Question

a warehouse safety technician uses the equation below to model the height of a hammer after it is dropped from the roof of a tall building. in the model the height y (in feet) is a function of t, the number of seconds after the hammer is dropped.$y = 256 - 16t^{2}$complete the parts below.(a) graph the parabola $y = 256 - 16t^{2}$. to do so, plot five points on the parabola: the vertex, two points to the left of the vertex, and two points to the right of the vertex. then click on the graph-a-function button.(b) find the positive t-intercept of the model $y = 256 - 16t^{2}$. then choose the best description of this t-intercept.positive t-intercept: $square$description:the time in seconds when the hammer hits the groundthe time in seconds when the hammer is droppedthe maximum height, when the hammer is hit. the hammer hits(c) write an inequality to represent all possibilities for the values of t for which the hammer is falling. then choose the best description of this inequality.inequality: $square$ $t$ $square$description:

Explanation:

Step1: Find the vertex

The given function is $y=256-16x^2$, which can be rewritten as $y=-16x^2+256$. For a quadratic function $y=ax^2+bx+c$, the x-coordinate of the vertex is $x=-\frac{b}{2a}$. Here $a=-16$, $b=0$, so $x=0$. Substitute $x=0$ into the function: $y=256-16(0)^2=256$. Vertex: $(0, 256)$

Step2: Find left vertex points

Choose $x=-2$ and $x=-4$:

• For $x=-2$: $y=256-16(-2)^2=256-64=192$, point $(-2, 192)$
• For $x=-4$: $y=256-16(-4)^2=256-256=0$, point $(-4, 0)$

Step3: Find right vertex points

Choose $x=2$ and $x=4$:

• For $x=2$: $y=256-16(2)^2=256-64=192$, point $(2, 192)$
• For $x=4$: $y=256-16(4)^2=256-256=0$, point $(4, 0)$

Step4: Find positive x-intercept

Set $y=0$:
$0=256-16x^2$
$16x^2=256$
$x^2=16$
$x=\pm4$. Positive intercept: $x=4$

Step5: Solve $256-16x^2>0$

$16x^2<256$
$x^2<16$
$-4

Answer:

Part (a) Points to plot:

Vertex: $(0, 256)$; Left points: $(-4, 0)$, $(-2, 192)$; Right points: $(2, 192)$, $(4, 0)$

Part (b)

Positive x-intercept: $4$
Description: The time in seconds when the hammer hits the ground.

Part (c)

Inequality: $0\leq x<4$
Description: The time interval (in seconds) when the hammer is falling (above the ground).