Related rates how long is the ladder

At least not nearly as easily. So: keep practicing!! Log in. Home » Calculus 1 » Related Rates. Calculus Related Rates Problem: At what rate does the angle change as a ladder slides away from a house?

I'd like to be. I agree to the Terms and Privacy Policy. The comment form collects the name and email you enter, and the content, to allow us keep track of the comments placed on the website. Please read and accept our website Terms and Privacy Policy to post a comment. Inline Feedbacks. This often seems like a silly step but can make all the difference in whether we can find the relationship or not. Okay, we should probably start off with a quick sketch probably not to scale of what is going on here.

Showing the 3D nature of the tank is liable to just get in the way. So here is the sketch of the tank with some water in it. This is actually easier than it might at first look. If we go back to our sketch above and look at just the right half of the tank we see that we have two similar triangles and when we say similar we mean similar in the geometric sense.

Recall that two triangles are called similar if their angles are identical, which is the case here. When we have two similar triangles then ratios of any two sides will be equal. For our set this means that we have,. This gives us a volume formula that only involved the volume and the height of the water. If we now differentiate this we have,.

Recall from the first part that we have,. At this point all we need to do here is use the result from the first part to get,. Much easier that redoing all of the first part. In the second part of the previous problem we saw an important idea in dealing with related rates. Sometimes there are multiple equations that we can use and sometimes one will be easier than another.

Also, this problem showed us that we will often have an equation that contains more variables that we have information about and so, in these cases, we will need to eliminate one or more of the variables.

In this problem we eliminated the extra variable using the idea of similar triangles. This will not always be how we do this, but many of these problems do use similar triangles so make sure you can use that idea. Note that an isosceles triangle is just a triangle in which two of the sides are the same length. In our case sides of the tank have the same length. So, we need an equation that will relate these two quantities and the volume of the tank will do it.

The volume of this kind of tank is simple to compute. The volume is the area of the end times the depth. For our case the volume of the water in the tank is,. One for the tank itself and one formed by the water in the tank. Again, remember that with similar triangles ratios of sides must be equal.

Remember the chain rule. The answer is the Chain Rule. Remember that x and y are both functions of time t : both positions change as time passes and the ladder slides down the wall. The negative value indicates that the top of the ladder is sliding down the wall, in the negative- y direction.

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