A metal chain ladder, 20 meters long, has a linear density of 3 Newtons per meter, and hangs down from the roof along the side of a building 30 meters tall. How much work is needed to pull the chain ladder to the top of the building? How much work is done in pulling up the first half of the chain ladder?
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
![### Problem: Work Required to Pull Up a Chain Ladder
#### Given:
- A metal chain ladder, 20 meters long
- Linear density: 3 Newtons per meter
- Hanging from the roof of a building that is 30 meters tall
#### Questions:
1. How much work is needed to pull the chain ladder to the top of the building?
2. How much work is done in pulling up the first half of the chain ladder?
---
### Solution
To solve this problem, we need to understand the concept of work and how it applies to lifting a distributed load like a chain ladder.
#### 1. Calculating Total Work
##### Work when lifting the entire chain ladder:
1. **Determine the weight per unit length of the chain ladder:**
\[
\lambda = 3 \, \text{N/m}
\]
2. **Total length of the chain ladder:**
\[
L = 20 \, \text{m}
\]
3. **Divide the chain ladder into infinitesimal segments:**
Each tiny segment \( dy \) at a height \( y \) above the ground.
4. **Weight of an infinitesimal segment:**
\[
dW = \lambda \cdot dy = 3 \, \text{N/m} \cdot dy
\]
5. **Integrated work to pull each infinitesimal segment to the top of the building:**
\[
dW = \lambda \cdot y \cdot dy = 3y \, dy
\]
The limits are from \(0\) to \(20\) since the height ranges from ground level to the top of the chain ladder.
6. **Integrate to find the total work:**
\[
W = \int_{0}^{20} 3y \, dy
\]
7. **Calculate the integral:**
\[
W = \left[ \frac{3y^2}{2} \right]_{0}^{20} = \frac{3}{2} \left[ 20^2 - 0^2 \right] = \frac{3}{2} \cdot 400 = 600 \, \text{J}
\]
#### 2. Calculating Work for the First Half
##### Work when lifting the first half of the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F26122c3d-d5aa-49d8-9e35-9e12e57916d1%2F6965603c-52b8-423c-8d4c-a388f07b7eb7%2F5jm9d7p_processed.jpeg&w=3840&q=75)
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