2 Answers

Farini · Answer 1 · 2019-11-16T04:55:42+0000

Answers:

a) What is the slope and y-intercept and write an equation for this situation?

In order to solve this problem, the figure attached will be helpful.

We already know the initial altitude
of the balloon in this case is
1800ft . If we draw a coordinate system (figure attached) where the y-axis represents the altitude of the balloon in feets and the x-axis represents the time in minutes, we will be able to find how the altitude changes with time, in other words the rate of altitude decay of the balloon.

This rate is found by calculating the slope
of the line that describes the variation of the position in time:

(1)

where
y_(2)-y_(1)=Y and
x_(2)-x_(1)=X

While the equation of the line that describes the movement of the balloon is:

(2)

where
is the y-interccept (the point in which the line crosses the y-axis)

Note the negative sign in the equation is because the altitude of the balloon is decreasing, this means the slope is negative. If the line were increasing the equation would be with a positive sign:

Y=mX+b (3)

Then, according to this the slope is:

Rate of altitude decay

Where the negative sign is only to indicate it is decreasing.

Now, if we take
y_(2)=1800ft and
y_(1)=0ft when the balloon finally approaches the ground, and
x_(2) the initial time and
x_(1)=0min as the final time, when the balloon is at altitude 0. We will be able to find
from equation (1), and then the y-intercept from equation (2), by substituting the values we know:

From (1):

300(ft)/(min)=(1800ft-0)/(x_(2)-0)

${x_(2)=(1800ft)/(300(ft)/(min))$

${x_(2)=6min$ (4)

Substituting (4) in (2) we will find b or the y-intercept:

1800ft=-300(ft)/(min)(6min)+b

1800ft=-1800ft+b

b=1800ft+1800ft

This is the y-intercept

b) Write the equation of the line that models this situation in slope-intercept form:

Here we have to use equation (2), which is also the equation of the line in the slope-intercept form, and substitute the values (
and
)we calculated above:

if
Y=-mX+b

then

Y=-300X+3600

c) How high off the ground is the balloon after 2 minutes?

Here we will use again the equation of the line in the slope-intercept form with the values we calculated and taking into account the following condition:

If
represents the time, then we have to apply the equation (2) when
to find the height
of the balloon after 2 minutes.