a wrench is dropped by a worker at a construction site. four seconds later the worker hears it hit the ground below. How high is the worker above the ground? (The velocity of sound is 1100 ft/s, and the distance the wrench falls as a function of time is s=6t^2
so, you have (assuming the real world)
d = 16t^2, so t = √(d/16)
That's how long it takes to fall a distance d.
The sound takes another d/1100 seconds to get back. Thus, you have
√d/4 + d/1100 = 4
That's just a quadratic equation, so finding √d is easy
To find the height of the worker above the ground, we need to determine the time it took for the wrench to fall.
Given that the equation for the distance the wrench falls as a function of time is s = 6t^2, we can set s = 0 to represent the distance fallen when the wrench hits the ground.
Therefore, 0 = 6t^2
Dividing both sides by 6, we get:
0 = t^2
Since t represents time and time cannot be negative, this equation implies that t = 0.
However, we know that it took four seconds for the worker to hear the wrench hit the ground. Since sound travels at 1100 ft/s, it takes four seconds for the sound to reach the worker.
Therefore, the actual time it took for the wrench to fall is (4 + 4) = 8 seconds.
Using this time, we can find the height of the worker above the ground:
s = 6t^2
s = 6(8^2)
s = 6(64)
s = 384 feet
Therefore, the worker is 384 feet above the ground.
To determine the height of the worker above the ground when the wrench is dropped, we need to find the time it takes for the sound of the wrench hitting the ground to reach the worker.
We know that the velocity of sound is 1100 ft/s, so it travels at this speed from the ground to the worker. The time it takes for the sound to travel is the same as the time it takes for the wrench to fall.
To find this time, we can equate the distance the wrench falls to the distance covered by the sound:
s(wrench) = s(sound)
Using the distance equation for the wrench: s(wrench) = 6t^2
And the distance equation for the sound: s(sound) = velocity * time = 1100t
Setting the two equations equal and solving for t:
6t^2 = 1100t
Rearranging the equation:
6t^2 - 1100t = 0
Factoring out t:
t(6t - 1100) = 0
Now we have two possible values for t: t = 0 or t = 1100/6.
Since t cannot be 0 (we're looking for the time it takes for the wrench to fall), we take the second solution:
t = 1100/6
Now that we have the time it takes for the sound to reach the worker, we can calculate the distance the worker is above the ground using the equation for the wrench's distance:
s(wrench) = 6t^2
Substituting t = 1100/6:
s(wrench) = 6 * (1100/6)^2
s(wrench) = 6 * (1100^2/6^2)
s(wrench) = 6 * (1100^2/36)
s(wrench) ≈ 200 * 1100^2
Calculating this value:
s(wrench) ≈ 200 * 1210000
s(wrench) ≈ 242,000,000 ft
Therefore, the worker is approximately 242,000,000 ft above the ground.