"Published in Newark, California, USA"
A weight is attached to one end of a 33-foot rope which passes over a pulley 18 feet above the ground. The other end is attached to a truck at a point 3 feet above the ground. If the truck moves away at a rate of 2 feet per second, how fast is the weight rising when the truck is 8 feet from the spot directly under the pulley?
Solution:
To visualize the problem, let's draw the figure as follows
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| Photo by Math Principles in Everyday Life |
As a truck moves away from the weight, a right triangle is formed between the weight, pulley, and a truck
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| Photo by Math Principles in Everyday Life |
By Pythagorean Theorem
In this problem, y is a constant because the height of a pulley from the ground is already fixed. As a truck moves away from the weight, the length of a rope from the truck to the pulley increases. Take the derivative of the above equation with respect to time t as follows
In order to get the value of dc/dt, we need to solve for the value of c first because it is not given in the problem. Since the distance of a truck from the spot directly under the pulley, the height of a truck from the ground, and the height of a pulley from the ground are given, we can solve for the value of c which is the length of a rope from the truck to the pulley as follows
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| Photo by Math Principles in Everyday Life |
By Pythagorean Theorem
The rate of increasing the length of a rope as a truck moves away from the spot directly under the pulley is also the same as the rate of rising the weight from the ground. Therefore, the rate of rising the weight from the ground is
