Respuesta :
The height of the balloon with time is given by a linear function, given that
the rate at which the balloon descends is constant.
Correct responses:
The equation that models when the balloon will reach an altitude of 75 meters is; 75 = 120 - 4.5·t
The time it takes for the balloon to reach 75 meters is 10 minutes
Methods used to arrive at the above responses
Given parameters:
Initial height of the hot air balloon, h = 120 meters
Rate at which the balloon descends, m = 4.5 meters per minute
Required:
Details of how an equation that models when the balloon reaches an altitude of 75 meters can be set up.
Solution:
Given that the rate at which the balloon descends is constant (4.5 m/min), we have the equation is a linear equation of the form;
y = b
Where:
y = h = The height of the balloon
m = The rate at which the balloon descends = -4.5 m/min (The negative sign indicates that the height is decreasing).
x = t = The time the balloon has been descending.
c = The height of the balloon when x = t = 0, which is the initial height of the balloon = 120 meters
Therefore, we have;
h = m·t + c
Which gives;
Height of the balloon, h = -4.5·t + 120
h = 120 - 4.5·t
When the height, h = 75 meters, we get;
75 = 120 - 4.5·t
The equation that models when the balloon will reach 75 meters above the ground is therefore;
- 75 = 120 - 4.5·t
The solution of the above equation is found as follows;
4.5·t = 120 - 75 = 45
[tex]\displaystyle t = \frac{45}{4.5} = \mathbf{ 10}[/tex]
- The time after which the balloon will be at an altitude of 75 meters is t = 10 minutes
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