A strees light L hangs H feet above a flat straight street. A person h feet tall at point O directly under the street light at time t=0 seconds, moves in a straight line along the street a v feet/second. If H=20 feet, h=6 feet, and v=4 miles per hour, find the velocity v of the tip of the person's shadow at t=3 seconds, how long is the person's shadow
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To find the velocity of the tip of the person's shadow at t=3 seconds, we can use similar triangles and basic trigonometry. Here's how to calculate it:
1. Convert the velocity from miles per hour to feet per second:
There are 5280 feet in a mile, and 60 seconds in a minute. Therefore, to convert from miles per hour to feet per second, we can multiply the given velocity by (5280 feet/mile) / (60 seconds/minute):
v = 4 miles/hour * (5280 feet/mile) / (60 seconds/minute) = 352 feet/minute = 5.87 feet/second (rounded to two decimal places).
2. Calculate the length of the person's shadow:
Since the street light is H feet above the street, and the person is h feet tall, the length of the person's shadow can be found using similar triangles.
In the larger triangle formed by the street light, person, and shadow, the ratio of the person's height to the length of their shadow will be the same as the ratio of the height of the street light to the length of the person's shadow. Therefore, we can set up the following proportion:
(Length of person's shadow) / h = (Length of person's shadow + H) / H
Solving for the length of the person's shadow:
(Length of person's shadow) = h * (H / (H - h))
(Length of person's shadow) = 6 * (20 / (20 - 6)) = 140 / 7 = 20 feet.
The length of the person's shadow is 20 feet.
3. Find the displacement of the shadow over time:
The person is moving along the street with a velocity of v = 5.87 feet/second. The tip of the person's shadow will move with the same velocity but opposite direction. Therefore, the displacement of the shadow over time can be calculated as:
Displacement = -v * t
Displacement = -5.87 feet/second * 3 seconds = -17.61 feet.
The displacement of the tip of the person's shadow at t=3 seconds is -17.61 feet.
Hence, the velocity of the tip of the person's shadow at t=3 seconds is -5.87 feet/second (opposite direction) and the person's shadow is 20 feet long.