Find the maximum for the profit function,
P = 2x+10y
subject to the following constraints.
4x + 2y ≤ 5
-3x+y 2-2
X>0
(y ≥0
4x + 2y ≤ 5
-3x + y 2 -2
Round your answer to the nearest cent (hundredth).

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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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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Answer: z=3

Step-by-step explanation:

Answer:

A. 16.7%

B. 89.5%

Step-by-step explanation:

A) Inside the Square

length:4

The square has an area of 16 square units. The rectangle has an area of 96 square units. The probability of a point chosen randomly inside the rectangle being in the square is:

(area of square)/(area of rectangle)

= 16/96

= 0.16666666666666666

This is equal to 16.7%.

B) Outside the right angles Triangle

base:4 height 5

The triangle has an area of 10 square units. The rectangle has an area of 96 square units. The probability of a point chosen randomly inside the rectangle being outside the triangle is:

(area of rectangle - area of triangle)/(area of rectangle)

= (96 - 10)/96

= 0.8958333333

This is equal to 89.5%.

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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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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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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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 solution of the given equations is: (3,-1).

Here, we have,

from the given graph we get,

1st line passes through ( 0,2 ) and (2,0)

so, equation: y = -x+2 .....1

2nd line passes through ( 0,-4 ) and (4,0)

so, equation: y = x-4 .....2

now, from 1 and 2 we get,

-x + 2 = x -4

or, 2x = 6

or, x = 3

so, we have, y = -1

so, the point of intersection is (3, -1)

Hence, The solution of the given equations is: (3,-1).

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The amount of fabric required is 400.551 ft².

We have,

CB= 8 feet

CF= 13 feet

AM = 8 feet

Using Pythagoras

AC² = AM² + CM²

AC = √64+16 = √80 = 4√5 feet

Now, the formula for Triangular prism is

= (Sum of three sides of triangle face)l + base area

= (4√5 + 4√5 + 8)13 + 8 x 8

= 104√5 + 104 + 64

= 104√5 + 168

= 232.551 + 168

= 400.551 ft²

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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 degrees of freedom for a sample size of 27 is 26, the alpha or significance level for a 98% confidence level is 0.02, and the t-critical value can be found using the t-table corresponding to a 98% confidence level and 26 degrees of freedom.

To find the degrees of freedom, alpha (significance level), and the t-critical value using the t-table, we need to consider the given information:

n = 27: This represents the sample size, which is 27 in this case.

CL = 98%: CL stands for the confidence level.

It indicates the level of confidence we want to have in our interval estimate.

In this case, the confidence level is 98%.

Degrees of Freedom (df): For a t-distribution, the degrees of freedom depend on the sample size.

Since we have a sample size of 27, the degrees of freedom would be n - 1.

Therefore, the degrees of freedom would be 27 - 1 = 26.

Alpha (α): Alpha, or the significance level, represents the probability of making a Type I error, which is rejecting a true null hypothesis.

The alpha level is determined by subtracting the confidence level from 1. In this case, the alpha level would be 1 - 0.98 = 0.02.

T-Critical Value: To find the t-critical value corresponding to a specific confidence level and degrees of freedom, we can consult the t-table. Since we have a confidence level of 98% and degrees of freedom of 26, we need to find the value that corresponds to the area of 0.01 (half of 1 - 0.98) in the t-distribution table.

The t-critical value would be the value from the table that matches these criteria.

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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:

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

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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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: 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:

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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Answer:

Step-by-step explanation:

Given the following quadratic sequences, you want the value of x and the expression for the n-th term.

One way to determine x is to look at the differences between terms. The "second difference" is constant for a quadratic sequence, and the third difference is zero.

The quadratic equation for the n-th term can be found by solving for its coefficients. The three known values of the sequence can give rise to three linear equations in the three unknown coefficients. These can be solved by your favorite method. We use this approach in the following.

First differences are the differences between each term and the one before:

  {6, x-5, 35-x}

Second differences are the differences of these:

  {x -11, 40 -2x}

Third differences are zero:

  51 -3x = 0   ⇒   x = 17

The value of x is 17.

The expression for the n-th term of the sequence can be written as ...

  an = a·n² +b·n +c

We are given values of a1, a2, and a4. This lets us write 3 equations for a, b, and c. The solution of those is shown on the first line of the first attachment. (The second line shows the evaluation of this quadratic equation for n=3. It gives 17, which we already knew.)

  an = 3n² -3n -1

The last line of the first attachment shows us the expression for the third differences. The value of that is zero, so ...

  -3x +60 = 0   ⇒   x = 20

The value of x is 20.

As in the above problem, the matrix of equations for the quadratic coefficients can be reduced to give the coefficient values. That tells us the n-th term of this sequence is ...

  an = 3n² -4n +5

The last line in the second attachment tells us this expression for the n-th term properly computes the 3rd term (x), as above.

__

Additional comments

You can also use quadratic regression to find the coefficients of the formula for the quadratic sequence. This is shown in the 3rd attachment.

If you're trying to avoid using a calculator, you can write the equations out and solve them in an ad hoc way. In case you cannot tell, the equations for the coefficients of an = a·n² +b·n +c for the first problem are ...

You can also use the first values of the sequence (p), first difference (q), second difference (r) to write the quadratic:

  an = p +(n -1)(q +(n -2)/2(r))

For (p, q, r) = (-1, 6, 6), this is an = -1 +(n -1)(3n) . . . . . . for the first sequence.

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Answer:

402.12 or just 402

Option (D) D. Points Q and R is used to removed to make this relation a function.

To determine which two points could be removed to make the relation a function, we need to check if there are any repeated x-values (inputs) in the given set of points. In a function, each input should have a unique output.

Let's analyze the given options:

A. Points R and S: (R, 3), (S, 5)

B. Points Q and T: (Q, 2), (T, 5)

C. Points P and Q: (P, 1), (Q, 2)

D. Points Q and R: (Q, 2), (R, 3)

In this case, if we remove points Q and R, we eliminate the repeated x-value of 2, which ensures that each input has a unique output. Therefore, the answer is:

D. Points Q and R.

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The relationship between angles 8 and 7 is that they are supplementary. option B is correct.

Given that a quadrilateral, with three parallel lines, we need to find the relation between angles 8 and 7,

We know that the adjacent angles between the parallel lines are supplementary,

We know that,

Supplementary angles are a pair of angles that add up to 180 degrees. In other words, if you have two angles that are supplementary, the sum of their measures will always be 180 degrees.

So,

The relation between the angles is that they are Supplementary angles.

Hence the option B is correct.

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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:

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

The slope of the line that is perpendicular to the line y = 3x - 8 is -1/3.

The given line has an equation in slope-intercept form, which is y = 3x - 8. In this form, the coefficient of x represents the slope of the line.

Therefore, the slope of the given line is 3.

To find the slope of a line perpendicular to the given line, we need to take the negative reciprocal of the slope.

The negative reciprocal of 3 is -1/3.

Therefore, the slope of the line that is perpendicular to the line y = 3x - 8 is -1/3.

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Answer: There are 240 people inside the auditorium.

Step-by-step explanation:

30 x 8 = 240

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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Answer:

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

Given a right triangle with two legs known.

Find the missing angle using tangent function:

Substitute values to get: