What is the sum of the infinite series:

72 - 36 + 18 - 9 +...

The volume of the triangular pyramid is 26 2/3 yd³

The formula for calculating the volume of a triangular pyramid is expressed with the equation

A = 1/3bh

Such that the parameters in the equation are;

The base area of the pyramid is;

Area = 1/2 × 4 × 5

Multiply the values

Base area =20/2

Base area = 10 yd

Substitute the value

Volume = 1/3 × 10 × 8

Multiply the values

Volume = 80/3

Divide the values

Volume = 26 2/3 yd³

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Answer: 11/9 or 1 2/9 or 11 over 9

Step-by-step explanation:

Firstly, you need to know what a fraction is.

A fraction is a segment (part of) a whole. To express a fraction, you write the numerator over the denominator. A numerator is the top part of the fraction, and it shows how much it is out of. The denominator shows how much it is out of.

Example: [tex]\frac{1}{3}[/tex]

That means it is 1 part out of 3 parts

Now you know what a fraction is, now let's add them.

Next, convert the word form into the number form:

2 over 3 is 2/3

plus is +

5 over 9 =is 5/9

Combine them:

2/3 + 5/9

Steps of the problem:

Firstly, you have to find the L.C.M (Lowest Common Multiple) of 3 and 9 because to add fractions, you need a common (same) denominator.

9 is the L.C.M of 3 and 9, as 9x1 = 9 and 3x3 = 9.

Mulitiply the numerator and denominator by 3:

2x3/3x3 = 6/9

Now, the equation is 6/9+5/9.

Add the numerators:

11/9

The answer is 11/9

If the question tells you to convert into a mixed number, then here's the following steps:

You have to find the highest number 11 goes into 9

It's 1.

Take the remainder (2) into the fraction:

2/9

Now, add 1 in front of it:

1 2/9

As a decimal, it is 1.2222222 (recurring).

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Answer:

11 over 9 [tex](\frac{11}{9})[/tex]

Step-by-step explanation:

A fraction is a fragment of a whole number, used to define parts of a whole. The whole can be a whole object, or many different objects. The number at the top of the line is called the numerator, whereas the bottom is called the denominator.

To solve for [tex]\frac{2}{3} +\frac{5}{9}[/tex], we need to first convert these fractions so that they share a common denominator.

A common denominator consists of two or more fractions that have the same denominator. This makes it easier to perform numeric equations, and to solve them.

If a common denominator just refers to 2 fractions having the same denominator, we can simply multiply the numerator and denominator of the fraction [tex]\frac{2}{3}[/tex] by a number. To solve for that number, we can divide 9 by 3.

Now that we know that the number is 3, multiply both by 3.

Now the fraction looks like this: [tex]\frac{6}{9}[/tex].

We can now change the expression to this:

When adding fractions with the same denominator, we can just add the 2 numerators and keep the common denominator the same.

Now just insert 11 in as the numerator:

Therefore 2 over 5 plus 5 over 9 is 11 over 9.

according the given question the exact value of given expression is [tex]$\cos\frac{x}{2} = -\sqrt{\frac{1}{2(1 - \left(-\frac{160}{81}\right)^2)}} = -\sqrt{\frac{81^2}{2(81^2 - 160^2)}} = \boxed{-\frac{81\sqrt{239}}{319}}$[/tex]

First, we need to find [tex]$\sin x$[/tex] using the identity[tex]$\cos^2x + \sin^2x = 1$:$\sin^2x = 1 - \cos^2x = 1 - \left(-\frac{4}{5}\right)^2 = \frac{9}{25}$[/tex]

Since  [tex]$\frac{\pi}{2} < x < \pi$[/tex], we know that  [tex]$\frac{\pi}{4} < \frac{x}{2} < \frac{\pi}{2}$[/tex]. Therefore, we can use the

identity [tex]$\tan\frac{x}{2} = \frac{\sin x}{1 + \cos x}$[/tex]:

[tex]$\tan\frac{x}{2} = \frac{\sqrt{\frac{9}{25}}}{1 - \frac{4}{5}} = \frac{\frac{3}{5}}{\frac{1}{5}} = \boxed{3}$[/tex]

[tex]If $\tan x = \frac{40}{9}$ and $\pi < x < \frac{3\pi}{2}$, find $\cos\frac{x}{2}$.[/tex]

First, we need to find [tex]$\sin x$[/tex] using the identity [tex]$\tan^2x + 1 = \sec^2x$[/tex]:

[tex]$\sin x = \frac{\tan x}{\sec x} = \frac{\frac{40}{9}}{-\frac{9}{40}} = -\frac{160}{81}$[/tex]

[tex]Since $\pi < x < \frac{3\pi}{2}$, we know that $\frac{\pi}{2} < \frac{x}{2} < \frac{3\pi}{4}$[/tex]. Therefore, we can use the identity [tex]$\cos\frac{x}{2} = \pm\sqrt{\frac{1 + \cos x}{2}}$[/tex]:

[tex]$\cos\frac{x}{2} = -\sqrt{\frac{1 + \cos x}{2}} = -\sqrt{\frac{1 + \frac{\cos^2x}{\sin^2x}}{2}} = -\sqrt{\frac{\sin^2x + \cos^2x}{2\sin^2x}} = -\sqrt{\frac{1}{2(1 - \sin^2x)}}$[/tex]

Plugging in [tex]$\sin x = -\frac{160}{81}$[/tex] , we get:

[tex]$\cos\frac{x}{2} = -\sqrt{\frac{1}{2(1 - \left(-\frac{160}{81}\right)^2)}} = -\sqrt{\frac{81^2}{2(81^2 - 160^2)}} = \boxed{-\frac{81\sqrt{239}}{319}}$[/tex]

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Answer:

Step-by-step explanation:

Domain is now called the "source" which means before a change or transformation. In mathematics, the term "source" is often used to refer to the set of all possible inputs or values that can be fed into a function or transformation, before any changes or transformations take place. The set of all possible outputs or resulting values from the function or transformation is called the "range" or "codomain".

The two statements that are both true are as follows: line

AC is perpendicular to line HB and line AC is parallel to FG. That is option A.

A perpendicular line is defined as the line that forms angle 90° where it meets with another line in a plane.

A line is said to be parallel to each other when they do not intercept as they are both on the same plane.

From the given diagram, line AC is perpendicular to line HB because they form angle 90° at the point of intersection.

Also, line AC is parallel to FG, because they can never intersect till infinity.

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The best linear equation that describe given graph is y=2.5x-35 i.e. C.

What is a linear equation ?

A linear equation is a mathematical equation that, when plotted on a graph, represents a straight line. It is an equation of the following form:

y = mx + b

where y = dependent variable, x = independent variable, m= slope of the line, and b = y-intercept (the point where the line crosses the y-axis).

Now,

Lets take a point on the given graph that is on line (50,90)

then the equation of line should give the same values.

So,

For A,  y=0.4x-35

y=0.4*50-35

y=-15  Hence, It does not follow.

For B, y= y=0.4x-70

y=0.4*50-70

y=-50  Hence, It does not follow.

For C, y= y=2.5x-35

y=2.5*50-35

y=125-35

y=90

 Hence, It does follow the given point.

Therefore, The best linear equation that describe given graph is y=2.5x-35

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Answer:

48 cookies were in the bag.

Step-by-step explanation:

It said 12 was 25%, so you want to multiply it by 4 to get 100% and 100% of the bag was 48 cookies.

Answer:

1. 57/16 or 3.5625 ounces of water

8 : 4.75

1 : 19/32  -----> divide both by 8

19/32x6=57/16 = 3.56 ounces

2. 2.4375 ounces of other ingredients

6-3.5625=2.4375

Answer: 64°

Step-by-step explanation:

Had this problem and got it right

The equation y = -1/2(x-3)² + 5  reduces the graph, which means that the graph decreases as we move away from the vertex (3, 5) in both directions along the x-axis.

In mathematics, a graph is a visual representation of a set of objects (called vertices or nodes) and the connections (called edges) between them.

The equation y = -1/2(x-3)² + 5 is a quadratic function in vertex form. The vertex of this parabola is at the point (3, 5), and the coefficient of the x²  term is negative (-1/2), which tells us that the parabola opens downwards. This means that the graph of this equation reduces.

To see why this is the case, consider the behavior of the y-values as x moves away from the vertex. Since the leading coefficient is negative, the y-values will decrease as x moves to the left or right from the vertex. Additionally, the squared term inside the parentheses means that the graph will be symmetric around the x-coordinate of the vertex, which is 3 in this case.

Thus, as x moves away from the vertex to the left or right, the y-values decrease in a symmetric manner, resulting in a graph that reduces. This can be seen in the shape of the parabola as it curves downwards from the vertex.

Therefore, the equation y = -1/2(x-3)² + 5  reduces the graph, which means that the graph decreases as we move away from the vertex (3, 5) in both directions along the x-axis.

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Answer:

Below

Step-by-step explanation:

3.15 can be read as  3 and 15 hundredths =   3   15/100  = 3  3/20

Answer:  63/20

Step-by-step explanation:

Answer:

450

Step-by-step explanation:

In response to the stated question, we may state that Since [tex]\theta[/tex]is in the trigonometry fourth quadrant, [tex]\theta/2[/tex] is also in the fourth quadrant.

The study of the connection between triangle side lengths and angles is known as trigonometry. The concept first originated in the Hellenistic era, during the third century BC, due to the application of geometry in astronomical investigations. The subject of mathematics known as exact techniques deals with certain trigonometric functions and their possible applications in calculations. There are six commonly used trigonometric functions in trigonometry. Sine, cosine, tangent, cotangent, secant, and cosecant are their separate names and acronyms (csc). The study of triangle characteristics, particularly those of right triangles, is known as trigonometry. As a result, geometry is the study of the properties of all geometric forms.

a) We can use the double angle formula for sine to find sin 2theta:

[tex]sin 2\theta = 2sin \theta cos \theta\\sin^2 \theta + cos^2 \theta = 1\\sin^2 \theta + (3/10)^2 = 1\\sin^2 \theta = 1 - (9/100)\\sin \theta = \sqrt(91)/10 \\sin 2\theta = 2sin \theta cos \theta\\sin 2\theta = 2(\sqrt(91)/10)(3/10)\\sin 2\theta = 3\sqrt(91)/50[/tex]

b) To find the quadrant in which [tex]\theta/2[/tex] is located, we need to find [tex]\theta/2[/tex] first:

[tex]\theta/2 = (3\pi/2 + \theta)/2\\\theta/2 = 3\pi/4 + t\heta/2\\\theta/2 - \theta/2 = 3\pi/4\\\theta/2 = 3\pi/4\\\theta = 3\pi/2[/tex]

Since theta is in the fourth quadrant, [tex]\theta/2[/tex] is also in the fourth quadrant.

c) To find cos theta/2, we can use the half angle formula for cosine:

[tex]cos(theta/2) = \sqrt((1 + cos theta)/2)\\cos(theta/2) = \sqrt((1 + 3/10)/2)\cos(theta/2) = \sqrt(13/20)\\cos(theta/2) = \sqrt(13)/2sqrt(20)\\cos(theta/2) = \sqrt(13)/10[/tex]

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The lumpsum amount invested in 18 years ago for the given value Future value and interest is = Rs 2763,18.3268

What about lumpsum?

In mathematics, a lump sum refers to a single, fixed amount of money or quantity of something that is paid or received all at once, rather than being paid or received in installments or over a period of time.

The term "lump sum" can be used in various mathematical contexts, such as in finance, where it can refer to a single payment or investment, or in statistics, where it can refer to a single value or data point.

Define future value:

In mathematics, future value (FV) refers to the value of an investment or cash flow at a specified date in the future, assuming a certain rate of return.

The future value of an investment is the amount of money that an investor would expect to have at a specified date in the future if they were to invest a certain amount of money today, and if that investment earns a certain rate of return over time.

⇒ The formula for calculating the future value of an investment is:

[tex]FV = PV x (1 + r)^n[/tex]

According to the given information:

Interest Rate = 5%

Time Period =  18 years

Future Value = Rs 6649,93.

Using the concept of TVM calculation we have that,

Present value = [tex]\frac{Future Value}{(1+r)^{t} }[/tex]

So, the Present Value = [tex]\frac{664993}{(1+0.05)^{18} } = 2763,18.3268[/tex]

Hence, the Lumpsum Amount is Rs 2763,18.3268

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The vector equations of L₂ when expressed as Cartesian equations are

y = 0

x = (z ± √(z² - 4(z-2))) / 2

z = [(4 - ((x-1)/(-z))²) ± √((4 - ((x-1)/(-z))²)² + 64)] / 8

To find the vector equation of the line L₂, we need to find a vector that is perpendicular to L₁ and passes through the point (1,0,2). Let's start by finding the vector equation of L₁.

Let P(x,y,z) be a point on L₁. Then the vector equation of L₁ is given by:

r₁ = P + t * d₁

where d₁ is the direction vector of L₁ and t is a scalar parameter.

Since L₂ is perpendicular to L₁, its direction vector must be perpendicular to d₁. Thus, we can find a vector that is perpendicular to d₁ by taking the cross product of d₁ with any non-zero vector that is not parallel to d₁. Let's choose the vector (0,1,0):

v = d₁ x (0,1,0) = (-z,0,x)

Note that we can choose any non-zero vector that is not parallel to d₁, and we will still get a vector that is perpendicular to d₁.

Now we have a point on L₂ (1,0,2) and a direction vector (v), so we can write the vector equation of L₂:

r₂ = (1,0,2) + s * v

where s is a scalar parameter.

To express the Cartesian equations of L₂, we can write the vector equation as a set of three parametric equations:

x = 1 - sz

y = 0

z = 2 + sx

We can eliminate the parameter s by solving for it in two of the equations and substituting into the third equation:

s = (x - 1) / (-z)

s = (z - 2) / x

Setting these two expressions equal to each other and solving for x, we get:

[tex]x^2 - zx + z - 2 = 0[/tex]

This is a quadratic equation in x, so we can solve for x using the quadratic formula:

[tex]x = (z \± \sqrt{(z^2 - 4(z-2)})) / 2[/tex]

Substituting this expression for x into one of the parametric equations, we get:

y = 0

And substituting the expressions for x and s into the other parametric equation, we get:

[tex]z = 2 + [(z \± \sqrt{(z^2 - 4(z-2)})) / 2] * [(1 - sz) / (-z)][/tex]

Simplifying this equation, we get:

[tex]4z^2 - (4 - s^2)z - 4 = 0[/tex]

Again, this is a quadratic equation in z, so we can solve for z using the quadratic formula:

[tex]z = [(4 - s^2) \± \sqrt((4 - s^2)^2 + 64)] / 8[/tex]

z = [(4 - s²) ± √((4 - s²)² + 64)] / 8

Finally, we can substitute these expressions for x and z into one of the parametric equations to get:

[tex]y = 0\\x = (z \± \sqrt{(z^2 - 4(z-2)})) / 2\\z = [(4 - ((x-1)/(-z))^2) \± \sqrt{((4 - ((x-1)/(-z))^2)^2} + 64)] / 8[/tex]

These are the Cartesian equations of L₂.

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The baby born in week 40 weighs relatively less since its z-score -0.26, is smaller than the z-score of -0.16 for the baby born in week 32.

To find the z-score, we use the formula:

z = (x - μ) / σ

where x is the individual value, μ is the mean, and σ is the standard deviation.

For the 32-week gestation period baby (weight = 2575 grams):

For the 40-week gestation period baby (weight = 3075 grams):

Now, we round the z-scores to two decimal places:

Since the z-score for the 40-week baby (-0.26) is smaller than the z-score for the 32-week baby (-0.16), the correct answer is:

C. The baby born in week 40 weighs relatively less since its z-score -0.26, is smaller than the z-score of -0.16 for the baby born in week 32.

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Miles is going to spend less than the hours that were spent last week since 2/3 is a fraction.

A fraction represents a part of a whole, and is therefore always less than the whole. For example, 2/3 represents two out of three equal parts of a whole.

The implication of this is that the time that Miles would have to spend on biking in the coming week would be wo out of three equal parts of a whole time that was spent in the last week and this would be less than the time spent last week.

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Answer:

subtract 20; 80, 60, 40

The pattern is to subtract 20 from the previous term.

Next three terms: 20, 0, -20.

Therefore, the sequence continues as follows:

80, 60, 40, 20, 0, -20.

Answer:

Step-by-step explanation:

12 to the second power (or 12 squared) is equal to 144.

Mathematically, it can be represented as:

12^2 = 12 x 12 = 144

The unknown side lengths of the given right triangle ∆LJK where LJ 30° and JK 5 are LJ = 6, and JK = 4.

Triangle is a three-sided polygon with three angles. It is one of the basic shapes in geometry. A triangle has three sides and three angles, usually measured in degrees. The angles always add up to 180°. Triangles can be classified according to the lengths of their sides or the sizes of their angles. Types of triangles include right, acute, obtuse, scalene, isosceles, and equilateral. Triangles are also used to form more complex shapes in geometry, such as polygons and circles.

To find the unknown side lengths of a right triangle, we can use the Pythagorean Theorem. The Pythagorean Theorem states that the square of the hypotenuse (the longest side of a triangle) is equal to the sum of the squares of the other two sides. In this case, the hypotenuse is JK and the other two sides are LJ and JK.

We can use the Pythagorean Theorem to find the unknown side length of JK. We know that the angle LJ is 30°, so we can use the sine rule to calculate the length of side LJ. The sine rule states that the ratio of the length of a side of a triangle to the sine of its opposite angle is equal for all three sides. Therefore, we can calculate the length of LJ by dividing the sine of 30° (0.5) by the length of the opposite side (JK).

We can now use the Pythagorean Theorem to calculate the length of side JK. We know that the length of LJ is 6, and the length of JK is 5. We can plug these values into the equation a2 + b2 = c2, where a is the length of LJ (6), b is the length of JK (5), and c is the hypotenuse (JK). This yields the equation 36 + b2 = c2, which simplifies to b2 = c2 - 36. Since we know that c2 = 52, we can solve for b2 by subtracting 36 from 52, which gives us 16. The square root of 16 is 4, so the length of JK is 4.

Therefore, the unknown side lengths of the given right triangle ∆LJK where LJ 30° and JK 5 are LJ = 6, and JK = 4.

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Answer:0.25

Step-by-step explanation:

Answer: LN = 15

Make a proportional relationship:

[tex]\dfrac{LN}{JL}=\dfrac{LM}{KL}[/tex]

Insert the values:

[tex]\longrightarrow\dfrac{LN}{30} =\dfrac{6}{30-18}[/tex]

cross multiply:

[tex]\longrightarrow LN=\dfrac{30(6)}{12}[/tex]

Simplify:

The distance the plane traveled from point A to point B is approximately 8.15 miles or 43056 feet (rounded to the nearest tenth of a foot).

An angle is a geometric figure formed by two rays, called the sides of the angle, that share a common endpoint, called the vertex of the angle. Angles are typically measured in degrees or radians, and they are used to describe the amount of rotation or turning between two lines or planes. In a two-dimensional plane, angles are usually measured as the amount of rotation required to move one line or plane to coincide with the other line or plane.

Let's first draw a diagram to visualize the problem:

                    /  |

                  /     |

                 /       |P (plane)

                /        |

               /         |

              /          | h = 6900 ft

            /            

           / θ2.        |  

         /                |

       /                  |

   B ___/θ1__  _|___ A

           d

We need to find the distance the plane traveled from point A to point B, which we'll call d. We can use trigonometry to solve for d.

From point A, we have an angle of elevation of 16° to the plane. This means that the angle between the horizontal and the line from point A to the plane is 90° - 16° = 74°. Similarly, from point B, we have an angle of elevation of 27° to the plane, so the angle between the horizontal and the line from point B to the plane is 90° - 27° = 63°.

Let's use the tangent function to solve for d:

x = h / tan(74°) = 19906.5 ft

d - x = h / tan(63°) = 23205.2 ft

So,

d = x + h / tan(63°) ≈ 43111.7 ft ≈ 8.15 miles.

Therefore, the distance the plane travelled from point A to point B is approximately 8.15 miles or 43056 feet (rounded to the nearest tenth of a foot).

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The expression represents the distributive property from the list of options is (b) a(b + c) = ab + ac

The distributive property is a mathematical property that applies to multiplication and addition or subtraction.

It states that when you multiply a number (or variable) by a  sum or difference inside parentheses, you can distribute the multiplication across the terms inside the parentheses.

This is represented by the following formula:

a(b + c) = ab + ac

This is represented by Option (B)

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Answer:i don't know the answer

Step-by-step explanation: i don't konw

Step-by-step explanation:

given quadratic equation is

simplification,

by using,

[tex]y^\frac{1}{2} -6y^\frac{1}{4} +8=0

\\ y^\frac{1}{2} -6y^\frac{1}{4} = - 8 \\

[/tex]

[tex]square[/tex]

[tex] \\ {(y^\frac{1}{2} -6y^\frac{1}{4})}^{2} = {(- 8)}^{2}[/tex]

[tex]y + 36 {y}^{ \frac{1}{2}} \: [/tex]

The value οf the variable y is 9 when k is -3 in the given question.

In mathematics and statistics, a variable is a quantity οr a characteristic that can take οn different values οr attributes. Variables can be classified as either quantitative οr categοrical, depending οn the type οf data they represent.

A quantitative variable is a variable that represents a numerical measurement οr quantity. Examples οf quantitative variables include height, weight, temperature, and incοme.

A categοrical variable is a variable that represents a grοup οr categοry. Examples οf categοrical variables include gender, race, and type οf car.

Given: y=k x

where, x= -3 and k= -3

we can find the value of y by multiplying k with x,

so, y=k x

now, putting values as follows:

we get, y = -3 × -3

               = 9

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Answer:

The function g(x) = (x + 2)³ − 6 is obtained by applying three transformations to the parent function f(x) = x³.

First, g(x) is shifted 2 units to the left by subtracting 2 from x inside the parentheses:

g(x) = (x + 2 - 2)³ − 6 = (x)³ − 6

This shows that option C, "g(x) is shifted 2 units to the left and 6 units down," is not correct.

Next, g(x) is shifted 6 units down by subtracting 6 from the entire function:

g(x) = (x + 2)³ − 6 - 6 = (x + 2)³ - 12

This shows that option A, "g(x) is shifted 2 units to the right and 6 units down," is correct.

Finally, g(x) is not shifted left or right by any additional units, but it is shifted 2 units up by adding 2 to the constant term:

g(x) = (x + 2)³ − 6 + 2 = (x + 2)³ - 4

This shows that option B, "g(x) is shifted 6 units to the right and 2 units up," is not correct.

Therefore, the correct answer is option A: g(x) is shifted 2 units to the right and 6 units down.