Question
Ted popped a baseball straight up into the air at a speed of 48 ft/s. The height of the ball after t seconds is given by the function h(t) = -16t2 + 48t + 3
When does the baseball hit the ground?
When does the baseball hit the ground?
Answers
GPT 3.5
To find when the baseball hits the ground, we need to determine the value of t when the height of the ball is 0.
Setting h(t) equal to 0, we have:
-16t^2 + 48t + 3 = 0
Now, we can solve this quadratic equation for t. We can either use the quadratic formula or factor the equation, but in this case, the quadratic formula will be most straightforward:
t = (-b ± √(b^2 - 4ac)) / 2a
Substitute a = -16, b = 48, and c = 3 into the formula:
t = (-48 ± √(48^2 - 4(-16)(3))) / 2(-16)
t = (-48 ± √(2304 + 192)) / -32
t = (-48 ± √2496) / -32
Now, calculate the values of t:
t = (-48 + √2496) / -32
t = (-48 + 49.96) / -32
t = 1.96 / -32
t ≈ -0.06
t = (-48 - √2496) / -32
t = (-48 - 49.96) / -32
t = -97.96 / -32
t ≈ 3.06
Since time cannot be negative, the baseball hits the ground at approximately 3.06 seconds after it was thrown.
Setting h(t) equal to 0, we have:
-16t^2 + 48t + 3 = 0
Now, we can solve this quadratic equation for t. We can either use the quadratic formula or factor the equation, but in this case, the quadratic formula will be most straightforward:
t = (-b ± √(b^2 - 4ac)) / 2a
Substitute a = -16, b = 48, and c = 3 into the formula:
t = (-48 ± √(48^2 - 4(-16)(3))) / 2(-16)
t = (-48 ± √(2304 + 192)) / -32
t = (-48 ± √2496) / -32
Now, calculate the values of t:
t = (-48 + √2496) / -32
t = (-48 + 49.96) / -32
t = 1.96 / -32
t ≈ -0.06
t = (-48 - √2496) / -32
t = (-48 - 49.96) / -32
t = -97.96 / -32
t ≈ 3.06
Since time cannot be negative, the baseball hits the ground at approximately 3.06 seconds after it was thrown.