Question

emily took her mom on a hot air balloon ride for mothers day. the altitude of the hot air balloon, in feet, after t minutes can be modeled by the expression -0.2(t - 90)² + 1,620. what does the number 1,620 represent?
the amount of time it took the hot air balloon to reach 50 feet
the amount of time it took the hot air balloon to reach its maximum altitude
the altitude of the hot air balloon after 50 minutes
the maximum altitude of the hot air balloon

Explanation:

Step1: Analyze the expression form

The expression for the altitude is $-0.2(t - 90)^2+1620$, which is a quadratic function in vertex form $y = a(x - h)^2 + k$, where $(h,k)$ is the vertex of the parabola. For a quadratic function with $a<0$ (here $a=- 0.2<0$), the parabola opens downwards, and the vertex represents the maximum point.

Step2: Identify the vertex meaning

In the given function, $h = 90$ and $k=1620$. The vertex form of the quadratic function for altitude (in feet) as a function of time $t$ (in minutes) tells us that the vertex $(90,1620)$ means when $t = 90$ minutes, the altitude $y$ reaches its maximum value of $1620$ feet. So the number $1620$ represents the maximum altitude of the hot air balloon.

Answer:

the maximum altitude of the hot air balloon