The function y = –16t2 + 224 models the height y in feet of a stone t seconds after it is dropped from the edge of a vertical cliff. How long will it take the stone to hit the ground? Round to the nearest hundredth of a second.
(1 point)
Responses
0.25 seconds
0.25 seconds
5.29 seconds
5.29 seconds
7.48 seconds
7.48 seconds
3.74 seconds
To find the time it takes for the stone to hit the ground, we need to find when the height y is equal to 0.
Setting the equation y = -16t^2 + 224 equal to 0, we have:
-16t^2 + 224 = 0
Divide both sides of the equation by -16:
t^2 - 14 = 0
Add 14 to both sides of the equation:
t^2 = 14
Take the square root of both sides of the equation:
t = ±√14
Since time cannot be negative in this context, we only consider the positive solution:
t ≈ √14 ≈ 3.74 seconds
Therefore, it will take the stone approximately 3.74 seconds to hit the ground.
To find the time it takes for the stone to hit the ground, we need to set y equal to zero in the function y = -16t^2 + 224 and solve for t.
0 = -16t^2 + 224
To solve for t, we can divide both sides of the equation by -16:
0 = t^2 - 14
Next, we can subtract 224 from both sides:
-224 = t^2 - 14
Now, we can add 14 to both sides:
-210 = t^2
Finally, we can take the square root of both sides:
sqrt(-210) = t
However, we cannot take the square root of a negative number, so we can conclude that the stone will not hit the ground.
Therefore, the correct answer is None of the above.
To find the time it takes for the stone to hit the ground, we need to determine when the height y is equal to zero. In other words, we need to solve the equation -16t^2 + 224 = 0.
To solve this quadratic equation, we can use the quadratic formula, which states that if we have an equation of the form ax^2 + bx + c = 0, the solution for x is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, the equation is -16t^2 + 224 = 0, which means a = -16, b = 0, and c = 224. Substituting these values into the quadratic formula, we get:
t = (-0 ± √(0^2 - 4(-16)(224))) / (2(-16))
Simplifying further:
t = ± √(64) / (-32)
t = ±8 / (-32)
t = ± 1/4
So the possible solutions for t are 1/4 and -1/4. However, in this context, a negative time does not make sense, so we can discard the negative solution.
Therefore, the stone will hit the ground at t = 1/4 seconds. Rounded to the nearest hundredth of a second, the answer is approximately 0.25 seconds.
So the correct answer is: 0.25 seconds.