The following number is a BASE-7 NUMBER. Convert this to a base 10 number and enter your answer in the box.
463421
(Remember, the above number is in Base 7!)

Number in base 10:

Answer:

[tex]x^2+2y=1[/tex]

Step-by-step explanation:

The equation of a parabola with a vertical axis of symmetry,

focus (h, k+p), and directrix x=h-p is given by:

[tex](x - h)^2 = 4p(y - k)[/tex]

In this case, the focus is (0, 1) and the directrix is x =3.

Comparing this to the general equation,

we have

h = 0, k = 1, and x = h - p = 3.

From x = h - p, we can solve for p:

3 = 0 - p

p = -3

Substituting the values of h, k, and p into the equation, we get:

[tex](x - 0)^2 = 4(-3)(y - 1)[/tex]

Simplifying further:

[tex]x^2 = -12(y - 1)[/tex]

[tex]x^2=-12y+1[/tex]

[tex]x^2+12y=1[/tex]

Therefore, the parabola equation is [tex]x^2+2y=1[/tex]

Answer:

-7/4

Step-by-step explanation:

we can see that every time the value of x increases by 4, the value of y decreases by 7.

let's pick two sets of coordinates (first two will be fine).

that is (-4, 6) and (0, -1)

Slope = (change in y values) / (change in x values)

= (-1 - 6) / (0 - -4)

= -7 / (0 + 4)

= -7/4.

so our slope (gradient) is -7/4

Answer: There are 240 people inside the auditorium.

Step-by-step explanation:

30 x 8 = 240

The equation 6y + y = 5 can be written in a slope-intercept form as y = 5/7.

We have,

To write the equation 6y + y = 5 in slope-intercept form (y = mx + b), we need to simplify the equation and isolate the y variable on one side.

Starting with the equation 6y + y = 5:

Combining the like terms on the left side gives us:

7y = 5

To isolate the y variable, we divide both sides of the equation by 7:

y = 5/7

Now the equation is in the form y = mx + b, where m represents the slope and b represents the y-intercept.

In this case, since the equation only contains the variable y and no x, the slope (m) is not present, and the y-intercept (b) is 5/7.

Therefore,

The equation 6y + y = 5 can be written in a slope-intercept form as y = 5/7.

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The perimeter of the dilated rectangle C'D'E'F' is 200 units.

To find the perimeter of the dilated rectangle C'D'E'F', we need to determine the new coordinates of its vertices after the dilation.

Given that the scale factor is 5 and the dilation is centered at (0, 0), each coordinate of the original rectangle CDEF will be multiplied by 5 to obtain the corresponding coordinate of the dilated rectangle C'D'E'F'.

The original coordinates of CDEF are:

C (-10, 10)

D (5, 10)

E (5, 5)

F (-10, 5)

To find the coordinates of the dilated rectangle C'D'E'F', we multiply each coordinate by 5:

C' = (-10 × 5, 10 × 5) = (-50, 50)

D' = (5 × 5, 10 × 5) = (25, 50)

E' = (5 × 5, 5 × 5) = (25, 25)

F' = (-10 × 5, 5 × 5) = (-50, 25)

Now, we can calculate the perimeter of the dilated rectangle C'D'E'F' by summing the lengths of its sides.

Length of side C'D':

√[(-50 - 25)² + (50 - 50)²] = √[(-75)² + 0²] = √[5625] = 75

Length of side D'E':

√[(25 - 25)² + (50 - 25)²] = √[0² + 625] = √[625] = 25

Length of side E'F':

√[(25 - (-50))² + (25 - 25)²] = √[75² + 0²] = √[5625] = 75

Length of side F'C':

√[(-50 - (-50))² + (25 - 50)²] = √[0² + 625] = √[625] = 25

Now, we add up the lengths of all four sides to find the perimeter:

Perimeter = C'D' + D'E' + E'F' + F'C'

= 75 + 25 + 75 + 25

= 200

Therefore, the perimeter of the dilated rectangle C'D'E'F' is 200 units.

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The probability of event B would likely be greater when event A is true, reflecting the tendency of professional basketball players to be taller.

To determine the probabilities in question, we need to consider the information provided and make some assumptions based on general knowledge about the population of the United States.

P(B) when you have no other information:

Without any other information, we cannot accurately determine the probability of event B, which represents "The woman is taller than 5 feet 4 inches." We would need additional data on the height distribution of women in the United States to calculate this probability.

P(B) when you know A is true:

If we know that event A is true, meaning "The woman is a professional basketball player," we can make some assumptions based on the nature of professional basketball players.

Generally, professional basketball players tend to be taller than the average population due to the physical requirements of the sport. Therefore, the probability of event B, "The woman is taller than 5 feet 4 inches," would likely be greater when we know event A is true.

P(B) when you know A is false:

If event A is false, meaning "The woman is not a professional basketball player," we cannot make any definitive conclusions about the probability of event B, "The woman is taller than 5 feet 4 inches." The height of an individual is not solely determined by their profession, so without further information, we cannot determine if event B is more or less likely when event A is false.

In summary, based on the given information, we can conclude that the probabilities of event B are not equal under different scenarios. The probability of event B would likely be greater when event A is true, reflecting the tendency of professional basketball players to be taller. However, without any other information, we cannot determine the probability of event B or make comparisons when event A is false.

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Answer: To rewrite the fractions 2/3 and 4/15 with the least common denominator (LCD), we need to find the smallest multiple that both denominators, 3 and 15, divide into evenly.

The prime factorization of 3 is 3, and the prime factorization of 15 is 3 * 5.

To find the LCD, we take the highest power of each prime factor that appears in either denominator. In this case, the highest power of 3 is 3, and the highest power of 5 is 5.

The LCD is the product of these highest powers: LCD = 3 * 5 = 15.

Now, we can rewrite the fractions with the least common denominator:

2/3 = (2/3) * (5/5) = 10/15

4/15 = (4/15) * (1/1) = 4/15

Therefore, the fractions 2/3 and 4/15 can be rewritten with the least common denominator as 10/15 and 4/15, respectively.

Step-by-step explanation:

Answer:

2. C. 50.3 ft²

3. A. 75.4 ft²

Step-by-step explanation:

The lateral surface area of a cylinder is the area of the curved surface of the cylinder. It is calculated by multiplying the circumference of the base by the height of the cylinder. The formula for the lateral surface area of a cylinder is:

Lateral Surface Area = 2πrh

Where:

The total surface area of a cylinder is the area of the lateral surface plus the area of the two circular bases. The formula for the total surface area of a cylinder is:

Total Surface Area = 2πrh + 2πr^2

Where:

2.

r=2 ft

h=4 ft

Lateral Surface Area = 2πrh=2*22/7*2*4=50.3 ft²

3.

Total Surface Area = 2πrh + 2πr^2=2*22/7*2*4+2*22/7*2

=50.3+25.1=75.4 ft²

Answer: Let's try x = 1 as a potential solution:

Substituting x = 1 into the inequality:

3(1) - 7 ≥ -10

3 - 7 ≥ -10

-4 ≥ -10

Since -4 is greater than or equal to -10, x = 1 is a solution to the inequality.

Let's try x = -3 as a potential solution:

Substituting x = -3 into the inequality:

3(-3) - 7 ≥ -10

-9 - 7 ≥ -10

-16 ≥ -10

Since -16 is not greater than or equal to -10, x = -3 is not a solution to the inequality.

Therefore, x = 1 is a solution to the inequality 3x - 7 ≥ -10, while x = -3 is not a solution.

Step-by-step explanation:

Using trigonometric identities sin(x)/[1 - cos(x)] - sin(x)cos(x)/[1 + cos(x)] = csc (x)(1 + cos² (x)),

Trigonometric identities are equations that contain trigonometric ratios.

To verify the trigonometric identity

sin(x)/[1 - cos(x)] - sin(x)cos(x)/[1 + cos(x)] = csc (x)(1 + cos² (x)), we need to show that Left Hand Side, L.H.S equals Right Hand Side R.H.S. We proceed as follows.

L.H.S = sin(x)/[1 - cos(x)] - sin(x)cos(x)/[1 + cos(x)]

Taking the L.C.M, we have that

{sin(x)[1 + cos(x)] - sin(x)cos(x)[1 - cos(x)]}/[1 - cos(x)][1 + cos(x)]

Expanding the brackets, we have that

{sin(x) + sin(x)cos(x)] - sin(x)cos(x) + sin(x)cos²(x)]}/[1 - cos(x)][1 + cos(x)]

Simplifying, we have that

= {sin(x) + 0 + sin(x)cos²(x)]}/[1 - cos²(x)] Since ([1 - cos(x)][1 + cos(x)] = [1 - cos²(x)]

= {sin(x) + sin(x)cos²(x)]}/sin²(x)  [since sin²(x) = 1 - cos²(x)]

Factorizing out sinx in the equation, we have that

= {sin(x)(1 + cos²(x)]}/sin²(x)

= (1 + cos²(x)]}/sin(x)

= cosec(x)(1 + cos²(x)]}  (since cosec(x) = 1/sin(x))

= R.H.S

Since L.H.S = R.H.S, we have that

sin(x)/[1 - cos(x)] - sin(x)cos(x)/[1 + cos(x)] = csc (x)(1 + cos² (x))

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Answer:

C. 50 cm²

Step-by-step explanation:

The volume of a triangular pyramid is calculated using the following formula:

Volume = (1/3) * Area of the base * Height

For Question:

length: 5cm

Breadth:5cm

Height : 6 cm

Now,

Volume = ⅓* Area of the base * Height

Volume = ⅓*length*breadth*height

Volume= ⅓*5*5*6=50 cm³

It implies that the collection of all ordered pairs (x, y) formed by taking an element from the set x and an element from the set y is a subset of the set containing all possible subsets of the Cartesian product of sets X and Y.

The expression "y^x ⊂ p(X x Y)" represents a subset relationship between two sets.

Let's break it down:

"y^x" represents the set of all possible ordered pairs (x, y) where x is an element of the set x and y is an element of the set y. This set represents the Cartesian product of the sets x and y.

"⊂" denotes a subset relationship. If we have two sets A and B, A ⊂ B means that every element in A is also an element of B. In other words, A is a subset of B.

"p(X x Y)" represents the power set of the Cartesian product of sets X and Y. The power set of a set is the set of all possible subsets of that set.

Therefore, "y^x ⊂ p(X x Y)" means that the set of all possible ordered pairs (x, y) where x is an element of the set x and y is an element of the set y is a subset of the power set of the Cartesian product of sets X and Y.

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The equation is written as z = 7 + (20 -4x²/2)

From the information given, we have that the equations as;

x²-2y=5   ( 1)

4y+z=7     (2)

From equation (1), make y the subject of formula, we have;

-2y= 5 - x²

Divide both sides by the coefficient of the variables, we have;

y = 5 - x²/-2

y = -5 + x²/2

Now, substitute the value of y in (2), we have;

4 (-5 + x²/2) + z = 7

expand the bracket

-20 + 4x²/2 + z = 7

collect the like terms, we have;

z = 7 + (20 -4x²/2)

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The two other positive angles of rotation that take Point A to Point B on the unit circle is (5π/6) and (5π/6) + 2π .

Given data ,

To find two other positive angles of rotation that take Point A to Point B, we need to consider the angle values that yield the same coordinates as (1, 0) after rotating counterclockwise.

The position of Point A is (1, 0) on the unit circle.

Now, let's find the coordinates of Point B after rotating (7π/6) radians counterclockwise.

To rotate counterclockwise by (7π/6) radians, we can subtract (7π/6) from the angle of Point A. So, the angle for Point B would be:

Angle of Point B = 0 - (7π/6) = - (7π/6)

Now , for positive angles of rotation, we can add multiples of 2π to the angle of Point B while keeping the same coordinates. Adding 2π to the angle gives us:

Angle of Point B = - (7π/6) + 2π = (5π/6)

Hence , two other positive angles of rotation that take Point A to Point B are (5π/6) and (5π/6) + 2π. Both of these angles yield the same coordinates as Point B, which is (1, 0) on the unit circle.

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Although the rectangles have the same base lengths, the heights and areas are not directly proportional in this case.

Let's denote the base length of both rectangles as 'b'. If one rectangle has a height that is twice the height of the other rectangle, we can denote the heights as 'h' and '2h', respectively.

The area of a rectangle is calculated by multiplying the base length by the height. Therefore, the area of the first rectangle with height 'h' would be A₁ = b * h, and the area of the second rectangle with height '2h' would be A₂ = b * (2h) = 2b * h.

Comparing the two areas, we have A₁ = b * h and A₂ = 2b * h. It is evident that the areas are not proportional because the area of the second rectangle is twice the area of the first rectangle.

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In the given triangle value of b is,

⇒ b = 16

We have to given that,

A triangle with three angles (b + 20), (b + 32) and 6b.

Now, WE know that;

Sum of all interior angles of triangle is 180 degree.

Hence., We can formulate;

⇒ (b + 20) + (b + 32) + 6b = 180

Solve for b,

⇒ 8b + 52 = 180

⇒ 8b = 180 - 52

⇒ 8b = 128

⇒ b = 128/8

⇒ b = 16

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Answer:

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

Given a right triangle with two legs known.

Find the missing angle using tangent function:

Substitute values to get:

The line of reflection that would make A'B'C'D' the image of ABCD is line 3

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

Rectangles ABCD and A'B'C'D'

Also, we can see that

Both rectangles are in opposite quadrants

This means that the line of reflection must be slant line in the adjacent quadrants

In this case, the line is line 3

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Answer:

2400 N (Newtons)

Step-by-step explanation:

The weight of an object can be calculated using the equation:

Weight = mass * gravitational acceleration

Given:

Mass of the bed (m) = 120 kg

Gravitational acceleration (g) = 20 m/s²

Using the weight equation:

Weight = mass * gravitational acceleration

Weight = 120 kg * 20 m/s²

Weight = 2400 kg·m/s²

The unit of the weight is kilogram-meter per second squared (kg·m/s²), which is equivalent to the unit of force called Newton (N).

Therefore, the weight of the bed is 2400 Newtons (N).

Answer:

carnival one
elevation = SWT
depression = RTW

deer one
elevation = ABC
depression = DCB

Step-by-step explanation:

Answer: B

Step-by-step explanation: 63.3 in^2

The equation of the axis of symmetry is x = -7.

Given is an equation of a parabola, y = -3x² - 42x - 159, we need to find the equation of the axis of the symmetry.

To find the equation of the axis of symmetry of a parabola in the form of y = ax² + bx + c, you can use the formula x = -b / (2a).

In this case, the given equation is y = -3x² - 42x - 159.

Comparing it to the general form, we have a = -3 and b = -42.

Applying the formula, we can calculate the x-coordinate of the vertex (the axis of symmetry):

x = -b / (2a)

x = -(-42) / (2(-3))

x = 42 / (-6)

x = -7

Therefore, the x-coordinate of the vertex is -7.

To find the equation of the axis of symmetry, we use the value of x in the form x = h, where (h, k) is the vertex.

Hence, the equation of the axis of symmetry is x = -7.

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Therefore, the missing expression in Step 1 is [-8 - 1] + (2 + 3).

In order to find the missing expression in Step 1, let's analyze the given steps and the final expression.

Step 1: ?

Step 2: -9 + 2 + 3

Step 3: -7 + 3

To find the missing expression in Step 1, we need to work backwards from Step 3 to Step 1.

In Step 3, the expression "-7 + 3" gives us a result of -4.

In Step 2, the expression "-9 + 2 + 3" gives us a result of -4.

So, the missing expression in Step 1 should also evaluate to -4 when performed correctly.

Let's check the available options:

[-10 + -1 + 2] + (2 + 3) = -11 + 2 + 5 = -4

[-8 - 1] + (2 + 3) = -9 + 5 = -4

[-10 + 1] + (2 + 3) = -9 + 5 = -4

[8 + 1] + (2 + 3) = 9 + 5 = 14

Out of the given options, only option 2, [-8 - 1] + (2 + 3), correctly evaluates to -4. Therefore, the missing expression in Step 1 is [-8 - 1] + (2 + 3).

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The x-intercepts of the graph of f(x) are x = 3/2 and x = 1,the Vertex of the graph of f(x) is (5/4, 3/8), and it is a minimum point, The vertex is at (5/4, 3/8). This is the minimum point of the graph.

Part A: To find the x-intercepts of the graph of f(x), we set f(x) equal to zero and solve for x.

2x^2 - 5x + 3 = 0

To factor this quadratic equation, we look for two numbers that multiply to give 3 (the coefficient of the constant term) and add up to -5 (the coefficient of the linear term). These numbers are -3 and -1.

2x^2 - 3x - 2x + 3 = 0

x(2x - 3) - 1(2x - 3) = 0

(2x - 3)(x - 1) = 0

Setting each factor equal to zero, we get:

2x - 3 = 0   -->   x = 3/2

x - 1 = 0   -->   x = 1

Therefore, the x-intercepts of the graph of f(x) are x = 3/2 and x = 1.

Part B: To determine whether the vertex of the graph of f(x) is a maximum or a minimum, we look at the coefficient of the x^2 term, which is positive (2 in this case). A positive coefficient indicates that the parabola opens upwards, so the vertex will be a minimum.

To find the coordinates of the vertex, we can use the formula x = -b/2a. In the equation f(x) = 2x^2 - 5x + 3, the coefficient of the x term is -5, and the coefficient of the x^2 term is 2.

x = -(-5) / (2*2) = 5/4

Substituting this value of x back into the equation, we can find the y-coordinate:

f(5/4) = 2(5/4)^2 - 5(5/4) + 3 = 25/8 - 25/4 + 3 = 3/8

Therefore, the vertex of the graph of f(x) is (5/4, 3/8), and it is a minimum point.

Part C: To graph f(x), we can use the information obtained in Part A and Part B.

- The x-intercepts are x = 3/2 and x = 1. These are the points where the graph intersects the x-axis.

- The vertex is at (5/4, 3/8). This is the minimum point of the graph.

We can plot these points on a coordinate plane and draw a smooth curve passing through the x-intercepts and the vertex. Since the coefficient of the x^2 term is positive, the parabola opens upwards, and the graph will be concave up.

Additionally, we can consider the symmetry of the graph. Since the coefficient of the linear term is -5, the line of symmetry is given by x = -(-5) / (2*2) = 5/4, which is the x-coordinate of the vertex. The graph will be symmetric with respect to this line.

By connecting the plotted points and sketching the curve smoothly, we can accurately graph the function f(x).

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Answer:

Therefore, the expression becomes:

7/1 - 2/6

Now, we need to find a common denominator for the fractions, which is 6:

7/1 - 2/6 = (76)/(16) - 2/6

= 42/6 - 2/6

= (42 - 2)/6

= 40/6

Finally, we can simplify the fraction:

40/6 = 20/3

So, the difference between 5 2/6 and -2 4/6 is 20/3.

Step-by-step explanation:

First, let's subtract the whole numbers: 5 - (-2) = 5 + 2 = 7.

Next, let's subtract the fractions: 2/6 - 4/6 = (2 - 4)/6 = -2/6.

Combining the whole number and fraction results, we have:

7 - 2/6

Now, to simplify this result further, we can express 7 as a fraction with a common denominator of 6:

7 = 7/1

A. A system of equations in standard form that can be solved to find the number of adults and children who attended the performance is:

x + y = 282

3x + 5y = 1046

B. The number of adults who attended the performance is 182 adults.

The number of children who attended the performance is 100 children.

In order to write a system of linear equations to describe this situation, we would assign variables to the number of adult tickets sold and number of children tickets sold, and then translate the word problem into an algebraic equation as follows:

Since 282 people attended the recent performance by Cinderella, a linear equation that models the situation is given by:

x + y = 282     ....equation 1.

Additionally, adult tickets sold for $5 while children tickets sold for $3 each with a total revenue was $1046, a linear equation that models the situation is given by:

3x + 5y = 1046   .......equation 2.

Part B.

By multiplying equation 1 by 3, we have:

3x + 3y = 846     .......equation 3.

By subtracting equation 3 from equation 2, we have:

2y = 200

y = 100 children.

For the x-value, we have:

x = 282 - y

x = 282 - 100

x = 182 adults.

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The correct equations are:

1. Two one-step equations:

[tex]\[3x + 2 = 11\]\[5y - 7 = 18\][/tex]

2. Two equations that contain fractions:

[tex]\[\frac{2}{3}x - \frac{1}{4} = \frac{5}{6}\]\[\frac{3}{5}y + \frac{2}{7} = \frac{1}{3}\][/tex]

3. One equation with distributive property:

[tex]\[2(4x - 3) = 10\][/tex]

4. One equation with decimals:

[tex]\[0.5x + 0.25 = 1.75\][/tex]

5. Real-world problem solved by an equation:

A bakery sells cakes for $[tex]15[/tex] each. Let's say the total cost of cakes sold in a day is $[tex]180[/tex]. We can use the equation [tex]\(15x = 180\)[/tex] to find the number of cakes sold, represented by the variable [tex]x[/tex]. Solving the equation, we find [tex](x = 12\)[/tex]), indicating that the bakery sold [tex]12[/tex] cakes that day.

Here's a basic explanation for the real-world problem:

Imagine there is a bakery that sells cakes for $[tex]15[/tex]each. We want to find out how many cakes the bakery sold in a day if the total revenue from cake sales is $[tex]180[/tex]. To solve this problem, we can use an equation. Let's represent the number of cakes sold as [tex]x[/tex].

The equation [tex]\(15x = 180\)[/tex] is used to express that the total cost of the cakes sold [tex](\$15\ per \ cake)[/tex] is equal to $[tex]180[/tex]. To solve the equation, we divide both sides by [tex]15[/tex] to isolate the variable [tex]x[/tex]. The equation simplifies to [tex]\(x = 12\),[/tex] which means that the bakery sold [tex]12[/tex] cakes that day.

By using the equation, we can determine the number of cakes sold based on the given information and calculate the desired result.

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The value of x in the given figure is 15.

In the given figure

The length of section of chords are given

We have to find the value of x

In order to find the value of x

Apply the intersecting chord theorem,

The intersecting chords theorem, often known as the chord theorem, is a basic geometry statement that defines a relationship between the four line segments formed by two intersecting chords within a circle. It asserts that the products of the line segment lengths on each chord are equal.

Therefore,

From figure we get,

⇒ 10/x = 8/12

⇒ x = 120/8

⇒  x = 15  

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