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Answer:
a) initial height = 0 m
b) 8 s
c) vertex = (4, 32)
maximum height = 32 m
Step-by-step explanation:
<u>Given equation</u>:
where:
- h is the height of the arrow about the ground (in metres)
- t is the time (in seconds)
The initial height of the arrow will be at the beginning of its journey, so when t = 0
Substitute t = 0 into the equation and solve for h:
Therefore, the initial height of the arrow is 0 m.
<h3><u>Part (b)</u></h3>The arrow will hit the ground when the height is 0 m.
Substitute h = 0 into the given equation:
Factor out -2t:
Therefore:
And:
Therefore, the arrow will hit the ground at 8 seconds.
<h3><u>Part (c)</u></h3>The vertex is the <u>turning point</u> of a parabola - it's minimum or maximum point.
As the leading coefficient for the given equation is <u>negative</u>, the parabola <u>opens downwards</u> and so its <u>vertex</u> is its <u>maximum point</u>.
There are different ways to find the vertex of a parabola. The easiest way to find the x-coordinate is by calculating the midpoint of the x-intercepts.
We have determined that the arrow has a height of 0 m at 0 seconds and 8 seconds. Therefore, t = 0 and t = 8 are the x-intercepts. The midpoint is halfway between zero and 8, so the midpoint is t = 4
To find the y-coordinate, substitute t = 4 into the equation and solve for h:
Therefore, the coordinates of the vertex are (4, 32)
The maximum height is the y-coordinate of the vertex.
Therefore, the maximum height is 32 m