For a typical basketball shot, the ball's height in feet will be a function of time in flight in seconds, modeled by an equation such as h = -16t^2 + 40t + 6. Find when the shot might reach the height of the basket which is 10 feet.
well, just solve
-16t^2 + 40t + 6 = 10
You'll get two answers, one going up and one coming back down.
To find when the shot might reach the height of 10 feet, you need to solve the equation h = 10.
The given equation that models the shot's height is h = -16t^2 + 40t + 6.
Substitute h with 10 in the equation: 10 = -16t^2 + 40t + 6.
Now, we have a quadratic equation. To solve it, we can set the equation equal to zero by subtracting 10 from both sides: 0 = -16t^2 + 40t - 4.
Next, we can rearrange the equation to put it in standard quadratic form: -16t^2 + 40t - 4 = 0.
Since this quadratic equation cannot be factored easily, we can use the quadratic formula: t = (-b ± √(b^2 - 4ac)) / (2a).
In our case, a = -16, b = 40, and c = -4.
Plugging these values into the quadratic formula, we get:
t = (-40 ± √(40^2 - 4(-16)(-4))) / (2(-16)).
Simplifying further:
t = (-40 ± √(1600 - 256)) / (-32),
t = (-40 ± √1344) / -32.
Now, we can calculate the two possible values for t:
t1 = (-40 + √1344) / -32,
t2 = (-40 - √1344) / -32.
Using a calculator:
t1 ≈ 0.194 seconds,
t2 ≈ 2.556 seconds.
Therefore, the shot might reach the height of 10 feet around 0.194 seconds and 2.556 seconds in flight.