 km per minute. So the practical meaning: the slope is the rate of change of altitude with respect to time. Since it's negative, it's a descent. So for each minute (change in x = 1), the altitude changes by -1 km (descends 1 km). But wait, the options: option c says "For every 2 minutes that passes, the airplane descends 1 kilometer". Wait, that would be a slope of -0.5 km per minute. But our calculation is -1 km per minute. Wait, no, wait the table: at x=0, y=12; x=2, y=10 (2 min later, 2 km descent); x=4, y=8 (another 2 min, another 2 km descent); so every 2 minutes, it descends 2 km? No, 12 to 10 is 2 km in 2 min, so 1 km per minute. Wait, maybe the options are misread. Wait the options:

a. For every minute that passes, the airplane descends 1 kilometer.

b. For every minute that passes, the airplane ascends 1 kilometer.

c. For every 2 minutes that passes, the airplane descends 1 kilometer.

d. For every minute that passes, the airplane ascends 2 kilometers.

Wait, let's recalculate the slope correctly. Let's take two points: (0,12) and (2,10). The change in y (altitude) is 10 - 12 = -2 km (descend 2 km) over a change in x (time) of 2 - 0 = 2 minutes. So the rate is -2 km / 2 min = -1 km per minute. So that means for each minute (x increases by 1), y (altitude) decreases by 1 km (descends 1 km). But option a says that. But wait, let's check with (2,10) and (4,8). Change in y: 8 - 10 = -2 km over 2 minutes (4 - 2 = 2 min), so -2/2 = -1 km per minute. So same rate. So the slope is -1 km per minute, meaning the airplane descends 1 km every minute. But wait, the options: option a is "For every minute that passes, the airplane descends 1 kilometer." But wait, maybe I made a mistake. Wait, no, the calculation shows that. But wait, let's check the table again. At x=0, 12 km; x=1, what would it be? If slope is -1, at x=1, y=12 -1(1)=11? But the table has x=2, y=10. So from x=0 to x=2, 2 minutes, y goes from 12 to 10, which is 2 km descent in 2 minutes, so 1 km per minute. So the slope is -1 km per minute, meaning descent of 1 km per minute. So option a is correct? But wait, maybe the options are different. Wait, no, maybe I miscalculated. Wait, no, 12 to 10 is 2 km in 2 min, so 1 km per min. So option a is correct. But wait, let's check the options again. Wait, maybe the original problem has a typo, but according to the calculation, the slope is -1 km per minute, so the practical meaning is that for each minute, the airplane descends 1 km, which is option a. But wait, maybe I made a mistake. Wait, no, let's see:

Wait, time (x) in minutes, altitude (y) in km.

Slope m = (y2 - y1)/(x2 - x1)

Take (0,12) and (2,10):

m = (10 - 12)/(2 - 0) = (-2)/2 = -1 km/min

So the slope is -1 km per minute. The negative sign indicates descent (since altitude is decreasing as time increases). So the rate of change of altitude with respect to time is -1 km per minute, meaning for each minute that passes (x increases by 1), the altitude (y) decreases by 1 km (descends 1 km). So option a is correct. But wait, let's check the options again. Wait, maybe the user made a mistake in the options, but according to the calculation, option a is correct. But wait, maybe I misread the options. Wait, option c says "For every 2 minutes that passes, the airplane descends 1 kilometer." Let's see: if in 2 minutes, it descends 1 km, the slope would be -1/2 km per minute. But our calculation is -1 km per minute. So there's a contradiction. Wait, maybe the points are (0,12), (2,11)? No, the table says (2,10). Wait, the table is:

Time, x (min.): 0, 2, 4, 6, 8, 10

Altitude, y (km): 12, 10, 8, 6, 4, 2

So from x=0 to x=2: 2 min, y=12 to 10: 2 km descent. So 2 km / 2 min = 1 km per min. So slope is -1 km per min. So option a is correct. But maybe the answer is c? Wait, no, let's check again. Wait, 12 to 10 is 2 km in 2 min, so 1 km per min. So option a: "For every minute that passes, the airplane descends 1 kilometer." is correct. But let's check the options again. Wait, maybe the problem is that the slope is calculated as (y2 - y1)/(x2 - x1), but maybe the question is about the practical meaning, not the numerical slope. Wait, the slope is the rate of change of altitude with respect to time. So if slope is -1, it means for each unit increase in x (1 minute), y (altitude) decreases by 1 unit (1 km). So the practical meaning is that the airplane descends 1 km every minute. So option a is correct.

Step2: Analyze the options

- Option a: Matches the slope calculation (descends 1 km per minute).
- Option b: Ascends, but altitude is decreasing, so incorrect.
- Option c: Descent of 1 km per 2 minutes, but our calculation shows 1 km per minute, so incorrect.
- Option d: Ascends 2 km per minute, but altitude is decreasing, so incorrect.

Answer:

a. For every minute that passes, the airplane descends 1 kilometer.