Mastering Related Rates: Solving Tricky Problems

[scorpa]

Hello,

I'm having some troubles with some more rates questions and was wondering if someone could help me out.

A construction worker pulls a 5m plank up the side of a building under construction by means of a rope tied to the end of a plank. The opposite end of the plank is being dragged along the ground. If the worker is pulling at a rate of 15 cm/s, how fast is the end of the plank sliding along the ground when it is 2m from the wall of the building?

A fish is being reeled in at a rate of 30cm/s from a bridge 4m above water. At what rate is the angle (in rad/s) between the line and the water chaning when there is 8m of line out?

The relation between distance s and velocity v is given by v = 150s/(3+s). Find acceleration in terms of s.

I had no clue how to do the first two, but I thought I could do the second so I found the first and second derivative of the equation thinking that it would give me the answer but it did not, so I guess I am more lost than I though. Any help would be greatly appreciated.

 

[whozum]

You only need to use the chain rule and the first derivative, there's no need for third derivatives in the fisrt two. Try to come up with equations relating the distance from the ladder base to the wall, the angle theta, and the height the ladder reaches. Drawing a triangle is very helpful

For the last however, use the relationship

$$a = \frac{dv}{dt}$$

 

[thepatientmental]

Hi there,

Related rates questions can definitely be tricky, but with some practice and understanding of the concepts, they can become easier to solve. Let's break down each of these questions and see if we can come up with a solution.

For the first question, we can use the relationship between the rates of change for similar triangles. Since the plank is being pulled up the building, we can create a right triangle with the plank as the hypotenuse and the vertical distance being pulled as one of the legs. The rate of change for the vertical distance is 15 cm/s, and the length of the plank is 5m. We can use the Pythagorean theorem to find the horizontal distance being pulled, which is the same as the rate of change for the distance along the ground. Then, we can use the chain rule to find the rate of change for the distance along the ground when the plank is 2m from the wall.

For the second question, we can use the Law of Sines to find the angle between the line and the water. Then, we can use the chain rule to find the rate of change for this angle when there is 8m of line out.

For the third question, we can use the quotient rule to find the acceleration in terms of s. Remember that acceleration is the rate of change of velocity, so we can use the given relationship between velocity and distance to find the acceleration.

I hope this helps! Remember to always set up a diagram and use the given information to create a relationship between the rates of change. Good luck!