The height of a projectile fired upward is given by the formula
s = v0t − 16t2,
where s is the height in feet,
v0
is the initial velocity, and t is the time in seconds. Find the time for a projectile to reach a height of 96 ft if it has an initial velocity of 128 ft/s. Round to the nearest hundredth of a second.
so you are solving
96 = 128 - 16t^2
16t^2 = 32
t = √2
I will leave the rounding to you.
should have been
96 = 128t - 16t^2
16t^2 - 128t + 96 = 0
t^2 - 8t + 6 = 0
t = 4 + √10 or 4 - √10
There are two possible answers, since the height of 96 feet is reached on its upwards flight, and then again when it comes back down.
To find the time for a projectile to reach a height of 96 ft, we need to solve the equation s = v0t - 16t^2 for the given values of s and v0.
Plugging in the values, we have:
96 = 128t - 16t^2
Now, let's rearrange the equation in standard quadratic form:
16t^2 - 128t + 96 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, factoring is more convenient.
Dividing the equation by 16, we get:
t^2 - 8t + 6 = 0
Now, let's factor the equation:
(t - 2)(t - 6) = 0
Setting each factor equal to zero, we have:
t - 2 = 0 or t - 6 = 0
Solving for t in each case, we get:
t = 2 or t = 6
Therefore, the projectile will reach a height of 96 ft at two different times: 2 seconds and 6 seconds.
To find the time for a projectile to reach a height of 96 ft, we can use the given formula for the height of the projectile:
s = v0t - 16t^2
We need to find the value of t when s = 96 ft and v0 = 128 ft/s.
Plugging in these values into the formula, we get:
96 = 128t - 16t^2
To solve this equation, we can rearrange it into a quadratic equation:
16t^2 - 128t + 96 = 0
Next, we can solve this quadratic equation to find the time when the projectile reaches a height of 96 ft. We can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 16, b = -128, and c = 96. Plugging in these values, we get:
t = (128 ± √((-128)^2 - 4 * 16 * 96)) / (2 * 16)
Simplifying further, we get:
t = (128 ± √(16384 - 6144)) / 32
t = (128 ± √(10240)) / 32
Now, we can calculate the two values of t:
t1 = (128 + √(10240)) / 32
t2 = (128 - √(10240)) / 32
Using a calculator, we can evaluate these expressions:
t1 ≈ 1.91 seconds
t2 ≈ 4.09 seconds
Therefore, the time it takes for the projectile to reach a height of 96 ft is approximately 1.91 seconds (rounded to the nearest hundredth of a second).