what is a collection of numbers and mathematical operations

Answer:

y = 5x + 11

Step-by-step explanation:

Step 1:  When two lines are parallel, they have the same slope, as indicated by the following equation as m2 = m1, where

Thus, since the slope of line 1 is 5, the slope of line 2 is also 5.

Step 2:  Now we can plug in (-2, 1) for x and y and 5 for m to solve for b, the y-intercept of the other line:

1 = 5(-2) + b

1 = -10 + b

11 = b

Thus, the equation of the line parallel to y = 5x + 2 and passing through (-2, 1) is y = 5x + 11

The value of tan (A-B) is equal to 715/2052.

To find the value of tan(A - B), we can use the trigonometric identity:

tan(A - B) = (tan(A) - tan(B))/(1 + tan(A)tan(B))

Given that tan(A) = 40/9 and sin(B) = 45/53, we can determine the values of cos(B) and tan(B) using the Pythagorean identity:

sin^2(B) + cos^2(B) = 1

cos(B) = sqrt(1 - sin^2(B))

cos(B) = sqrt(1 - (45/53)^2)

cos(B) = sqrt(1 - 2025/2809)

cos(B) = sqrt(784/2809)

cos(B) = 28/53

tan(B) = sin(B)/cos(B)

tan(B) = (45/53)/(28/53)

tan(B) = 45/28

Now we can substitute the values into the formula for tan(A - B):

tan(A - B) = (tan(A) - tan(B))/(1 + tan(A)tan(B))

tan(A - B) = (40/9 - 45/28)/(1 + (40/9)(45/28))

tan(A - B) = [(1120/252 - 405/252)] / (1 + (1800/252))

tan(A - B) = (715/252) / (2052/252)

tan(A - B) = 715/2052

Therefore, tan(A - B) is equal to 715/2052.

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Answer:

Step-by-step explanation:

Answer:

3



Step-by-step explanation:

Area = 7pi/5

56/360 × pi r² = 7pi/5

Pi is canceled on both sides.

r² = 7/5 ÷ 56/360 = 9

r = root 9 = 3

The equation of the line that passes through point (-3,-7) and point (-3,10) is x = -3.

The formula for equation of line is expressed as;

y = mx + b

Where m is slope and b is y-intercept.

Given the points through which the line passes: (-3,-7) and (-3,10).

The two given points (-3, -7) and (-3, 10) have the same x-coordinate -3

Hence, the two lines  lie on a vertical line.

since the slope of the vertical line is undefined.

The equation of the line passing through these two points is simply the equation of the vertical line:

x = -3

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All the expressions after completing the square each equation are,

⇒ (x - 1/2)² = 5/4

⇒ (x - 5/2)² + 3/4 = 0

⇒ (2x + 3/2)² = 57/4

Given that;

Expressions are,

⇒ 2x² - 2x + 7 = 5

⇒ -2x² + 10x = 14

⇒ 4x² + 6x = 12

Now, We can completing the square each equation as;

⇒ 2x² - 2x + 7 = 5

⇒ x² - x + 7/2 = 5/2

⇒ x² - x + 1/4 - 1/4 + 7/2 = 5/2

⇒ (x - 1/2)² = 1 + 1/4

⇒ (x - 1/2)² = 5/4

⇒ -2x² + 10x = 14

⇒ - x² + 5x = 7

⇒ x² - 5x = - 7

⇒ x² - 5x + 25/4 - 25/4 = - 7

⇒ (x - 5/2)² = - 7 + 25/4

⇒ (x - 5/2)² = - 3/4

⇒ (x - 5/2)² + 3/4 = 0

⇒ 4x² + 6x = 12

⇒ (2x)² + 2×2x×3/2 + 9/4 -9/4 = 12

⇒ (2x + 3/2)² = 12 + 9/4

⇒ (2x + 3/2)² = 57/4

Thus, All the expressions after completing the square each equation are,

⇒ (x - 1/2)² = 5/4

⇒ (x - 5/2)² + 3/4 = 0

⇒ (2x + 3/2)² = 57/4

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Answer:

(a) 5 ft^2

Step-by-step explanation:

The formula for area of a triangle is given by the formula,

A = 1/2bh, where

Thus, we can plug in 5 for b and 2 for h in the formula to find the area of the triangle in square feet:

A = 1/2(5)(2)

A = (5/2)(2)

A = 10/2

A = 5

Thus, the area of a triangle whose base is 5 ft and whose height is 2 ft is 5 ft^2

Answer:

[tex]m = \frac{1 - ( - 3)}{0 - ( - 4)} = \frac{4}{4} = 1[/tex]

The calculated value of the mode(s) of the data is (a) 1

From the question, we have the following parameters that can be used in our computation:

1, 0, 2, 2, 3, 0, 1, 1, 4, and 5

By definition, the mode of a data is the data that has the highest frequency

Using the above as a guide, we have the following:

The data element 1 has the highest frequency of 3

Other data elements have lesser frequencies

Hence, the mode(s) of the data is (a) 1

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The total volume of the air space of spherical ball is A = 12.265625 inches³

Given data ,

Since each tennis ball has a diameter of 2.5 inches, the radius of each ball is 1.25 inches.

The air space around the balls can be thought of as a cylinder with a height equal to the diameter of one ball and a radius equal to the radius of one ball.

The height of the cylinder is 2.5 inches, and the radius is 1.25 inches.

The formula for the volume of a cylinder is:

V = πr²h

V = ( 3.14 ) ( 1.25 )² ( 7.5 )

V = 36.796875 inches³

where V is the volume, r is the radius, and h is the height.

So, the volume of the one ball is:

V₁ = ( 4/3 )π(1.25)³

V₁ = 8.177083 inches³

The total volume of three balls is = volume of 3 spherical balls

V₂ = 3V₁ = 3(8.177083) ≈ 24.53125 cubic inches

Therefore, the total volume of the air space around the three tennis balls is approximately A = 36.796875 inches³ - 24.53125 inches³

A = 12.265625 inches³

Hence , the volume of air space is A = 12.265625 inches³

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t = (6 + 3k)/(k - 2) is the formula for t in the equation k=2(t+3)/t-3.

The given equation is k=2(t+3)/t-3.

We need to isolate t on one side of the equation.

Let's start by cross-multiplying:

k(t - 3) = 2(t + 3)

Expanding the brackets:

kt - 3k = 2t + 6

Now, let's gather all the terms with t on one side and all the constant

terms on the other side:

kt - 2t = 6 + 3k

Factoring out t on the left side:

t(k - 2) = 6 + 3k

Finally, we can solve for t by dividing both sides of the equation by (k - 2):

t = (6 + 3k)/(k - 2)

Therefore, t is the subject of the formula, and it is given by t = (6 + 3k)/(k - 2).

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The measure,

⇒∠s = 124 degree

In the given figure of kite,

PQRS,

Measure of angle p = 22 degree

And angle R is a right angle

Therefore,

∠R = 90 degree

Now we know that for a kite PQRS

Angle Q and Angle S are equal

Now consider,

Angle s is equal to x degree

Therefore,

∠S = ∠ Q = x degree

We know that,

For a kite the sum of interior angle is equal to 360 degree.

Therefore,

⇒ ∠P + ∠Q + ∠R + ∠S = 360

⇒ 22 + x + 90 + x = 360

⇒ 22 + x + 90 + x = 360

⇒                     2x  = 248

⇒                        x  = 124 degree.

Thus,

Measure of angle s = 124 degree.

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The coordinates of vertex D in the parallelogram are (-7, -10).

We have,

To find the coordinates of vertex D of the parallelogram, we need to use the properties of a parallelogram.

One of these properties states that opposite sides of a parallelogram are parallel and have equal lengths.

Given that A = (8, 2), B = (6, -4), and C = (-5, -4), we can find the coordinates of D as follows:

Find the vector representing one of the sides of the parallelogram.

We can use the vector AB.

Vector AB = (x-coordinate of B - x-coordinate of A, y-coordinate of B - y-coordinate of A)

= (6 - 8, -4 - 2)

= (-2, -6)

Add this vector to point C to find the coordinates of D.

Coordinates of D = (x-coordinate of C + x-coordinate of AB, y-coordinate of C + y-coordinate of AB)

= (-5 - 2, -4 - 6)

= (-7, -10)

Therefore,

The coordinates of vertex D are (-7, -10).

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Based on the information provided, it seems that you have a rectangular grid with a shaded area, and the grid contains  option D.  48 square units.

To calculate the area of the shaded part, we would need to know the dimensions of the rectangle as well as the specific location and size of the shaded region within it.

If you can provide more details about the dimensions of the rectangular grid and the shaded region, I would be happy to help you calculate the area of the shaded part. Once we have that information, we can apply the formula for calculating the area of a rectangle,

which is Area = length × width. The result will give us the area of the shaded part in square units.

The area in square units, of the shaded part of the rectangle It has 48. Therefore the correct option  D

The Question was Incomplete, Find the full content below :

The grid shown below is in the shape of a rectangle. What is the area, in square units, of the shaded part of the rectangle? a 14 b 24 c 28 d 48

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Answer:

a) r(t) = (10, 5, -5) + (5, 5, 0)*t

b) (0, -5, -5)

Step-by-step explanation:

a) 2x + 4 -2y -5 = -3z -6

2x - 2y +3z +5 =0

(10, 5, -5)

(15, 10, -5)

(5, 5, 0)

r = (10, 5, -5) + (5, 5, 0)*t

b) The yz plane is given by the equation x = 0.

x = 0  in the vector equation of a straight line if and only if  t = -2, than r ( - 2) = (0, -5, -5) is the desired intersection point.

A single log statement from the many log statements to 1 is: [tex]log(7wy^{(-8)}/x^3)[/tex]

The exponent that indicates the power to which a base number is raised to produce a given number are called logarithm.

Use the logarithmic identity:

log[tex](a^n)[/tex] = n*log(a)

to convert the coefficients 2, 3, and 8:

log w + log 7 - 3log x - 8log y

= log w + log 7 - log [tex]x^3[/tex] - log [tex]y^8[/tex]

Combine the terms on the right-hand side using the logarithmic identity:

log(a) + log(b) = log(ab)

log w + log 7 - log[tex]x^3[/tex] - log [tex]y^8[/tex]

= log([tex]7wy^{-8}/x^3[/tex])

Therefore, the single log statement is from the many log statements to 1 is: log[tex](7wy^{-8}/x^3)[/tex]

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Answer:

12

Step-by-step explanation:

206

5x10 12x13=206 it may be incorrect but i not sure if it is incorrect

The equation of the line is y = x/7 - 1/7

The equation of a straight line is y=mx+c . Where m is the gradient or slope and c is the height at which the line crosses the y -axis, also known as the y -intercept.

The equation of a line can be expressed as;

y-y1 = m(x -x1)

where m is the slope

The slope = 1/7

x1 = 8 and y1 = 1

Therefore;

y -1 = 1/7( x -8)

7(y-1) = x-8

7y -7 = x-8

collecting like terms

7y = x - 8+7

7y = x -1

y = x/7 -1/7

The intercept is -1/7 and the slope is 1/7

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The equation of the plane pi is: -6x-7y-9z=0.

To find the equation of the plane that contains the given line L and the origin as well, we first need to find two vectors that lie on the plane. One vector can be the direction vector of the line L, which is (2i - j + 3k). Now to find the second vector, we can take the vector from the origin to any point on the line L, and this vector will lie on the plane.

Let us now take t=0, and find the point on the line L:

r = 3i + 3j - k + 0(2i - j + 3k)

= 3i + 3j - k

So, the vector from the origin to this point is simply (3i + 3j - k). We can just take (3i + 3j - k) as our second vector.

Now, we can find the normal vector of the plane by taking the cross-product of two vectors that we just found, we get:

n = (2i - j + 3k) * (3i + 3j - k)

= -6i - 7j - 9k

Therefore, the equation of the plane pi is: -6x-7y-9z=0.

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Solving a system of equations we can see that Gavin worked 7 hours tutoring.

Let's define the variables:

x = number of hours tutoring.

y = number of hours land scaping.

We know that he worked for 10 hours and earned $137, then we can write a system of equations:

x + y = 10

14x + 13y = 137

Isolating y on the first equation we get:

y = 10 - x

Replace that in the second one to get:

14x + 13*(10 - x) = 137

14x + 130 - 13x = 137

x = 137 - 130 = 7

He worked 7 hours tutoring.

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In a rectangle, the value of x is,

⇒ x = 5

We have to given that;

ABCD is a rectangle.

And, AC = 5x + 2

BD = x + 22

Since, We know that;

Diagonal of a rectangle are equal in length.

Hence, We get;

AC = BD

(5x + 2) = (x + 22)

5x - x = 22 - 2

4x = 20

x = 20/4

x = 5

Thus, the value of x is,

⇒ x = 5

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37 million can be written as 3.7 × 10⁶ in the power of ten exponent notation.

In scientific notation, a number is written as the product of a coefficient and a power of ten.

To convert 37 million to scientific notation, we need to move the decimal point to the left until we have a number between 1 and 10.

Starting with 37 million, we can move the decimal point six places to the left to obtain the number 3.7:

= 37,000,000 -> 3.7

Now, we express this number as a product of the coefficient (3.7) and 10 raised to the power of the number of places we moved the decimal point.

In this case, since we moved the decimal point six places to the left, the exponent of ten is 6:

37 million = 3.7 × 10⁶

Therefore, 37 million can be written as 3.7 × 10⁶ in the power of ten exponent notation.

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Answer: 0 , 5

Step-by-step explanation:

Imagine a plane taking off, first it drives to gain speed then flies, thats what you can use for coordinates.

X is the driving and Y is the flight.

The X is at 0, and the Y is at 5.

The triangles are not similar

Given data ,

Let the first triangle be ΔJKL

Let the second triangle be ΔMNP

Now , the corresponding sides are

JK / JL ≠ NM / MP

where the corresponding sides of similar triangles are not in the same ratio

And , the common angle to both the triangles is ∠J = ∠M

So , ∠J = ∠M and JK / JL ≠ NM / MP

Hence , the triangles are not similar

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Answer:

  (x +4)²/25 +(y -3)²/16 = 1

Step-by-step explanation:

You want the equation of the ellipse with center (-4, 3) and semi-axes 5 and 4 in the x- and y-directions, respectively.

The standard form equation for an ellipse with center (h, k) and sem-axes 'a' and 'b' in the x- and y-directions, respectively, is ...

  (x -h)²/a² +(y -k)²/b² = 1

Using the given values, we find the equation to be ...

  (x +4)²/25 +(y -3)²/16 = 1

__

Additional comment

The longer of the two axes is the "major" axis, and its end points are the "vertices". For the purpose of an ellipse with center, vertices, and co-vertices specified, the equation is not affected by which axis is longer.

Effectively, this is the equation of a circle with different scale factors in the x- and y-directions.

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The standard deviation of the random variable x is 1.1145

Since the Probability distribution of a random variable is the collection of value and its probability pair for values of that considered random variable.

The probability of a value of x is never neg.

The sum of the probabilities is always 1.

The sum of all values of P(X = x) equals 1, which is, considering n values

There are no negative values of P(X = x).

Since we are given the probability distribution as below, we will go and calculate the mean and standard deviation using the formula:

E(X)= ∑xP(X=x)

Standard deviation = √Variance(x)

Var(x) = E(x²) - [E(x)]²

X           0          1            2        3        4          5

p(x=x)   0.032   0.152   0.316  0.316  0.152  0.032

First we have to find E(x)

E(x) = (0x0.032) + (1x0.152) + (2x0.316) + (3x0.316) + (0.152) + (5x0.032)

The mean E(x) = 2.5

To find variance, we use Var(x) = E(x²) - [E(x)]²

But we have to calculate E(x²) first, since we already have found E(x)

So,

E(x²) = (0²x0.032) + (1²x0.152) + (2²x0.316) + (3²x0.316) + (4²x0.152) + (5²x0.032)

therefore, E(x²) = 7.492

Now we have all the values we can find the variance:

Var(x) = E(x²) - [E(x)]²

=7.492 - [2.5]²

= 1.242

With variance now we can find the standard deviation;

S.D = √Var(x)

= √1.242

= 1.1145

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The area of the given figure with rectangle and triangle is 27.5 square centimeters.

The given figure has a rectangle and two triangles.

The area of the rectangle is length times width

Length = 5 cm

Width = 4 cm

Area of rectangle = 5×4

=20 square centimeters

Area of triangle =1/2 base ×height

=1/2×2.5×3

=3.75  square centimeters

As there are two triangles, 2(3.75)=7.5   square centimeters

Total area = 20+7.5

=27.5 square centimeters

Hence, the area of the given figure with rectangle and triangle is 27.5 square centimeters.

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Answer:

Step-by-step explanation:

The numerical value of the given log expression is -73.

From definitions of logarithms, we know that :

[tex]log(a*b) = log(a) + log(b)\\log(a / b) = log(a) - log(b)[/tex]

[tex]log(a^n) = n*log(a)[/tex]

Therefore,

[tex]log(\frac{\sqrt[3]{b^4}}{a^8c^5 }) = \frac{4}{3}*log(b) - 8*log(a) - 5*log(c)[/tex]

substituting the given values

[tex]log(a) = 10\\log(b) = 9\\log(c) = 1\\[/tex]

we get the value of the given expression

[tex]log(\frac{\sqrt[3]{b^4}}{a^8c^5 }) = \frac{4}{3}*9 - 8*10 - 5*1[/tex]

[tex]log(\frac{\sqrt[3]{b^4}}{a^8c^5 }) = 12 - 80 - 5[/tex]

[tex]log(\frac{\sqrt[3]{b^4}}{a^8c^5 }) = -73[/tex]

Therefore the numerical value of the given log expression is -73.

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