A high diver is standing on a platform above a pool. Her diving path can be modeled by the function h(t)=-16t^2+8t+12, where h is her height in feet above the water and t is time in seconds.
a) How high is the platform (initial height of the diver)?
b) After the diver leaves the platform, what is her maximum height?
c) How long does it take for the diver to reach her maximum height?
d)How long is the diver in the air before landing in the water?
I don't know where to start with these questions! Please help.
Use what you know about parabolas:
(a) h(0)
(b,c) max height at the vertex, where t = 8/32
(d) solve for t when h(t) = 0
idl
No problem! I can help you step-by-step with these questions.
a) To find the initial height of the diver (the height of the platform), we can look at the equation h(t). The constant term in the equation represents the initial height. In this case, the constant term is 12. Therefore, the initial height of the diver is 12 feet.
b) To find the maximum height of the diver, we need to find the vertex of the quadratic function h(t)=-16t^2+8t+12. The vertex is the highest or lowest point of a parabola. The formula for finding the x-coordinate of the vertex of a quadratic function in the form ax^2+bx+c is given by x = -b/2a.
In our case, a = -16 and b = 8. Plugging these values into the formula, we get x = -8 / 2(-16) = -8 / -32 = 1/4.
Now we can find the maximum height by substituting the x-coordinate back into the original function: h(1/4) = -16(1/4)^2 + 8(1/4) + 12.
Simplifying, we get h(1/4) = -1 + 2 + 12 = 13. Therefore, the maximum height of the diver is 13 feet.
c) The time it takes for the diver to reach her maximum height is half of the total time it takes for the parabolic motion. Since the equation h(t) is in the form -16t^2 + 8t + 12, we can see that the coefficient of the t^2 term is -16. The formula for finding the time it takes for an object to reach its maximum height in a parabolic motion is given by t = -b/2a.
In our case, a = -16 and b = 8. Plugging these values into the formula, we get t = -8 / 2(-16) = -8 / -32 = 1/4.
Therefore, it takes the diver 1/4 seconds to reach her maximum height.
d) To find the time it takes for the diver to land in the water, we need to find the time when h(t) = 0. In other words, we need to find the solutions for the equation -16t^2 + 8t + 12 = 0.
We can solve this quadratic equation by factoring or by using the quadratic formula. In this case, the equation can be factored as (-2t - 2)(8t - 6) = 0.
Setting each factor equal to zero, we get -2t - 2 = 0 or 8t - 6 = 0.
Solving these equations, we find t = -1 or t = 3/4. Since time cannot be negative in this context, we discard the -1 solution.
Therefore, the diver is in the air for 3/4 seconds before landing in the water.
To answer these questions, we need to understand the function h(t)=-16t^2+8t+12, which represents the height of the diver above the water at any given time t.
a) To find the initial height of the diver (height of the platform), we need to determine the value of h(0). Substitute t=0 into the function:
h(0) = -16(0)^2 + 8(0) + 12
Simplifying, we get:
h(0) = 12
Therefore, the initial height of the diver (the height of the platform) is 12 feet.
b) To find the maximum height reached by the diver, we can look for the vertex of the function h(t). The vertex represents the highest point on the parabolic path.
The general form of a quadratic function is ax^2 + bx + c, where the vertex is given by (-b/2a, f(-b/2a)). In this case, a=-16 and b=8. We can find the x-coordinate of the vertex, which represents the time when the diver reaches her maximum height, using the formula -b/2a.
t(max) = -8 / (2 * -16)
= 8 / 32
= 0.25
Substituting t=0.25 into h(t), we can find the maximum height:
h(0.25) = -16(0.25)^2 + 8(0.25) + 12
Simplifying, we get:
h(0.25) = -1 + 2 + 12
= 13
Therefore, the maximum height reached by the diver is 13 feet.
c) We have already found the time at which the diver reaches her maximum height, which is t(max) = 0.25 seconds.
d) To determine how long the diver is in the air before landing in the water, we need to find the time when the height (h) equals zero. This represents the moment when the diver hits the water.
Set h(t) = 0 and solve for t:
-16t^2 + 8t + 12 = 0
To solve this quadratic equation, you can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a=-16, b=8, and c=12. Plugging these values into the quadratic formula and simplifying, we find two solutions:
t = (-8 ± √(8^2 - 4*(-16)*12)) / (2*(-16))
Simplifying further, we get:
t = (-8 ± √(64 + 768)) / (-32)
= (-8 ± √832) / (-32)
Since we are looking for the time when the diver hits the water, we can discard the negative value:
t = (-8 + √832) / (-32)
Calculating this expression will give you the time in seconds when the diver lands in the water.