How would I work this?
The equation h= -16t^2 +112t gives the height of an arrow shot upward from the ground with an initial velocity of 112 ft/s, where t is the time after the arrow leaves the ground. Find the time it takes for the arrow to reach a height of 120 ft.
set your h equal to 120 and solve that routine quadratic equation.
You will get two answers for t, one will be the time to reach 120 feet on its way up, and the other the time when it was 120 feet high on its way down.
To find the time it takes for the arrow to reach a height of 120 ft, we can set the equation h = 120 and solve for t.
Given equation: h = -16t^2 + 112t
Height: h = 120 ft
Substitute the value of h into the equation:
120 = -16t^2 + 112t
Rearrange the equation to obtain a quadratic equation:
-16t^2 + 112t - 120 = 0
To solve this equation, we can use factoring or the quadratic formula. We'll use factoring here.
First, divide the equation by -8 to simplify it:
2t^2 - 14t + 15 = 0
Now, factor the equation:
(2t - 3)(t - 5) = 0
Set each factor equal to zero and solve for t:
2t - 3 = 0 => 2t = 3 => t = 3/2
t - 5 = 0 => t = 5
Therefore, the arrow reaches a height of 120 ft at two different times:
1) t = 3/2 seconds
2) t = 5 seconds
To find the time it takes for the arrow to reach a height of 120 ft, we can set up the equation and solve for t.
The given equation is h = -16t^2 + 112t, where h represents the height and t represents time.
We are given that h = 120 ft. Substitute this value into the equation:
120 = -16t^2 + 112t
To solve for t, we can rearrange the equation to get it in quadratic form:
16t^2 - 112t + 120 = 0
Now, we can solve this quadratic equation using either factoring, completing the square, or the quadratic formula. In this case, factoring is the most straightforward approach.
Divide the equation by 4 to simplify:
4t^2 - 28t + 30 = 0
Now, look for two numbers that multiply to give the product of the coefficient of t^2 (4) and the constant term (30), which is 4 * 30 = 120. These numbers should also add up to give the coefficient of t (-28). In this case, the numbers are -2 and -15.
So, we can rewrite the equation as:
(2t - 2)(2t - 15) = 0
Now, set each factor equal to zero and solve for t:
2t - 2 = 0 or 2t - 15 = 0
For the first factor:
2t - 2 = 0
2t = 2
t = 1
For the second factor:
2t - 15 = 0
2t = 15
t = 15/2 = 7.5
So, the time it takes for the arrow to reach a height of 120 ft is 1 second or 7.5 seconds.