Question
During a softball game, a batter hits a ball upward from an initial height of 3 feet. The height, in feet, of the softball is given by s(t)=−16t2+80t+3
, where t is time in seconds and t≥0
. Which is closest to the time when the softball will be 50 feet above the ground?(1 point)
Responses
To find the time when the softball will be 50 feet above the ground, we need to set the height function equal to 50 and solve for t:
s(t) = -16t^2 + 80t + 3
Setting this equal to 50, we get:
-16t^2 + 80t + 3 = 50
Rearranging the equation:
-16t^2 + 80t - 47 = 0
To solve this quadratic equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = -16, b = 80, and c = -47. Plugging these values into the quadratic formula, we get:
t = (-80 ± √(80^2 - 4(-16)(-47))) / 2(-16)
Simplifying the equation:
t = (-80 ± √(6400 - 3016)) / (-32)
t = (-80 ± √(3384)) / (-32)
Since the problem specifies that t ≥ 0, we can ignore the negative value from the ± symbol. So, we have:
t = (-80 + √3384) / (-32)
Calculating the square root:
t = (-80 + √(4 * 846)) / (-32)
t = (-80 + 2√846) / (-32)
Approximating the value of square root of 846:
t ≈ (-80 + 2 * 29.09) / (-32)
t ≈ (-80 + 58.18) / (-32)
t ≈ -22.18 / (-32)
t ≈ 0.693 seconds
Therefore, the time when the softball will be 50 feet above the ground is closest to 0.693 seconds.
To find the time when the softball will be 50 feet above the ground, we need to solve the equation s(t) = 50.
The equation for the height of the softball is given by s(t) = -16t^2 + 80t + 3. To find the time when the softball is 50 feet above the ground, we set s(t) equal to 50:
-16t^2 + 80t + 3 = 50
To solve this equation, we can subtract 50 from both sides:
-16t^2 + 80t + 3 - 50 = 0
Simplifying further, we have:
-16t^2 + 80t - 47 = 0
Now, we can solve this quadratic equation using methods such as factoring, completing the square, or using the quadratic formula. Once we find the values of t that satisfy this equation, we can determine which value is closest to the desired height of 50 feet.
To find the time when the softball will be 50 feet above the ground, we need to solve the equation s(t) = 50.
-16t^2 + 80t + 3 = 50
Subtracting 50 from both sides, we get:
-16t^2 + 80t - 47 = 0
To solve this quadratic equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = -16, b = 80, and c = -47.
Plugging in these values, we get:
t = (-80 ± √(80^2 - 4(-16)(-47))) / (2(-16))
Simplifying further:
t = (-80 ± √(6400 - 2992)) / -32
t = (-80 ± √3408) / -32
We can take the positive value for t since time cannot be negative.
Calculating:
t ≈ (-80 + √3408) / -32 ≈ 0.6608 seconds
Therefore, the closest time when the softball will be 50 feet above the ground is approximately 0.6608 seconds.