What is the volume of the triangular prism?


PLS HELP!!

Addition of the 3 projections will be: (-2.268, 0.168, -1.536)

A matrix is a rectangular array of values that are defined for addition, subtraction, and multiplication among other mathematical operations. The number of rows and columns in a matrix, also known as the order of the matrix, defines the size of the matrix. The order of a matrix with 6 rows and 4 columns is shown when written as a 6 4 and read as 6 by 4.

To find the projection of b onto a3, we use the projection formula:

proj a3(b) = (b dot a3 / [tex]||a3||^2[/tex]) * a3

where "dot" represents the dot product and "|| ||" represents the magnitude or norm.

We first find the dot product:

b dot a3 = (0)(23) + (3)(-13) + (0)(23) = -39

Next, we find the norm of a3:

[tex]||a3||^2 = (23)^2 + (-13)^2 + (23)^2 = 1388[/tex]

Using these values, we can compute the projection:

proj a3(b) = (-39 / 1388) * (23, -13, 23) = (-0.768, 0.436, -0.768)

Similarly, we can find the projections of b onto a1 and a2:

proj a1(b) = (-3 / 14) * (2, 4, 4) = (-3/7, -6/7, -6/7)

proj a2(b) = (3 / 18) * (-3, 3, 0) = (-1/2, 1/2, 0)

Finally, we add the three projections:

(-3/7, -6/7, -6/7) + (-1/2, 1/2, 0) + (-0.768, 0.436, -0.768) = (-2.268, 0.168, -1.536)

To see why P = a1aT1 +a2aT2 +a3aT3, we note that each projection can be written in matrix form:

proj_a1(b) = a1(aT1b)

proj_a2(b) = a2(aT2b)

proj_a3(b) = a3(aT3b)

where aT represents the transpose of a. Using these expressions, we can write:

P = a1(aT1b) + a2(aT2b) + a3(aT3b)

= [a1 a2 a3] [aT1b aT2b aT3b] (by distributing the dot product)

= [a1 a2 a3] [b] (since the aT's are row vectors that multiply with b to form a scalar)

= abT

where abT is the outer product of a and b, which is a matrix. This matrix is the projection matrix that projects vectors onto the subspace spanned by the columns of A.

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Answer:

243

Step-by-step explanation:

Going in times 3 so 81 x 3 = 243

Answer:

540.7

Step-by-step explanation:

You need to find the circumference of the pond. You can find the circumference using the diameter.

The formula for circumference is d×3.14

The diameter is 169

so, 169×3.14=540.66 rounded to 540.7

The solutions to the system are -1 and 3.

What is system of equations?

equations simultaneously, or a system of equations Multiple equations in algebra must be solved concurrently (i.e., the solution must satisfy all the equations in the system). There must be an equal number of equations and unknowns for a system to have a singular solution. Even then, a solution may not be assured.

Given : y = x + 5  ... equation 1

           y = x² – x + 2 ...equation 2

Putting  y=x+5 in equation 2

We get, x+5 =  x² – x + 2

             x² - x - x +2 -5 = 0

             x² -2x - 3 = 0

It can written be solved as :

                x² -2x - 3 = 0

                x² -3x +x - 3 = 0

                x ( x-3) + 1(x-3) = 0

                (x+1) (x-3) = 0

So, x = -1 and 3.

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From the given scenario, AB = [17 + 28, 47 + 58] = [23, 64]

To calculate the product of matrices XY, the number of columns in X must match the number of rows in Y. In this case, X has 1 column and Y has 2 rows, so the product XY is undefined.

A general example of this with matrices is:

Let A be a 2x3 matrix:

A = [1 2 3; 4 5 6]

Let B be a 1x2 matrix:

B = [7 8]

To calculate the product AB, we need the number of columns in A to match the number of rows in B.

In this case, A has 3 columns and B has 1 row, so the product AB is:

AB = [17 + 28, 47 + 58] = [23, 64]

P.S Your question is incomplete, so I showed how to calculate the product of matrices XY,

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Answer:

A part of basic arithmetic, long division is a method of solving and finding the answer and remainder for division problems that involve numbers with at least two digits. Learning the basic steps of long division will allow you to divide numbers of any length, including both integers (positive,negative and zero) and decimals. This process is an easy one to learn, and the ability to do long division will help you sharpen and have more understanding of mathematics in ways that will be beneficial both in school and in other parts of your life.[1]

Step-by-step explanation:

A part of basic arithmetic, long division is a method of solving and finding the answer and remainder for division problems that involve numbers with at least two digits. Learning the basic steps of long division will allow you to divide numbers of any length, including both integers (positive,negative and zero) and decimals. This process is an easy one to learn, and the ability to do long division will help you sharpen and have more understanding of mathematics in ways that will be beneficial both in school and in other parts of your life.[1]

The y-value of the solution of the system of equations is given as follows:

y = -1.

The system of equations for this problem is defined as follows:

Adding the two equations, the value of y can be obtained as follows, as the x-variable is eliminated:

6y = -6

y = -6/6

y = -1.

Hence option F is correct.

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The line of equation which passes through (-5, 3) and (-2, -3) is 2x+y+7=0.

A line of equation is an equation that represents a straight line on a coordinate plane.

The Straight line equation is: y = mx + b

Here y is the dependent variable, x is the independent variable, m is the slope of the line, and constant b is the y intercept.

The equation of line is:

[tex]y-y_{1} = \frac{y_{2}-y_{1}}{x_{2}-x_{1}} (x-x_{1} )[/tex]

Given that, ([tex]x_{1}, y_{1}[/tex]) = (-5, 3) and ([tex]x_{2}, y_{2}[/tex]) = (-2, -3)

By putting the values in above equation,

[tex]y-3 = \frac{-3-3}{-2+5} (x+5)\\[/tex]

y-3 = -2x-10

2x+y+7 = 0

Therefore the line equation is 2x+y+7 = 0.

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Answer:

[tex]71.43[/tex]

Step-by-step explanation:

[tex]5 b+ 2 b\geqslant 500 \\ 7b \geqslant 500 \\ \frac{7b}{7} \geqslant \frac{500}{7} \\ b \geqslant 71.43 \\ [/tex]

[tex] b= 71.43 < \infty [/tex]

Therefore, the length of AC is √(145) inches.

A triangle is a three-sided polygon, which means it is a flat shape with straight sides. It is the simplest polygon, consisting of three line segments connected end-to-end. The corners where the sides meet are called vertices, and the line segments themselves are called edges or sides. The interior angles of a triangle always add up to 180 degrees. Triangles can be classified according to the lengths of their sides and the sizes of their angles. Common types of triangles include equilateral triangles, isosceles triangles, scalene triangles, acute triangles, right triangles, and obtuse triangles.

Here,

        A

       / \

      /   \

     /     \

    / B    C

   /_|\___/ \

  D       E   \

Since triangle ABC is right-angled at B, we know that angle BAC is a right angle. We can use this fact to find the length of AC by applying the Pythagorean theorem. Let x be the length of AC:

AC² = AB² + BC² (by the Pythagorean theorem)

AC² = AD² + BD² + BC² (by the definition of right triangles)

AC² = AD² + CD² + BC² (since BCD = DCA)

AC² = AE² + EC² + CD² (since BAE = EAC)

AC² = 9² + EC² + 8² (since AE = 9 inches and CD = 8 inches)

Now we need to find EC. We can use the fact that BAE = EAC to find the relationship between AE and EC. Let y be the length of BE:

AE/BE = AC/BC (by the angle bisector theorem)

9/y = x/(BC-y) (substituting the given values)

9(BC-y) = xy

BCy - y² = (9/ x) y²

Next, we can use the fact that BCD = DCA to find the relationship between BD and DC. Let z be the length of AD:

BD/DC = BA/AC

(BD/DC) = (AB/AC) (since B is the right angle)

BD/8 = (x²)/(x² + 8²)

Now we have two equations with two variables (y and z). We can use substitution to solve for one variable in terms of the other. Let's solve for y in terms of z from the first equation:

BCy - y² = (9/x) y²

BCy = y² (9/x + 1)

y(BC - y) = y² (9/x + 1)

y² (BC - y - (9/x + 1)) = 0

Since y cannot be zero, we must have:

BC - y - (9/x + 1) = 0

BC - y = 9/x + 1

Now let's substitute this expression for y into the second equation:

BD/8 = (x²)/(x² + 8²)

BD = (8x²)/(x² + 64)

Since BCD = DCA, we have:

BD/CD = BC/AC

(8x²)/(x² + 64) / 8 = BC/x

BC = x² / (x² + 64)

Now we can substitute this expression for BC into the equation we derived earlier:

BC - y = 9/x + 1

x²/(x² + 64) - y = 9/x + 1

Substituting the value we derived for BD into the equation, we get:

x²/(x² + 64) - 8x²/(x² + 64) = 9/x + 1

Simplifying, we get:

x² = 145

x = √(145)

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Therefore, using the rule of 72, it would take approximately 9 years for Sofia's $18,000 investment to double at a rate of 8% in interest.

Percent is a term used to describe a value that is a fraction of 100. It is often represented by the symbol "%".  Percentages are commonly used to express rates, ratios, and proportions in many fields, including finance, science, and everyday life. They are often used to describe changes in value, such as increases or decreases in prices, population, or other quantities over time.

Here,

The rule of 72 is a simple formula used to estimate the time it takes for an investment to double in value based on its rate of return. To use this rule, divide 72 by the annual percentage yield (APY) to get an estimate of the number of years it would take for the investment to double.

In this case, the APY is 8%, so we can apply the rule of 72 as follows:

Number of years to double = 72 / APY

= 72 / 8

= 9 years

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The length of the measurements are;

EF = 7. 1 units

DE = 7. 0 units

Using the sine trigonometric identity, we have that;

sin θ = opposite/hypotenuse

Given that;

Opposite side = DE

Hypotenuse = DF

Substitute the values

sin 45 = DE/10

Find the value and substitute

DE = 0. 7071(10)

DE = 7. 0

Using the Pythagorean theorem;

EF² = 10² - 7²

find the square value

EF = √ 51

EF = 7.1

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The mode of the data set is 7

The median is 6

the mean is 5.2

The range is 7

This is the number with the highest frequency or occurrence. This is 7

arrange based on ascending order

1, 2, 3, 4, 5, 7, 7, 7, 8, 8

median = 5 + 7 / 2

= 6

add all the numbers / total number

52 /10

= 5.2

The range = highest - lowest

= 8 - 1

= 7

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A sequence of transformations to rectangle JKLM that will result in rectangle J'K'L'M' is a translation by 3 units and -4 units, a reflection about the y-axis, and a rotation of 90 degrees counterclockwise about the origin.

A rectangle is a quadrilateral with four right angles and four sides of equal length. It is one of the most common shapes in nature and is found in everything from architecture to floor patterns. A rectangle is a two-dimensional shape that has four sides and four angles, all of which are right angles, and two pairs of opposite sides which are parallel.

In order to transform rectangle JKLM into J'K'L'M', a series of transformations must be applied. The first transformation is a translation of the rectangle along the x-axis by 3 units and along the y-axis by -4 units. This will move the original rectangle from its original position, JKLM, to the new position, J'K'L'M'. Next, a reflection about the y-axis needs to be applied, which will flip the rectangle around the y-axis. This will result in the length of side K'L' being 6 units. Finally, a rotation about the origin needs to be applied. The rotation should be 90 degrees counterclockwise in order for the coordinates of vertex K' to be (-3,2) and the coordinates of vertex M' to be (3,-2). Therefore, a sequence of transformations to rectangle JKLM that will result in rectangle J'K'L'M' is a translation by 3 units and -4 units, a reflection about the y-axis, and a rotation of 90 degrees counterclockwise about the origin.

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The number of different types of movies that Lavar can buy according to the inequality m > 15 is action and comedy.

What is meant by inequality?

In mathematics, inequalities specify the connection between two non-equal numbers. Equal does not imply inequality. Typically, we use the "not equal sign" to indicate that two values are not equal. However several inequalities are utilised to compare the numbers, whether it is less than or higher than. An inequality symbol has non-equal expressions on both sides. It indicates that the expression on the left should be bigger or smaller than the expression on the right, or vice versa. Literal inequalities are relationships between two algebraic expressions that are expressed using inequality symbols.

The complete question is given below.

The inequality to be used is m > 15.

Now we will look into each genre to select which movies he could buy.

There are 18 action movies.

m = 18 > 15.

There are 18 comedy movies as well.

m = 18 > 15

There are 12 drama movies.

but m = 12< 15

There are 15 thriller movies.

But the given inequality allows only categories with a number of movies greater than 15 and not equal to 15.

Therefore the number of different types of movies that Lavar can buy according to the inequality m > 15 is action and comedy.

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Answer:

A square is a rhombus.

Step-by-step explanation:

When rotated 45 degrees, a square can sometimes be a rhombus (aka a diamond). A rhombus can have four equal sides with four right angles, just like a square.

A trapezoid cannot be a parallelogram. A parallelogram has two sides that are parallel with each other going horizontally and two that are also parallel but going diagonally.

A parallelogram opposite sides are equal while rectangles have opposite sides that are equal as well as all of its adjacent sides being perpendicular to form right angles. A parallelogram has no right angles. The diagonals of a rectangle are equal whereas the diagonals of a parallelogram are not.

A rectangle is a closed 2D shape with four straight edges and four right angles, also the diagonals are equal. A square on the other hand is a closed 2D shape with four straight edges, four right angles, and four equal sides. Squares can be rectangles but not all rectangles can be squares.

I hope this helps.

Have a great day!

Answer:

B. 2

Step-by-step explanation:

The standard form of a sine function is given by:

y = A sin(Bx - C) + D

where A is the amplitude, B is the frequency, C is the phase shift, and D is the vertical shift.

In the given function y = 1/2 sin(pi x) + 3, we can see that the amplitude is 1/2, the frequency is pi, the phase shift is 0, and the vertical shift is 3.

The period of a sine function is given by:

Period = 2π/B

In this case, B = pi, so the period is:

Period = 2π/pi = 2

Therefore, the period of the graph of y = 1/2 sin(pi x) + 3 is 2, which is option B.

Answer:

Step-by-step explanation:

The period is 2.

Normally the period of sin(x) is 2pi, but the pi inside the sin(pix) is a horizontal compression by a factor of 1/pi.  So 2pi·1/pi = 2

The "1/2" and "+3" do not impact the period.  Those just impact the amplitude (vertical aspect) of the graph.

The value of y when x=3 is y=7. Point A on the graph corresponds to x=3 and y=7.

What is graph ?

A graph is a visual representation of data or mathematical functions. Graphs are used to display data in a way that is easy to understand and interpret. There are many types of graphs, including line graphs, bar graphs, pie charts, scatter plots, and more. Each type of graph has its own unique features and is used for different purposes. Graphs are commonly used in many fields, including science, economics, finance, and engineering, among others, to visualize data and communicate information in a clear and concise way.

According to the question:

a) If x = 3, then:

y = x² - 2

y = 3² - 2

y = 9 - 2

y = 7

Therefore, when x = 3, y = 7.

b) From the graph, we can see that the point (3,1) corresponds to x = 3 and y = 1. Therefore, point A is the correct point on the graph that corresponds to x = 3 and y = 7.

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Se compraron 500 panes, pero solo se utilizó el 75%, lo que significa que se consumió el 100% - 75% = 25% de los panes. Para calcular la cantidad de panes que se consumió, podemos multiplicar la fracción de panes consumidos (25% o 0.25) por el número total de panes:

0.25 x 500 = 125

Por lo tanto, se consumieron 125 panes en el cóctel.

The area of a regular 8 sides polygon with radius 4 units long is 309 unit²

Area is defined as the total space taken up by a flat (2-D) surface or shape of an object.

The space enclosed by the boundary of a plane figure is called its area.

The area of a figure is the number of unit squares that cover the surface of a closed figure.

Given is a regular 8 sides polygon with radius 4 units long, we need to find its area,

We know that the regular 8-sided polygon is called an Octagon.

Area of an octagon =

A = 2(1+√2)a²

a = side

A = 2(1+√2)8²

A = 2(2.41)64

A = 309

Hence, the area of a regular 8 sides polygon with radius 4 units long is 309 unit²

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So the handwriting is a bit messy.

Answer:

2 3/8

Step-by-step explanation:

8 [tex]\frac{5}{8}[/tex]

-  4 [tex]\frac{1}{4}[/tex]

First you got to get the fractions to have the same denominator (or bottom number)

In this case it would be 8. You would multiply the 4 in 1/4 to get 8, and what you do to the bottom you have to do to the top, so the 1 would multiply by 2 as well to get to.

So the new equation would be,

 8 [tex]\frac{5}{8}[/tex]

- 4 [tex]\frac{2}{8}[/tex]

5-2= 3

And the denominator stays the same so the fraction would be 3/8.

8-4=4 so the whole number would be 4

Making the full answer 4 3/8

Answer:

A. 2.50x + 85 < 125.

Step-by-step explanation:

Let x be the number of cans of tennis balls that Joseph wants to buy.

The total cost of the tennis racket and x cans of tennis balls is:

Cost = 85 + 2.5x

Joseph wants to make sure that he spends less than $125, so we can set up the inequality:

85 + 2.5x < 125

Simplifying this inequality, we get:

2.5x < 40

x < 16

So Joseph can buy at most 15 cans of tennis balls if he wants to spend less than $125.

However, the question asks for the correct inequality, and the inequality that correctly represents this situation is:

2.50x + 85 < 125

Therefore, your answer is gonna be A. 2.50x + 85 < 125.

Answer: "B"

Step-by-step explanation:

Joseph wants to buy some tennis balls. Which costs 2.50$ and tennis racket costs 85$

then, 2.50x+85<125

Answer: Para resolver el ejercicio se plantea un diagrama de conjuntos, en el

   cual se representan un conjunto A y B ( Ver dibujo adjunto  ) .

        b +x = 46%           a + x = 60%           a + b = 58%

    Al sumar las dos ecuaciones :

       b + x = 46%

       a + x=  60%

--------------------------

   a + b +2x = 106%

      58% + 2x = 106%

                 2x = 106% - 58%

                 2x = 48%

                   x = 48%/2

                   x = 24%   no leen ninguna de las dos revistas

       c = 63000 personas   que leen A y B

       se llama al total de personas y :

       y * 18% = 63000 personas

       y = 63000 / 18%

       y = 63000/0.18 = 350000 son las personas hay en Trujillo

Answer:

it will cost $520.00 to install the gravel path.

Step-by-step explanation:

We can use the Pythagorean theorem to find the length of the diagonal of the rectangular garden. The Pythagorean theorem states that the square of the length of the hypotenuse (in this case, the diagonal) of a right triangle is equal to the sum of the squares of the lengths of the other two sides. Therefore, we have:

diagonal^2 = 24^2 + 32^2

diagonal^2 = 576 + 1024

diagonal^2 = 1600

Taking the square root of both sides, we get:

diagonal = sqrt(1600)

diagonal = 40 feet

Now, we need to calculate the area of the path that Jorge will install. Since the path is 2 feet wide, the width of the garden is reduced by 4 feet (2 feet on each side). Therefore, the length and width of the garden after the path is installed will be:

length = 24 - 4 = 20 feet

width = 32 - 4 = 28 feet

The area of the path is the difference between the area of the garden before and after the path is installed:

area of path = (24 * 32) - (20 * 28)

area of path = 768 - 560

area of path = 208 square feet

Finally, we can calculate the cost of the gravel path by multiplying the area of the path by the cost per square foot:

cost of path = 208 * 2.50

cost of path = $520.00

Therefore, it will cost $520.00 to install the gravel path.

Simple interest is based on principal only and Compound interest is based on principal and interest both. Then the correct option is C.

Let P be the principal, r be the rate of interest, and n be the number of years. Then the interest rate is given as,

The formula for simple interest is written as,

I = (P × r × n)/100

The compound interest is given as

A = P(1 + r)ⁿ

Simple interest is computed just on the principle. Compound interest is interest calculated on both the initial principal and any accrued interest.

Thus, the correct option is C.

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Answer:

Step-by-step explanation:

6^8 x 6^10 = 6^(8+10) = 6^18

Answer:

18

Step-by-step explanation:

a^x times a^y = a^x+y

Therefore, 6⁸×6¹⁰ -----> 8+10 = 6¹⁸