Farmer John has 1200 feet of fencing and wishes to use it to fence a rectangular plot divided into two subplots as in the figure below. Suppose we want to determine the dimensions w and h of the plot with maximum area. h If Farmer John uses all the fencing, then 3h + 2w 1200 and so the area of the region is Α(w) - υ 1200–2w 3 Determine the values of w and h that give the maximum area. The exact value of w s... The exact value of h is...
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.
Answer both questions for the values of w and h
If Farmer John uses all the fencing, then \( 3h + 2w = 1200 \) and so the area of the region is
\[ A(w) = w \left( \frac{1200 - 2w}{3} \right) \]
Determine the values of \( w \) and \( h \) that give the maximum area.
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**The exact value of \( w \) is:**
[Input box for \( w \) value]
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**The exact value of \( h \) is:**
[Input box for \( h \) value]
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### Diagram Explanation:
- The diagram shows a large rectangle divided into two equal smaller rectangular sections by a vertical line.
- The total width of the large rectangle is denoted by \( w \).
- The height of the large rectangle is denoted by \( h \).
- The perimeter constraint is given by the equation \( 3h + 2w = 1200 \).
### Steps to Solve the Problem:
1. **Setting up the Perimeter Equation:**
Since the total fencing used around the border and the dividing line is 1200 feet, we have the equation \( 3h + 2w = 1200 \).
2. **Express \( h \) in terms of \( w \):**
\[
h = \frac{1200 - 2w}{3}
\]
3. **Formulate the Area Function:**
Substituting \( h \) in terms of \( w \) into the area formula \( A = w \cdot h \):
\[
A(w) = w \left(\frac{1200 - 2w}{3} \right) = \frac{1200w - 2w^2}{3}
\]
4. **Maximizing the Area:**
Find the value of \( w \) that maximizes \( A(w) \). This involves differentiating \( A(w) \) with respect to \( w \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7c9fe6be-99fb-4863-a04f-7bda979173e6%2Fb960e714-0b55-478b-80ad-bb5132109d7c%2F0j8y91q_processed.png&w=3840&q=75)
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