Calculus Related Rates Math Problems

In summary, the problem involves a 10 ft ladder sliding down a vertical wall at a rate of 1.1 ft/s. Using the information given, the angle between the ladder and the ground can be found by setting up a right triangle and using the cosine function. After deriving the function, the angle can be found when the ladder is 8 ft from the wall. The answer should be -.183 rad/s, but there may be a possibility that the answer is expected in degrees/sec.
  • #1
gina4930
10
0
A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1.1 ft/s, how fast is the angle between the ladder and the ground changing when the bottom of the ladder is 8 ft from the wall?
 
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  • #2
You can't just ask a question here. You need to show an attempt. Label some variables and try it.
 
  • #3
dx/dt=1.1 ft/sec
x=8ft
cos(theta)=x/10
-sin(theta)*dtheta/dt=1/10 * dx/dt
-sin(.644) * dtheta/dt=1/10 * 1.1
dtheta/dt=-.183

I'm not getting the correct answer. I don't know what I'm doing wrong. Help!
 
  • #4
gina4930 said:
dx/dt=1.1 ft/sec
x=8ft
cos(theta)=x/10
-sin(theta)*dtheta/dt=1/10 * dx/dt
-sin(.644) * dtheta/dt=1/10 * 1.1
dtheta/dt=-.183

I'm not getting the correct answer. I don't know what I'm doing wrong. Help!

Did you draw a picture of the situation?
What does x represent? You have that x = 8ft. Does this mean that x is always 8 ft or just at a particular moment?
 
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  • #5
gina4930 said:
dx/dt=1.1 ft/sec
x=8ft
cos(theta)=x/10
-sin(theta)*dtheta/dt=1/10 * dx/dt
-sin(.644) * dtheta/dt=1/10 * 1.1
dtheta/dt=-.183

I'm not getting the correct answer. I don't know what I'm doing wrong. Help!

That actually looks pretty good unless I am missing something too. There is an easier and more accurate way to find sin(theta) than the way you did it, though. Do you know what the answer you are supposed to get is?
 
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  • #6
Mark44- x represents the length of the ground. I think it means that x=8 at that particular moment

Dick- What is the more accurate way to find sin(theta)? I don't know what the correct answer is supposed to be. It's an online problem so I would assume it needs to be as specific as possible. My previous answers include: -.183 , -.2 , and -.1375 all in rad/s. Would it be possible for the answer to be positive?
 
  • #7
gina4930 said:
Mark44- x represents the length of the ground. I think it means that x=8 at that particular moment

Dick- What is the more accurate way to find sin(theta)? I don't know what the correct answer is supposed to be. It's an online problem so I would assume it needs to be as specific as possible. My previous answers include: -.183 , -.2 , and -.1375 all in rad/s. Would it be possible for the answer to be positive?

You found theta by using cos(theta)=8/10 since the adjacent side is 8 and the hypotenuse is 10. What's the opposite side? Use it to find sin(theta) directly.
 
  • #8
I did what you said and got the answer of -.183, which I already tried and it was wrong. I tried .183 and it is still incorrect. I don't know what I'm doing wrong. Do you have any suggestions?
 
  • #9
gina4930 said:
I did what you said and got the answer of -.183, which I already tried and it was wrong. I tried .183 and it is still incorrect. I don't know what I'm doing wrong. Do you have any suggestions?

I would say your answer SHOULD be correct. My only other suggestion is that you can write sin(theta)=6/10 exactly. Do you see why?
 
  • #10
I tried that and it made no difference. I am going to approach my Professor about it and hopefully she will help me.
 
  • #11
Maybe they're looking for an answer in degrees/sec.
 
  • #12
I think you should try leaving the base of the triangle as 1.1t and not replace it with 8 until after you have derived.

So you should first get a function of θ(t), then derive and then replace time t with the time that the ladder would be 8 feet from the ground.
 

FAQ: Calculus Related Rates Math Problems

What is a "related rates" problem in calculus?

A related rates problem in calculus involves finding the rate of change of one quantity with respect to another related quantity. This is done by using the chain rule to take derivatives and setting up an equation that relates the two rates.

How do I approach solving a related rates problem?

The key to solving a related rates problem is to first identify the rates and variables involved, and then use the given information to set up a relationship between them. From there, you can take derivatives and solve for the unknown rate using algebraic manipulation. It is important to always double check your work and make sure the units are consistent.

What are some common tips for solving related rates problems?

Some tips for solving related rates problems include drawing a diagram to visualize the problem, using the chain rule and product rule to take derivatives, and setting up a proportion or equation that relates the rates. It is also helpful to assign variables to the rates and keep units consistent throughout the problem.

What are some real-world applications of related rates problems?

Related rates problems can be found in many real-world scenarios such as calculating the rate at which a balloon is inflating, the speed of an object falling from a building, or the rate at which a chemical reaction is occurring. These types of problems are useful in fields such as physics, engineering, and economics.

What are some common mistakes to avoid when solving related rates problems?

Some common mistakes to avoid when solving related rates problems include forgetting to take derivatives, not setting up the relationship between the rates correctly, and using incorrect units. It is also important to pay attention to the wording of the problem and not assume any information that is not explicitly given.

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