A ball is thrown straight up from a rooftop 240 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 120 feet and identify the solution as a point on the graph.
h equals negative 16 t squared plus 8 t plus 240
just solve 16t^2+8t+240 = 120
To determine when the ball's height will be 120 feet, we need to set the equation equal to 120 and solve for t.
The equation is:
h = -16t^2 + 8t + 240
Setting h equal to 120, we have:
120 = -16t^2 + 8t + 240
Next, we need to rearrange the equation to bring it into quadratic form (ax^2 + bx + c = 0):
-16t^2 + 8t + 240 - 120 = 0
Simplifying, we have:
-16t^2 + 8t + 120 = 0
Now, we can solve this quadratic equation using either factoring, the quadratic formula, or completing the square. Let's use factoring in this case. We notice that we can factor out a common factor of 8 from each term:
8(-2t^2 + t + 15) = 0
Now, we can factor the quadratic trinomial inside the parentheses:
8(-2t + 5)(t + 3) = 0
Setting each factor equal to zero, we have:
-2t + 5 = 0 or t + 3 = 0
Solving each equation for t, we get:
-2t = -5 --> t = 5/2
t = -3
Therefore, the two solutions for t are t = 5/2 and t = -3. However, since the ball is thrown upwards from the rooftop, we can ignore the negative value for t=-3 (since time cannot be negative in this context).
So the solution for when the ball's height will be 120 feet is t = 5/2 seconds.
To find the point on the graph, plug this value of t back into the equation:
h = -16(5/2)^2 + 8(5/2) + 240
Simplifying, we get:
h = -200/4 + 40/2 + 240
h = -50 + 20 + 240
h = 210
Therefore, the point on the graph when the ball's height is 120 feet is (5/2, 210).