What is the rate of change of angle in a 90 degree triangle?

In summary: This concludes the summary for question 1.In summary, the author doubts their response to question 1 and is looking for help.
  • #1
ardentmed
158
0
Hey guys,

I need some more help for this problem set I've been working on. I'm doubting some of my answers and I'd appreciate some help.

This is only for question 1. Ignore 2.
Question:
08b1167bae0c33982682_19.jpg


For the first one, drawing out the question clearly tells us that we are working with a 90 degree triangle. Thus, we can calculate the angle via cosØ and the Pythagorean formula.

The derivative thereof is:

-sinØ dØ/dt = [dx/dt (s) - ds/dt (x)] / s^2

We can substitute for values and solve for ds/dt, which leads to:

ds/dt = 1.73205 m/s (is this supposed to be an exact value?


Thus, dØ/dt can be calculated via substitution of the newly calculated ds/dt value. From isolation, we get:

dØ/dt = -0.055 m/s (or decreasing at a rate of 0.055 m/s. I'm highly doubtful of my response to this question.

Thanks in advance.
 
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  • #2
Beginning with a sketch is a very good first step:

View attachment 2841

As we can see, we may write:

\(\displaystyle \cot(\theta)=\frac{x}{50}\)

Now, if we implicitly differentiate with respect to time $t$, we get:

\(\displaystyle -\csc^2(\theta)\d{\theta}{t}=\frac{1}{50}\d{x}{t}\)

Now, since:

\(\displaystyle \csc(\theta)=\frac{L}{50}\)

we may write:

\(\displaystyle \d{\theta}{t}=-\frac{50}{L^2}\d{x}{t}\)

Now, what do you get when you plug in the given values for \(\displaystyle L,\,\d{x}{t}\)?
 

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  • #3
MarkFL said:
Beginning with a sketch is a very good first step:

View attachment 2841

As we can see, we may write:

\(\displaystyle \cot(\theta)=\frac{x}{50}\)

Now, if we implicitly differentiate with respect to time $t$, we get:

\(\displaystyle -\csc^2(\theta)\d{\theta}{t}=\frac{1}{50}\d{x}{t}\)

Now, since:

\(\displaystyle \csc(\theta)=\frac{L}{50}\)

we may write:

\(\displaystyle \d{\theta}{t}=-\frac{50}{L^2}\d{x}{t}\)

Now, what do you get when you plug in the given values for \(\displaystyle L,\,\d{x}{t}\)?

Plugging in the aforementioned values results in the following expression:

$\d{\theta}{t}$ = (-50*2)/(100^2)
$\d{\theta}{t}$ = -1/100 rad/minute (albeit I'm doubting the units for the final answer. Is it m/s, since the other units were also m/s? I just assumed rad/minute due to commonly seeing it used in related rates questions with angles.)

Thanks again.
 
  • #4
Yes, the angle is decreasing at a rate of 1/100 rad/s at that point in time. If you look at the right side of:

\(\displaystyle \d{\theta}{t}=-\frac{50}{L^2}\d{x}{t}\)

Performing a dimensional analysis, we see the units are:

\(\displaystyle \frac{\text{m}}{\text{m}^2}\cdot\frac{\text{m}}{\text{s}}=\frac{1}{\text{s}}\)

Because we measure angles in radians, and being the ratio of two lengths, are dimensionless, so we take the rate of change here to be radians per second.
 

FAQ: What is the rate of change of angle in a 90 degree triangle?

What are related rates and how are they calculated?

Related rates refer to the concept of how different variables are changing in relation to one another. They are calculated using derivatives, specifically the chain rule, to determine the rate of change of one variable with respect to another.

How are related rates and angles connected?

Angles are often used to describe the relationship between two variables in a related rates problem. They can be used to set up equations and determine the rates of change of these variables.

What are some common real-life applications of related rates and angles?

Related rates and angles can be used to solve problems involving motion, such as the rate at which a ladder is sliding down a wall or the rate at which a balloon is rising. They are also useful in calculating the rate of change of quantities in chemistry and physics problems.

What are some common challenges when solving related rates problems?

One of the main challenges with related rates problems is setting up the correct equation and identifying which variables are changing in relation to one another. It can also be difficult to visualize the problem and determine which geometric concepts and formulas are needed to solve it.

What are some tips for solving related rates problems more efficiently?

It can be helpful to draw a diagram and label all the given information before attempting to solve the problem. Additionally, it is important to carefully read the problem and identify the relationship between the variables. Breaking the problem into smaller, more manageable steps can also make it easier to solve. Practice and familiarity with different types of related rates problems can also improve efficiency in solving them.

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