A ball is tossed straight upward with an initial velocity of 80 ft per second from a rooftop that is 12 feet above ground level . The height of the ball in feet at time t seconds is given by h(t)= -16t^2 +80t+12 find maximum height above ground level the ball reaches
If you know Calculus,
h ' (t) = -32t + 80
= 0 for a max height
t = 80/32 = 5/2 or 2.5 seconds
h(2.5)= -16(2.5)^2 + 80(2.5) + 12
= 112 ft
If you don't know calculus, let's complete the square
h(t) = -16(t^2 - 5t + 25/4 -25/4 + 12
= -16( (t-5/2)^2 - 25/4) + 12
= -16(t-5/2)^2 + 100 + 12
= -16(t-5/2)^2 + 112
the vertex is (5/2 , 112)
so the max of h is 112 , when x = 5/2 as above
Well, if we take a closer look at the equation h(t) = -16t^2 + 80t + 12, we can see that it is a quadratic equation in the form ax^2 + bx + c. In this case, a = -16, b = 80, and c = 12.
To find the maximum height, we need to find the vertex point of the parabolic curve. The x-coordinate of the vertex can be found using the formula x = -b/2a. In this case, x = -80 / (2 * -16) = 80/32 = 2.5.
Now we can substitute this value back into the equation to find the maximum height. h(2.5) = -16(2.5)^2 + 80(2.5) + 12 = -100 + 200 + 12 = 112 feet.
So, the maximum height above ground level that the ball reaches is 112 feet. That's quite a high flyer! Just hope it doesn't mistake itself for a bird and start chirping.
To find the maximum height above ground level that the ball reaches, we need to determine the vertex of the quadratic equation representing the height of the ball.
The equation h(t) = -16t^2 + 80t + 12 represents the height of the ball at time t.
We can see that this is a downward-opening quadratic equation because the coefficient of t^2 (-16) is negative. The vertex of a downward-opening quadratic equation represents the maximum point.
The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a is the coefficient of t^2 (-16) and b is the coefficient of t (80).
Substituting the values, we have:
x = -80 / (2(-16))
x = -80 / (-32)
x = 2.5
Therefore, the maximum height is reached at t = 2.5 seconds.
To find the corresponding height, we substitute t = 2.5 into the equation h(t):
h(2.5) = -16(2.5)^2 + 80(2.5) + 12
Simplifying this equation, we get:
h(2.5) = -16(6.25) + 200 + 12
h(2.5) = -100 + 200 + 12
h(2.5) = 112
Therefore, the maximum height above ground level the ball reaches is 112 feet.
To find the maximum height the ball reaches, we need to determine the vertex of the parabolic function h(t) = -16t^2 + 80t + 12. The vertex of a parabola represents its maximum or minimum point.
The formula for the x-coordinate of the vertex of a parabola is given by x = -b / (2a), where a and b are the coefficients of the quadratic equation.
In this case, a = -16 and b = 80. Plugging these values into the formula, we get:
x = -80 / (2 * -16)
x = -80 / -32
x = 2.5
So, the ball reaches its maximum height at t = 2.5 seconds.
To find the maximum height above ground level, we substitute this value back into the equation h(t):
h(2.5) = -16(2.5)^2 + 80(2.5) + 12
h(2.5) = -16(6.25) + 200 + 12
h(2.5) = -100 + 200 + 12
h(2.5) = 112
Therefore, the maximum height above ground level the ball reaches is 112 feet.