An arrow is fired into the air with an initial velocity of 96 feet per second. The height in feet of the arrow t seconds after it was shot into the air is given by the function Find the maximum height of the arrow.
Straight up I hope.
well if so, I happen to have a hunch that the function is
h = Vi t - 16 t^2
and the velocity is given by
v = Vi - 32 t
at the top v = 0
so at top, t = Vi/32 = 96/32 = 3 seconds
then
h = 96 * 3 - 16 (9)
= 144
To find the maximum height of the arrow, we need to determine the vertex of the parabolic function.
The height of the arrow as a function of time can be represented by the equation:
h(t) = -16t^2 + 96t
The vertex of a parabola in the form ax^2 + bx + c is given by (-b/2a, f(-b/2a)).
In this case, a = -16 and b = 96. Therefore, the x-coordinate of the vertex is -b/2a.
x = -b/2a = -96/(2(-16)) = 96/32 = 3
To find the maximum height, we substitute t = 3 into the equation h(t):
h(3) = -16(3)^2 + 96(3)
= -16(9) + 288
= -144 + 288
= 144
Therefore, the maximum height of the arrow is 144 feet.
To find the maximum height of the arrow, we need to determine the peak point of the function. In this case, the function is not provided. However, we know the height of the arrow t seconds after it was shot can be represented by a quadratic function.
Let's represent the height of the arrow as h(t).
We can set up the quadratic function using the given information. The general form of a quadratic function is:
h(t) = at^2 + bt + c
Since the arrow is fired into the air, we know that the coefficient of t^2 (a) will be negative to create a parabolic shape that opens downwards. Additionally, we know that the initial velocity (v₀) of the arrow is 96 feet per second. Therefore, the initial velocity will be the value of the coefficient of t (b).
The initial height (h₀) of the arrow is not given, so we can assume it to be 0 since we are interested in finding the maximum height.
Now we have the following information:
a = -g/2 (where g is the acceleration due to gravity and is approximately 32 feet per second squared)
b = 96
c = 0
Substituting these values into the quadratic function, we get:
h(t) = (-g/2)t^2 + 96t + 0
Simplifying, we have:
h(t) = -16t^2 + 96t
To find the maximum height, we need to determine the vertex of the parabola represented by this function.
The t-coordinate of the vertex can be found using the formula:
t = -b / (2a)
In this case, a = -16 and b = 96:
t = -96 / (2 * -16) = -96 / -32 = 3
So, the time taken to reach the maximum height is 3 seconds.
To find the maximum height, substitute this value of t = 3 into the function h(t):
h(3) = -16(3)^2 + 96(3) = -144 + 288 = 144
Therefore, the maximum height of the arrow is 144 feet.