The height of an arrow shot in the air is given by the function h(t)=-16t^2+144t+24, where h(t) represents the height in feet, and t represents the time in seconds.
what is the maximum height of the arrow?
how long does it take to reach the maximum height?
how long does it take the arrow to hit the ground? (round to the nearest 10th of a second)
what height was the arrow shot?
Since you labeled it algebra rather than Calculus, I will use an algebraic solution,
completing the square
h(t) = -16(t^2 - 9t + 81/4 - 81/4) + 24
= -16( (t - 9/2)^2 - 81/4) + 24
= -16(t - 9/2)^2 + 324 + 24
= -16(t-9/2)^2 + 348
this is a downwards-opening parabola with vertex
(9/2 , 348)
It will have a maximum of 348 , when t = 9/2 or 4.5 seconds
or
the x of the vertex is -b/(2a) = -144/-32 = 4.5
h(4.5) = -16(20.25) + 144(4.5) + 24
= 348
To find the maximum height of the arrow, we can use the formula h(t)=-16t^2+144t+24. The maximum height occurs when the derivative of h(t) is zero. Let's find the derivative first.
The derivative of h(t) with respect to t is given by:
h'(t) = -32t + 144
To find when the derivative is zero, we set h'(t) = 0 and solve for t:
-32t + 144 = 0
-32t = -144
t = -144 / -32
t = 4.5 seconds
This means that it takes 4.5 seconds for the arrow to reach its maximum height.
Now, let's find the maximum height by plugging the value of t into the function h(t):
h(4.5) = -16(4.5)^2 + 144(4.5) + 24
h(4.5) = -16(20.25) + 648 + 24
h(4.5) = -324 + 648 + 24
h(4.5) = 348 feet
Therefore, the maximum height of the arrow is 348 feet.
To find how long it takes for the arrow to hit the ground, we need to find when h(t) = 0.
-16t^2 + 144t + 24 = 0
We can solve this quadratic equation using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values for a, b, and c, we get:
t = (-144 ± √(144^2 - 4*(-16)*(24))) / (2*(-16))
t = (-144 ± √(20736 + 1536)) / (-32)
t = (-144 ± √(22272)) / (-32)
t = (-144 ± 149.14) / (-32) [Rounded square root of 22272 is 149.14]
t ≈ (5.14 or -8.39) / (-32) [Rounding to the nearest 10th]
Since time cannot be negative in this case, we take the positive value:
t ≈ 0.16 seconds
Therefore, it takes approximately 0.16 seconds for the arrow to hit the ground.
The height at which the arrow was shot is given by h(0):
h(0) = -16(0)^2 + 144(0) + 24
h(0) = 0 + 0 + 24
h(0) = 24 feet
Therefore, the arrow was shot from a height of 24 feet.
To find the maximum height of the arrow, we need to determine the vertex of the function h(t). The vertex gives us the highest point on the graph, which represents the maximum height.
The vertex of a quadratic function in the form h(t) = at^2 + bt + c is given by the formula t = -b/(2a), where a, b, and c are coefficients of the quadratic function.
For the given function h(t) = -16t^2 + 144t + 24, we can identify a = -16, b = 144, and c = 24. Plugging these values into the formula, we get:
t = -144 / (2 * -16)
t = -144 / -32
t = 4.5
Therefore, the arrow reaches its maximum height at t = 4.5 seconds.
To find the maximum height, we substitute t = 4.5 into the function h(t):
h(4.5) = -16(4.5)^2 + 144(4.5) + 24
h(4.5) = -16(20.25) + 648 + 24
h(4.5) = -324 + 648 + 24
h(4.5) = 348
Hence, the maximum height of the arrow is 348 feet.
To determine how long it takes the arrow to hit the ground, we need to find the first positive root of the function h(t) = 0. This occurs when the height of the arrow is zero. We can solve for t by setting h(t) = 0 and then solving the resulting quadratic equation:
-16t^2 + 144t + 24 = 0
To solve this quadratic equation, we can use the quadratic formula: t = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = -16, b = 144, and c = 24. Plugging these values into the quadratic formula, we get:
t = (-144 ± √(144^2 - 4 * -16 * 24)) / (2 * -16)
t = (-144 ± √(20736 + 1536)) / -32
t = (-144 ± √22272) / -32
Now, taking the positive root (since we want the time it takes to hit the ground), we have:
t = (-144 + √22272) / -32
Calculating this expression using a calculator, we find t ≈ 8.38 seconds (rounded to the nearest 10th of a second).
Therefore, it takes approximately 8.38 seconds for the arrow to hit the ground.
Finally, to determine the height at which the arrow was shot, we need to find the value of h(0). Plugging t = 0 into the function h(t), we get:
h(0) = -16(0)^2 + 144(0) + 24
h(0) = 0 + 0 + 24
h(0) = 24
Thus, the arrow was shot at a height of 24 feet.