multiple function types in context quick check\na hot air balloons path is modeled using the function…
multiple function types in context quick check\na hot air balloons path is modeled using the function f(x)=-x² + 150. a mine shaft elevators path is modeled using the equation g(x)=-20x. in both instances, the domain is time, in minutes, and the range is height, in meters. using geogebra, graph the two situations. which statement correctly interprets the graphs? (1 point)\nthe balloon will reach the ground before the elevator car does.\nnegative range values dont make sense for either scenario.\nthe starting point of the balloon is higher than that of the elevator.\npositive domain values dont make sense for either scenario.
Answer
Explanation:
Step1: Analyze the balloon - function
For the hot - air balloon function $f(x)=-x^{2}+150$, when $x = 0$, $f(0)=150$, which is the starting height of the balloon.
Step2: Analyze the elevator - function
For the mine - shaft elevator function $g(x)=-20x$, when $x = 0$, $g(0)=0$, which is the starting height of the elevator.
Step3: Analyze each option
- Option 1: To find when the balloon reaches the ground, set $f(x)=0$, so $-x^{2}+150 = 0$, $x^{2}=150$, $x=\sqrt{150}\approx12.25$ minutes. To find when the elevator reaches the ground, set $g(x)=0$, so $-20x = 0$, $x = 0$ (this is wrong as we consider the non - zero descent, and in fact, the elevator is descending, and if we consider the height going to 0, for the elevator starting from some non - zero negative height in the mine, we need more context about its initial position below ground. But we can compare starting heights).
- Option 2: In the context of height, negative range values can make sense for the elevator (it's in a mine shaft), so this option is wrong.
- Option 3: Since $f(0)=150$ and $g(0)=0$, the starting point of the balloon is higher than that of the elevator. This option is correct.
- Option 4: Positive domain values (time) make perfect sense for both scenarios as time is a non - negative quantity in this context, so this option is wrong.