This question has a few parts to it. I tried but not sure if they are correct.
A boy standing on top of a building in Albany throws a water balloon up vertically. The height (h) in feet, of the water is given by the equation
h(t)=-16t^2+64t +192. where the time( in seconds) after he threw the water balloon. Answer the following. Show work and explain you answer.
a) How high is the building? h(t)=16(t^2 +4t+12)
h(t)=16(t+2)(t+6) this is as far as I can go, help please
b)What is the maximum height of the balloon?
h(t)16(t+2)(t+6)
0 t=-2 t=-6 so 0 is what I got
c)What is the value of t when the balloon hits the ground?
0=16(t+2)(t+6)
0=16(t+8t+12)
0=144t+192 t=-192/144
Please help since I tried but I don't think I did them correctly and I want to understand where I went wrong. thank you
a) the balloon is at the top of the building at time zero
... h = 192
b) the time for max height is at the vertex of the parabola
... on the axis of symmetry ... x = -b / 2a ... t = -64 / (2 * -16)
... plug the value of t into the equation to find max height
c) set h equal to zero, and use the quadratic formula to find t
... you want the positive solution
ok so the work for a)
h(0)=-16(0)^2 +64(0)+192
h(0)=192
b) h(2)=-16(2)^2+64(2)=192
h(2)=-64+128+192
h(2)=64+192 so h(2)=256
c) 0=-64plus/minus the square root64^2-4(-16)(192)/-32
o=-64plus/minus the square root of 4096+12288/-32
0=-64 plus/minus 128/-32 so it is -2 or 6 so the answer is 6
Thank you for your explanation. Hopefully these are correct now
Let's go through each part of the question and explain the steps to get the correct answers:
a) How high is the building?
To find the height of the building, we need to find the value of h(t) when t is equal to 0, because when the boy throws the water balloon, time starts at 0 seconds. So we substitute t = 0 into the equation for h(t):
h(0) = -16(0)^2 + 64(0) + 192
= 0 + 0 + 192
= 192
Therefore, the height of the building is 192 feet.
b) What is the maximum height of the balloon?
To find the maximum height of the balloon, we need to identify the vertex of the parabolic equation h(t) = -16t^2 + 64t + 192. The vertex represents the highest point of the parabola.
The equation h(t) = -16t^2 + 64t + 192 is in the form of ax^2 + bx + c, where a = -16, b = 64, and c = 192. The formula for the x-coordinate of the vertex is given by x = -b/(2a).
Using this formula, we can calculate the value of t at the vertex:
t = -64 / (2*(-16))
= -64 / (-32)
= 2
Substituting t = 2 back into the equation for h(t):
h(2) = -16(2)^2 + 64(2) + 192
= -16(4) + 128 + 192
= -64 + 128 + 192
= 256
The maximum height of the balloon is 256 feet.
c) What is the value of t when the balloon hits the ground?
To find the value of t when the balloon hits the ground, we need to find the roots of the equation h(t) = -16t^2 + 64t + 192, which occur when the height is equal to 0 feet.
Setting h(t) = 0 and solving for t:
0 = -16t^2 + 64t + 192
Divide through by -16 to simplify the equation:
0 = t^2 - 4t - 12
To solve this quadratic equation, we can use factoring or the quadratic formula. Let's use factoring in this case:
0 = (t + 2)(t - 6)
Now we can set each factor equal to zero and solve for t:
t + 2 = 0 --> t = -2
t - 6 = 0 --> t = 6
Therefore, the balloon hits the ground at t = -2 seconds and t = 6 seconds.
I hope this explanation helps you understand the steps and where you went wrong. Let me know if you have further questions!