Question

2. Ariana has created a paper airplane for her physics class at school where all the students will be testing their planes by sending it off the third floor of the building. The physics teacher is requiring her paper airplane to swoop down at a single point to graze the ground before it rises back into the air. Her initial attempt is modeled by the function, h(x)=x2‒12x+35 where h is the height in feet and x is the time in seconds of the paper airplanes path.

a) In how many seconds will the paper airplane crash into the ground?


3. A rocket is launched with an initial velocity of 100 ft/s. The height of the rocket in meters is modeled by the function shown below, where t is time in seconds.

h(t)=‒4t2 + 120t

Write a statement that describes the 1) domain of this function and 2) range of this function. Make sure to include an explanation as to why you know these values are included in the domain and in the range of the function.

Answer 1

2. The airplane will crash into the ground after 5 seconds.

3. The domain of the function is [0,30] and range of the function is
`[-\infty,900]`.

Explanation:

2.

Ariana initial attempt is modeled by the function,

`h(x)=x^2-12x+35`

where, h is the height in feet and x is the time in seconds of the paper airplanes path.

If airplane crash into the ground, then
`h(x)=0`,

`0=x^2-12x+35`

`0=x^2-7x-5x+35`

`0=x(x-7)-5(x-7)`

`0=(x-7)(x-5)`

Equate each factor equal to 0.

`x=5,7`

Therefore the airplane will crash into the ground after 5 seconds.

3.

The height of the rocket in meters is modeled by the function shown below, where t is time in seconds.

`h(t)=-4t^2+120t`

The value of t must be positive because time can not be negative.

Find the zeros of the function.

`0=-4t(t-30)`

`t=0,30`

The leading coefficient is negative, so the it is a downward parabola. Since the zeros of the function are 0 and 30, therefore the function is negative before 0 and after 30.

The height can not be negative, so the domain of the function is

`0\leq t\leq 30`

The vertex of a downward parabola is the maximum point.

Vertex of a parabola,

`((-b)/(2a),h((-b)/(2a)))`

`((-120)/(2(-4)),h((-120)/(2(-4))))`

`(15,h(15))`

Put t=15 it in the equation.

`h(15)=-4(15)^2+120(15)=900`

`(15,900)`

The range of the function is

`[-\infty,900]`

Therefore the domain of the function is [0,30] and range of the function is
`[-\infty,900]`.