I am a parent trying help my son, who has trouble understanding and remembering how to work his math. I never had this math, so it is hard for me to explain it. If I see it worked out. Then I can explain it, (I, Think) here is the problems. A ball is tossed into the air and it is modeled by the function h(t)=-16t^2+96t+8. This has a maximum height of? Also need to find the time it would return to the ground?
Your dealing with parabolas and quadratic equations.
If you have never learned this, I don't see how you can help him.
He must know how to find the vertex of the above parabola, either by completing the square or a short formula which he must memorize.
h(t) = -16t^2 + 96t + 8
= -16(t^2 - 6t + ......) + 8
= -16(t^2 - 6t + 9 - 9 ) + 8
= -16( (t-3)^2 - 9) + 8
= -16(t-3)^2 + 144 + 8
= -16(t-3)^2 + 152
the vertex is (3,152)
so the maximum height is 152 ft, after 3 seconds
when it hits the ground h(t) = 0
0 = -16t^2 + 96t + 8
divide by -8
2t^2- 12t - 1 = 0
by the formula ...
t = (12 ± √152)/4
= 6.0822 or a negative t, which makes no sense
It will take appr 6.08 seconds to return to the ground.
WHOOOUUUU
To find the maximum height of the ball, you need to find the vertex of the parabolic function h(t) = -16t^2 + 96t + 8. The vertex of a parabola is the highest or lowest point on the curve.
The vertex can be found using the formula t = -b / (2a), where a is the coefficient of the t^2 term, and b is the coefficient of the t term.
In this case, a = -16 and b = 96. Plugging these values into the formula gives:
t = -96 / (2 * -16) = -96 / -32 = 3
So, the ball reaches its maximum height after 3 seconds.
To find the maximum height itself, you can substitute t = 3 back into the original function:
h(t) = -16(3)^2 + 96(3) + 8
h(t) = -16(9) + 288 + 8
h(t) = -144 + 288 + 8
h(t) = 152
Therefore, the maximum height of the ball is 152 units.
To find the time it takes for the ball to return to the ground, you need to determine when the height is equal to zero. This represents the time at which the ball hits the ground.
To find the values of t when h(t) = 0, you can solve the quadratic equation -16t^2 + 96t + 8 = 0. You can either factor the equation or use the quadratic formula.
In this case, factoring would be a bit difficult, so using the quadratic formula is a better option. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation -16t^2 + 96t + 8 = 0, a = -16, b = 96, and c = 8. Plugging these values into the quadratic formula, we get:
t = (-96 ± √(96^2 - 4(-16)(8))) / (2(-16))
t = (-96 ± √(9216 + 512)) / -32
t = (-96 ± √9728) / -32
Now, calculate the square root of 9728:
√9728 ≈ 98.63
Plugging this back into the equation gives us:
t = (-96 ± 98.63) / -32
If we calculate with the positive square root:
t = (-96 + 98.63) / -32
t = 2.63 / -32
t ≈ -0.08
Therefore, the ball hits the ground after approximately -0.08 seconds.
In this case, the negative solution (-0.08 seconds) doesn't make physical sense for time since time cannot be negative. So, we ignore the negative solution.
Therefore, the ball takes approximately 2.63 seconds to return to the ground.