making linear equation from word problem
At low altitudes, the altitude of a parachute and time in the air are linearly related. A jump at 2,880 ft use the US army T-10 parachute system last 120 sec.
(A) find a linear model relating altitude a (in feet) and time in the air t (in seconds)
(B) the rate of descent is the speed at which the jumper falls. What is the rate of descent for a T-10 system?
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thank you
To create a linear equation based on the given word problem, we need to establish the relationship between the altitude (a) and the time in the air (t).
(A) To find the linear model, we can use the two data points provided: when the altitude is 2,880 ft, the time in the air is 120 sec. This gives us the coordinates (a1, t1) = (2,880, 120).
Since the problem states that the relationship between altitude and time is linear, we can use the slope-intercept form of a linear equation, y = mx + b. In this case, a represents the dependent variable (altitude) and t represents the independent variable (time in the air).
We can find the slope (m) of the linear equation using the formula:
m = (t2 - t1) / (a2 - a1)
Let's assume a second point for our linear model. For example, when the time in the air is 0 seconds, the altitude should also be 0 feet, giving us the point (a2, t2) = (0, 0):
m = (0 - 120) / (0 - 2,880)
m = -120 / -2,880
m = 1/24
Now that we have the slope (m) of the linear equation, we can find the y-intercept (b) by substituting one of the given points (a1, t1) into the equation and solving for b:
t = m * a + b
120 = (1/24) * 2,880 + b
120 = 120 + b
b = 0
Therefore, the linear equation relating altitude (a) and time in the air (t) is:
t = (1/24)a + 0
t = (1/24)a
(B) The rate of descent is the speed at which the jumper falls, which would be the slope of the linear equation. From the linear model we found, the rate of descent for a T-10 system is 1/24 feet per second. This means that for every 24 feet descent in altitude, the time in the air increases by 1 second.
if time at altitude x is y, then
y = ax+b
Now, b=0, since a jump from 0 feet takes 0 time. So,
y = ax
120 = 2880a
a = 120/2880 = 12/288 = 1/24
y = x/24
the speed is thus ft/s = x/y = 24 ft/s