Question
Graduate student Ryan Lukeman studied the behaviour of duck flocks swimming near Canada Place in Vancouver, BC. This figure from his PhD thesis shows his photography set-up. Here
meters is the height from sea level up to his camera aperture at the observation point,
meters is the width of a pier (a stationary platform whose size is fixed), and
is the distance from the pier to the leading duck in the flock (in meters).
is a visual angle subtended at the camera, as shown.
An observer at height H photographing ducks.
If the visual angle is increasing at the rate of 1/100 radians per second, at what rate is the distance
changing at the instant that
meters?
Hints
Hint 1
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Consider the angles ![{\displaystyle \theta +\alpha }](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/19fe2e67812e22362c25027ff09ffa62651a3fa8) and ![{\displaystyle \alpha }](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3) and write down trigonometric functions relating these angles to the distances in the diagram.
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Solutions
Solution
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Consider the angles in the diagram, and note the definition of the angle .
Consider the angles ![{\displaystyle \theta +\alpha }](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/19fe2e67812e22362c25027ff09ffa62651a3fa8) and ![{\displaystyle \alpha }](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3) and write down trigonometric functions relating these angles to the distances in the diagram.
Then .
.
We are told that the visual angle is increasing at the date of 1/100 radians per sec. This means that
Taking the first derivative:
meters per second.
If the visual angle is increasing at radians per second, then at the instant meters, meters per second.
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