A ball is thrown straight up from a rooftop 9696 feet high. The formula below describes the ball's height above the ground, h, in feet, t seconds after it was thrown. The ball misses the rooftop on its way down. The graph of the formula is shown. Determine when the ball's height will be 4848 feet and identify the solution as a point on the graph.
h equals negative 16 t squared plus 8 t plus 96h=−16t2+8t+96
I am going to assume that the roof is 96 ft high, not 9696, and you want to know when the ball is 48 ft high, not 4848
so 48 = -16t^2 + 8t + 96
(the equation implies it was tossed upwards with a velocity of 8 ft/sec)
16t^2 - 8t - 48 = 0
2t^2 - t - 6 = 0
(2t + 3)(t - 2) = 0
t = -3/2 or t = 2
but t ≥ 0
the ball was 48 ft high after 2 seconds
To determine when the ball's height will be 48 feet, we need to set the equation h = 48 and solve for t.
The given equation is h = -16t^2 + 8t + 96.
Substituting h = 48, we get:
48 = -16t^2 + 8t + 96.
Rearranging it to the standard quadratic equation form:
-16t^2 + 8t + 48 - 96 = 0,
-16t^2 + 8t - 48 = 0.
Dividing the equation by -8 to simplify it further, we have:
2t^2 - t + 6 = 0.
To solve this quadratic equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a,
where a = 2, b = -1, and c = 6.
Plugging in these values, we get:
t = (-(-1) ± √((-1)^2 - 4(2)(6))) / (2(2)),
t = (1 ± √(1 - 48)) / 4,
t = (1 ± √(-47)) / 4.
Since the square root of a negative number is imaginary, there are no real solutions for this quadratic equation. Therefore, the ball will never be at a height of 48 feet. Thus, there is no solution point on the graph for h = 48.
To determine when the ball's height will be 48 feet, we can substitute 48 for h in the equation and solve for t.
Starting with the equation: h = -16t^2 + 8t + 96
Substitute 48 for h: 48 = -16t^2 + 8t + 96
Rearrange the equation to set it equal to zero:
-16t^2 + 8t + 96 - 48 = 0
-16t^2 + 8t + 48 = 0
Divide the entire equation by -8 to simplify the equation:
2t^2 - t - 6 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Let's use factoring to solve it:
Factor the quadratic equation:
(2t + 3)(t - 2) = 0
Set each factor equal to zero and solve for t:
2t + 3 = 0 -> 2t = -3 -> t = -3/2
t - 2 = 0 -> t = 2
So there are two possible solutions: t = -3/2 and t = 2. However, since the ball's height cannot be negative, we discard the negative solution.
Therefore, the ball's height will be 48 feet at t = 2 seconds.
To identify this solution as a point on the graph, we can plot the point (2, 48) on the graph. The x-coordinate represents time (t) in seconds, and the y-coordinate represents height (h) in feet.