# Ladder problem with related rates

• Ingenium
In summary, the ladder is 24 ft long and is moving away from the house at a rate of 3 ft/s. To find the rate at which the slope of the ladder is decreasing when it is 14 ft away from the house, we use the derivative formula for slope (z = y/x) and the chain rule to calculate dz/dt.
Ingenium
1. The ladder is 24 ft long and is leaning against a house. The ladder is moving away from the house at a rate of 3 ft/s. I'm supposed to find the rate the slope of the ladder is decreasing when it is 14 ft away from the house.

3. I'm thinking its got something to do with the second derivative of the ladder, but i can think of how to do it in this context.

also: yay, first post

Last edited:
So, let x be the horizontal distance of the ladder, y be the vertical distance. We know x^2+y^2 = 24^2

The slope is z = y\x.
I think we are given that dx\dt = 3.

This should be enough information to find dz/dt.

agree with what grief said, one more hint. Chain rule!

where would the chain rule come in? i see where i use quotient rule but that's all I'm seeing.

since z=y/x

d/dt(z) = d/dt(y/x)
== dz/dt = 1/x dy/dt - y/x^2 dx/dt

we know dx/dt = 3ft/s
we know dy/dt = dy/dx*dx/dt

u know x^2+y^2=24^2

ok i got it now. thanks a lot guys for the help

## 1. What is the ladder problem with related rates?

The ladder problem with related rates is a mathematical problem that involves a ladder sliding down a wall at a constant rate while its base moves away from the wall at a different constant rate. The goal is to find the rate at which the top of the ladder is sliding down the wall at any given point.

## 2. What are the key concepts involved in solving the ladder problem with related rates?

The key concepts involved in solving the ladder problem with related rates are similar triangles, the Pythagorean theorem, and the chain rule in calculus. It is also important to understand the relationship between the changing variables and how they affect each other.

## 3. How do you set up the equations for solving the ladder problem with related rates?

To set up the equations for solving the ladder problem with related rates, you first need to identify the relevant variables and their rates of change. Then, use the Pythagorean theorem to create a right triangle with the ladder as the hypotenuse. Finally, use the chain rule to relate the changing variables in the triangle.

## 4. Can you provide an example of solving the ladder problem with related rates?

Sure, let's say we have a ladder that is 10 feet long and is sliding down a wall at a rate of 2 feet per second. The base of the ladder is also moving away from the wall at a rate of 1 foot per second. We can use the Pythagorean theorem to set up the equation: x^2 + 10^2 = (x+1)^2. Solving for x, we get x = 5. This means that at any given point, the top of the ladder is sliding down the wall at a rate of 5 feet per second.

## 5. What are some real-life applications of the ladder problem with related rates?

The ladder problem with related rates has many real-life applications, such as determining the speed of a shadow moving up a building, calculating the speed of a train passing by a station, or finding the rate at which a water tank is filling or draining. It is a useful concept in physics, engineering, and other fields that involve changing variables and rates of change.

Replies
11
Views
8K
Replies
1
Views
2K
Replies
5
Views
3K
Replies
2
Views
6K
Replies
4
Views
2K
Replies
5
Views
2K
Replies
4
Views
2K
Replies
2
Views
2K
Replies
3
Views
2K
Replies
6
Views
4K