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: Surface Area Calculation**
Amelia is decorating the outside of a box in the shape of a square pyramid. The figure below shows a net for the box.
[Diagram Description: The diagram features a net for a square pyramid, which consists of a central square connected to four triangles, one on each side of the square. The side length of the square is 3.7 meters and the slant height of the triangle is 5 meters.]
**Question:**
What is the surface area of the box, in square meters, that Amelia decorates?
**Answer:**
\[ A = \quad \boxed{\quad} \, \text{m}^2 \]
**Explanations and Calculations:**
To find the surface area of the pyramid, we need to calculate the area of the base (square) and the area of the four triangular faces.
1. **Area of the Base (Square):**
- Side length of the square is \( 3.7 \, \text{m} \).
- Area of the square = \( \text{side}^2 = (3.7 \, \text{m})^2 = 13.69 \, \text{m}^2 \).
2. **Area of the Triangular Faces:**
- There are 4 triangular faces.
- Each triangle has a base equal to the side of the square (base = \( 3.7 \, \text{m} \)) and height (slant height) \( 5 \, \text{m} \).
- Area of one triangle = \( \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3.7 \, \text{m} \times 5 \, \text{m} = 9.25 \, \text{m}^2 \).
- Total area of the four triangles = \( 4 \times 9.25 \, \text{m}^2 = 37 \, \text{m}^2 \).
3. **Total Surface Area:**
- Surface area of the pyramid = Area of the base + Area of the four triangles = \( 13.69 \, \text{m}^2 + 37 \, \text{m}^2 = 50](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe95140d0-b0ae-4b73-878d-570406dc1277%2F72479111-b8ae-435f-9da0-08b128d35fc4%2Fcrz41ph_processed.jpeg&w=3840&q=75)
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