A ball is thrown upward from a height of 240 feet. the height h of the object(in feet) t seconds after the ball is released is given by h=-16t^2+32t+240
a. how long does it take the ball to reach its maximum height?
b. what is the maximum height attained by the ball?
c. how long does it take the object to hit the ground?
(a) vertex of the parabola is at t = -b/2a
(b) plug in that value to get h
(c) solve for t when h=0
a) x = -b/2a
x = -32/2(-16) = 1
t = 1 sec to reach max height
b) h = -16(1)^2 + 32(1) + 240
h = -16 + 32 + 240 = 256
h = 256 ft is the max height
c) 0 = -16(t^2 -2t-15)
0 = -16(t-5)(t+ 3)
t = -3, t = 5
t = 5 seconds for the ball to hit the ground.
To find the answers to the given questions, we will use the equation for the height of the ball as a function of time:
h = -16t^2 + 32t + 240
a. How long does it take the ball to reach its maximum height?
The maximum height is reached at the vertex of the parabolic equation. We can find the x-coordinate of the vertex using the formula:
t = -b / (2a)
where a, b, and c are the coefficients of the quadratic equation in the form ax^2 + bx + c.
In this case, a = -16 and b = 32. Plugging these values into the formula, we get:
t = -32 / (2(-16))
t = -32 / (-32)
t = 1
So, it takes the ball 1 second to reach its maximum height.
b. What is the maximum height attained by the ball?
To find the maximum height, we substitute the value of t from part a into the equation:
h = -16(1)^2 + 32(1) + 240
h = -16 + 32 + 240
h = 256
Therefore, the maximum height attained by the ball is 256 feet.
c. How long does it take the object to hit the ground?
To find the time it takes for the object to hit the ground, we set the height function equal to zero and solve for t:
0 = -16t^2 + 32t + 240
This equation can be factored as:
0 = 16(-t^2 + 2t + 15)
Setting each factor equal to zero, we get:
-t^2 + 2t + 15 = 0
We can solve this by factoring or using the quadratic formula. Factoring gives us:
(t - 3)(-t - 5) = 0
Setting each factor equal to zero, we get:
t - 3 = 0 --> t = 3
-t - 5 = 0 --> -t = 5 --> t = -5
The negative root (-5) can be ignored since time cannot be negative in this context. Therefore, the object takes 3 seconds to hit the ground.
To find the answers to these questions, we need to analyze the given equation for the height of the ball as a function of time.
The equation given is:
h = -16t^2 + 32t + 240
a. To find the time it takes for the ball to reach its maximum height, we can use the fact that the vertex of a quadratic function gives the maximum or minimum point. The vertex of a quadratic function in the form ax^2 + bx + c can be found using the formula: x = -b/2a.
In this case, a = -16 and b = 32. Plugging these values into the formula, we have:
t = -32 / 2(-16)
t = -32 / -32
t = 1
Therefore, it takes 1 second for the ball to reach its maximum height.
b. To find the maximum height attained by the ball, we can substitute the value of t = 1 into the height equation and solve for h.
Using t = 1 in the equation h = -16t^2 + 32t + 240, we have:
h = -16(1)^2 + 32(1) + 240
h = -16 + 32 + 240
h = 256
Therefore, the maximum height attained by the ball is 256 feet.
c. To find the time it takes for the object to hit the ground, we need to determine when the height h becomes 0. We can set the equation h = -16t^2 + 32t + 240 equal to 0 and solve for t.
Setting h = 0 in the equation -16t^2 + 32t + 240 = 0, we have a quadratic equation. We can solve this equation using factoring, completing the square, or the quadratic formula. In this case, we'll use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values a = -16, b = 32, and c = 240, we have:
t = (-(32) ± √((32)^2 - 4(-16)(240))) / (2(-16))
t = (-32 ± √(1024 + 15360)) / (-32)
t = (-32 ± √16384) / (-32)
t = (-32 ± 128) / (-32)
Simplifying, we have:
t = (-32 + 128) / (-32) or t = (-32 - 128) / (-32)
t = 96 / -32 or t = -160 / -32
t = -3 or t = 5
Since time cannot be negative in this context, we disregard t = -3. Therefore, the object takes 5 seconds to hit the ground.
To summarize:
a. It takes 1 second for the ball to reach its maximum height.
b. The maximum height attained by the ball is 256 feet.
c. The object takes 5 seconds to hit the ground.