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Fall time calculations.
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PART A
Polly has decided to end herself. To be more funny with it, she will get someone to see how long it would take for her to fall off a 70 foot building?
The equation for Polly's fall is -16t² - 16t + 66. Using this equation, find how long it will take for her to fall. Round to the nearest thousandth.
PART B
Polly jumped off a building into a sinkhole, so it will take longer to fall down. Using a rock, you determined that the sinkhole is an extra 40 feet deep. How can you change the equation to account for this? Solve for when she will hit the bottom of the sinkhole when you have obtained the equation and round to the nearest thousandth.
PART C
Using the quadratic equation, find the negative time where she would have also been at the bottom of the sinkhole.
PART D [OPTIONAL]
If Polly had instead jumped off a 110 foot building into a 30 foot deep sinkhole, when will she reach the ground? Round to the nearest thousandth.
ChatGPT
PART A:
To find how long it will take for Polly to fall, we need to solve the equation -16t² - 16t + 66 = 0 for t. We can use the quadratic formula:
t = (-b ± sqrt(b² - 4ac)) / 2a
where a = -16, b = -16, and c = 66.
Plugging in the values, we get:
t = (-(-16) ± sqrt((-16)² - 4(-16)(66))) / 2(-16)
t = (16 ± sqrt(16² + 4224)) / (-32)
t = (16 ± sqrt(4224 + 256)) / (-32)
t = (16 ± sqrt(4480)) / (-32)
t ≈ 2.059 seconds or t ≈ -2.559 seconds
Since we're only interested in the positive time, we can discard the negative solution. Therefore, it will take Polly approximately 2.059 seconds to fall off the 70-foot building.
Answer: Approximately 2.059 seconds.
PART B:
If the sinkhole is an extra 40 feet deep, then the new equation would be -16t² - 16t + 106, where 106 is obtained by adding 40 to the original constant term of 66.
To find when she will hit the bottom of the sinkhole, we need to solve -16t² - 16t + 106 = 0 for t. We can use the quadratic formula as before:
t = (-b ± sqrt(b² - 4ac)) / 2a
where a = -16, b = -16, and c = 106.
Plugging in the values, we get:
t = (-(-16) ± sqrt((-16)² - 4(-16)(106))) / 2(-16)
t = (16 ± sqrt(16² + 4(16)(106))) / (-32)
t = (16 ± sqrt(6912)) / (-32)
t ≈ 3.130 seconds or t ≈ -2.005 seconds
Since we're only interested in the positive time, we can discard the negative solution. Therefore, it will take Polly approximately 3.130 seconds to hit the bottom of the sinkhole.
Answer: Approximately 3.130 seconds.