The equation y = -16t^2 + 60t describes the height (in feet) of a projectile launched from the ground at 60 feet per second upward. In how many seconds will the projectile first reach 56 feet in height? Express your answer as a decimal rounded to the nearest hundredth.
To find the time it takes for the projectile to reach a height of 56 feet, we need to solve the equation:
-16t^2 + 60t = 56
Let's set the equation equal to 0 by subtracting 56 from both sides:
-16t^2 + 60t - 56 = 0
To solve this quadratic equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, the coefficients are:
a = -16
b = 60
c = -56
Plugging in these values, we can calculate the solutions for t:
t = (-60 ± √(60^2 - 4(-16)(-56))) / (2(-16))
Simplifying the equation further:
t = (-60 ± √(3600 - 3584)) / (-32)
t = (-60 ± √16) / -32
Calculating the square root:
t = (-60 ± 4) / -32
This gives us two possible values for t:
t1 = (-60 + 4) / -32
t2 = (-60 - 4) / -32
Simplifying further:
t1 = -56 / -32 ≈ 1.75
t2 = -64 / -32 ≈ 2.00
Since time cannot be negative, we discard t2 = 2.00 and take t1 = 1.75 as the answer.
Therefore, the projectile will reach a height of 56 feet in approximately 1.75 seconds.
To find the time it takes for the projectile to reach a height of 56 feet, we need to solve the equation y = 56 for t.
Given the equation y = -16t^2 + 60t, we can substitute y with 56 to get:
56 = -16t^2 + 60t
This equation is a quadratic equation. To solve it, we need to set it equal to zero by subtracting 56 from both sides:
0 = -16t^2 + 60t - 56
Now we can use the quadratic formula to find the values of t that satisfy this equation. The quadratic formula is given by:
t = (-b ± sqrt(b^2 - 4ac)) / (2a)
In this case, a = -16, b = 60, and c = -56. Substituting these values into the quadratic formula:
t = (-60 ± sqrt(60^2 - 4(-16)(-56))) / (2(-16))
Simplifying:
t = (-60 ± sqrt(3600 - 3584)) / (-32)
t = (-60 ± sqrt(16)) / (-32)
t = (-60 ± 4) / (-32)
This gives us two possible values for t:
t1 = (-60 + 4) / (-32) = -56 / -32 = 1.75
t2 = (-60 - 4) / (-32) = -64 / -32 = 2
Both values are possible solutions, but we need to find the one where the projectile first reaches 56 feet in height. Since time cannot be negative, the answer is t = 1.75 seconds.
Therefore, the projectile will first reach a height of 56 feet after approximately 1.75 seconds.
-16t^2 + 60t = 50
(4t-5)(5-2t) = 0