A spotlight is located on the ground 20 meters from a wall. A person walks away from the wall at a constant rate of 1 meter per second. How fast is the person's shadow on the wall lengthening when they are 15 meters from the wall?
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Let's assume that the person is standing at point A, the spotlight is at point O, and the top of the wall is point B. We need to find the rate at which the length of the person's shadow (AB) is changing when they are 15 meters from the wall.
Using similar triangles, we can set up the following proportion:
AB / OA = OB / OA
Since OA is a constant 20 meters and the person is moving away from the wall at a rate of 1 meter per second, we can substitute the known values into the equation as follows:
AB / 20 = OB / (20 + 1t)
where t is the time in seconds.
To find the rate at which AB is changing when the person is 15 meters from the wall, we can substitute AB = 15 and solve for OB:
15 / 20 = OB / (20 + 1t)
Simplifying the equation, we have:
15(20 + 1t) = 20 x OB
Expanding and rearranging, we get:
300 + 15t = 20 x OB
Dividing both sides by 20:
15 + 0.75t = OB
To find the rate at which AB is changing, we can differentiate both sides of the equation with respect to time t:
d/dt [15 + 0.75t] = d/dt [OB]
0.75 = dOB/dt
Therefore, the rate at which the person's shadow on the wall lengthens is 0.75 meters per second when they are 15 meters from the wall.
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