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4 days ago

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What is the solution of sqrt 1 - 3x = x + 3?

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Which pair of undefined terms is used to define a ray?

Zina [12379]

A line and a point.

Additional explanation

A ray consists of a segment of a line emerging from one endpoint. It represents an infinite straight path extending in one direction from a starting point, exemplified by $\boxed{ \ \overrightarrow{PQ} \ }$ .

The arrow indicates the direction along which the ray extends. The ray's length isn't quantifiable.

In geometry, undefined terms refer to fundamental figures that are not defined by other figures. The primary undefined terms in geometry include a point, line, and plane.

These crucial concepts cannot be precisely defined in mathematical terms using other established words.

A point indicates a specific location without any size; it is marked with a capital letter and a dot.
A line comprises an infinite collection of points stretching in both directions with just one dimension; it is a straight path without thickness.
A plane represents a flat surface encompassing numerous points and lines, extending infinitely in all directions. It is two-dimensional, and any three points that are not collinear define a unique plane.

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Which points are both coplanar and noncollinear?
Definition of line segments
What are three collinear points on a line?

Keywords: which undefined terms define a ray, point, line, plane, endpoint, one direction, the ray's arrow

4 0

3 months ago

Read 2 more answers

What are the solutions of the equation (2x + 3)2 + 8(2x + 3) + 11 = 0? Use u substitution and the quadratic formula to solve.

lawyer [12517]

Answer:

$x=-2.38$

$x=-4.62$

Step-by-step explanation:

The question is $(2x+3)^2+8(2x+3)+11=0$

We let $u=2x+3$ , so the equation becomes:

$u^2+8u+11=0$

Where $a=1, b=8, c=11$

By applying the quadratic formula, we arrive at:

Quadratic formula: $\frac{-b+-\sqrt{b^2-4ac} }{2a}$

Substituting yields: $\frac{-8+-\sqrt{(8)^2-4(1)(11)} }{2(1)}\\=\frac{-8+-\sqrt{20} }{2}\\=\frac{-8+-2\sqrt{5} }{2}\\=-4+\sqrt{5}, -4-\sqrt{5} }$

We let $u=2x+3$ , thus x calculates to:

$u=2x+3\\(-4+\sqrt{5})=2x+3\\x=\frac{-7+\sqrt{5}}{2}=-2.38$

and

$u=2x+3\\(-4-\sqrt{5})=2x+3\\x=\frac{-7-\sqrt{5}}{2}=-4.62$

The solutions to the equation are $x=-2.38$ (rounded to 2 decimal places) and $x=-4.62$ (rounded to 2 decimal places)

6 0

2 months ago

Mai wants to make an open top box by cutting out corners of a square piece of cardboard and folding up the sides. The cardboard

Inessa [12570]

Answer:

$V(x)=(4x^{3}-40x^{2}+100x)\ cm^3$

The variable x lies within the interval of all positive real numbers less than 5 cm.

Detailed solution:

Problem statement:

Determine the volume of the open-topped box as a function of the side length x (in centimeters) of the square cutouts.

Refer to the provided diagram for clarity.

Define:

x → length in centimeters of each square cutout side

The volume of the box with open top can be written as:

$V=LWH$

Given this, we have:

$L=(10-2x)\ cm$

$W=(10-2x)\ cm$

$H=x)\ cm$

By substitution:

$V(x)=(10-2x)(10-2x)x\\\\V(x)=(100-40x+4x^{2})x\\\\V(x)=(4x^{3}-40x^{2}+100x)\ cm^3$

Determine the domain of x:

Because:

$(10-2x) > 0\\10> 2x\\ 5 > x\\x < 5\ cm$

Therefore:

Domain is the interval (0,5)

That means all real numbers strictly greater than zero and less than 5 cm are valid for x.

Hence, the volume V as a function of x is:

$V(x)=(4x^{3}-40x^{2}+100x)\ cm^3$

5 0

4 months ago