A trebuchet launches a projectile from a hilltop 30 feet above ground level on a parabolic arc at a velocity of 40 feet per second. The equation h = −16t2 + 40t + 30 models the projectile's h height at t seconds. How long will it take for the projectile to hit its target on the ground? (to the nearest tenth of a second)
A) 2.8
B) 3.1
C) 3.5
D) 3.9
plug in zero for h , and solve the quadratic for t
quadratic formula works
lik
3.1 seconds
To find out how long it will take for the projectile to hit its target on the ground, we need to find the time (t) when the height (h) of the projectile is equal to zero.
The equation given is h = -16t^2 + 40t + 30.
Setting h equal to zero, we have:
0 = -16t^2 + 40t + 30.
To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, factoring is not straightforward, so we'll use the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac)) / (2a),
where a, b, and c are the coefficients from the quadratic equation in the form ax^2 + bx + c = 0.
In our case, a = -16, b = 40, and c = 30.
Plugging in these values into the quadratic formula, we have:
t = (-40 ± sqrt(40^2 - 4(-16)(30))) / (2(-16)).
Simplifying further:
t = (-40 ± sqrt(1600 + 1920)) / -32,
t = (-40 ± sqrt(3520)) / -32.
Now, we can calculate the two possible values of t:
t1 = (-40 + sqrt(3520)) / -32,
t2 = (-40 - sqrt(3520)) / -32.
Calculating these values using a calculator:
t1 ≈ 3.118,
t2 ≈ -0.618.
Since time cannot be negative in this context, we can disregard t2.
Therefore, the projectile will hit its target on the ground approximately 3.1 seconds after launch.
So, the answer is B) 3.1.