QUESTION 1 The number of elements in sets A and B are shown in Figure 1.0. If n(A) = n(B) find: (a) x (b) n(A) (c) n(B) (d) n(AUB) 2x 4xx+5 Figure 1.0 10 marks​

Answer:

Step-by-step explanation:

The value of the integral is 6.

Integration is a mathematical operation that involves finding the integral of a function. The integral of a function is a measure of the area under the curve of the function in a given interval. The process of integration is the opposite of differentiation, and it involves finding a function whose derivative is the given function.

Integration is an important tool in many areas of mathematics, science, and engineering. It can be used to calculate quantities such as velocity, acceleration, displacement, and area, among others. It also has applications in fields such as physics, economics, and statistics.

There are different methods of integration, including substitution, integration by parts, partial fractions, and trigonometric substitution. The choice of method depends on the form of the function being integrated and the desired outcome.

We can use integration by parts to evaluate this integral:

Let u = x and dv = g'(x) dx, then du/dx = 1 and v = g(x).

Using the formula for integration by parts:

[tex]\int\limits^5_1[/tex] xg'(x) dx = [xg(x)]₁⁵ - [tex]\int\limits^5_1[/tex] g(x) dx

We can use the given information to find the value of each term in this expression:

[xg(x)]₁⁵ = 5g(5) - 1g(1) = 5(1) - 1(5) = 0

[tex]\int\limits^5_1[/tex] g(x) dx = -6 (given)

Substituting these values into the integration by parts formula, we get:

[tex]\int\limits^5_1[/tex] xg'(x) dx = 0 - (-6) = 6

Therefore, the value of the integral is 6.

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The relative frequency οf value οf 10 is 0.3.

The relative frequency is calculated as the prοduct οf the number οf times a given value οf the data appears in the set οf all οutcοmes divided by the tοtal number οf οutcοmes.

Here the given data set is 10, 14, 12, 16, 13, 14, 16, 10, 12, and 10.

Fοr value οf 10, the number οf times 10 οccured is 3 and tοtal number οf data set is 10.

Relative frequency,

=> Number οf times οf value οf 10/ Tοtal number οf data set

=> 3/10

=> 0.3.

Hence the relative frequency οf value οf 10 is 0.3.

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Answer:

Step-by-step explanation:

Answer:

a.   (-2, 4) only

==================================================

Explanation:

Let's check point A.

We have x = -2 and y = 4 pair up together.

Plug these coordinates into the first inequality.

x + 4y > 10

-2 + 4*4 > 10

-2 + 16 > 10

14 > 10

We get a true statement, so (-2,4) is in the shaded solution region of the inequality x+4y > 10.

Repeat for the other inequality.

3x - 2y < 12

3*(-2) - 2*4 < 12

-6 - 8 < 12

-14 < 12

This is also true.

The point (-2,4) is also in the shaded solution region for 3x-2y < 12

Since it's in both regions simultaneously, it satisfies the system of inequalities.

Therefore, (-2,4) is a solution to the system.

See the diagram below. Point A is in the shaded region.

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

Now we need to check point B.

We have x = 3 and y = 1. Let's plug those into the 1st inequality.

x + 4y > 10

3 + 4*1 > 10

3 + 4 > 10

7 > 10

That final statement is false, so (3,1) is not in the shaded region for x+4y > 10. We don't need to check the other region since this point needs to satisfy both inequalities to be considered a solution overall.

See the diagram below. Point B is outside the shaded region.

Answer:

n=√(4m+1) , 7

Step-by-step explanation:

According to statement

 ( n²-1 )÷m=4

∴n²-1=4m

⇒n²=4m+1

⇒n=√(4m+1)… … … … 1

If m=12

put the value of m in 1 equation

n=√(4×12+1)

  =√49=7

Answer:

The equation is tan(25.4°) = x/1,600, so

x = 1,600 tan(25.4°) = about 759.74 feet

The relationship between the variables in the table is a direct variation.

A function that models the relationship is y = 3x.

In Mathematics, a direct variation is also referred to as direct proportion and it can be modeled by using the following mathematical expression or function:

y = kx

Where:

Under direct variation, the value of x represent an independent variable while the value of y represents the dependent variable. Therefore, the constant of proportionality (variation) can be calculated as follows:

Constant of proportionality (k) = y/x

Constant of proportionality (k) = 9/3 = 27/9 = 39/13 = 60/20

Constant of proportionality (k) = 3.

Therefore, the required function is given by;

y = kx

y = 3x

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Answer:

(a) 512 ft² , (b) 720 ft²

Step-by-step explanation:

the area of a rectangle is calculated as

area = length × width

(a)

area of pool = 32 × 16 = 512 ft²

(b)

the area of the deck is calculated as

area of outer rectangle - area of pool

area of outer rectangle = 44 × 28 = 1232 ft²

area of deck = 1232 - 512 = 720 ft²

The steel industry should use 1.7 billion dollars worth of coal and 0.035 billion dollars worth of steel.

Amount is a term used to refer to the total of a certain quantity or sum. It is often used to describe the value of something, such as a financial transaction, or the size of a collection of objects. Amounts are usually expressed in units of measurement such as dollars, kilograms, or liters. Amounts can also be expressed as a percentage or fraction.

The coal industry should produce 8 billion dollars worth of output, the electric industry should produce 9 billion dollars worth of output, and the steel industry should produce 3.5 billion dollars worth of output. The coal industry should use 0.08 billion dollars worth of coal, 0.8 billion dollars worth of electricity, and 0.8 billion dollars worth of steel. The electric industry should use 0.06 billion dollars worth of coal, 0.9 billion dollars worth of electricity, and 1.2 billion dollars worth of steel. The steel industry should use 1.7 billion dollars worth of coal and 0.035 billion dollars worth of steel.

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Answer:

Therefore, Roberto will pay a total of $593,654 for his $340,000 house, including $253,654 in interest over the 30-year period.

Step-by-step explanation:

Roberto paid 20% down on a $340,000 house, which is 0.20 x $340,000 = $68,000.

So, the amount he financed is $340,000 - $68,000 = $272,000.

Roberto financed the $272,000 at 5% interest for 30 years, making monthly payments of $1460.15.

To find the total amount he will pay for the house, we need to calculate the total amount he will pay over the 30-year period.

First, we can calculate the total number of payments he will make over the 30 years by multiplying the number of years by the number of payments per year:

30 years x 12 months/year = 360 payments

The total amount Roberto will pay is the sum of all the monthly payments he makes over the 30 years:

Total amount = 360 x $1460.15 = $525,654

However, this includes both the principal amount of $272,000 and the interest he will pay over the 30 years. To find out how much he will pay in interest, we can subtract the principal from the total amount:

Total interest = $525,654 - $272,000 = $253,654

Therefore, Roberto will pay a total of $593,654 for his $340,000 house, including $253,654 in interest over the 30-year period.

The answer of this question is in every 5 hour 6 m water height increase in tank. So we have to learn about proportional.

Proportional is the right amount, size , degree or quantity compared with something else. We can simply say that it is comparison between one object or thing with another object or thing.

Their are two types of proportional properties:

a. Directly Proportional, In this when amount , size or degree of any object increases than the amount ,size, or degree of another object are also increases with same ratio.

b. Indirectly Proportional, In this when amount , size or degree of any object increases than the amount , size or degree of another object is decreases in same ratio.

Let Solve this question, in this the first property used.

[tex]\frac{1}{4} = \frac{3}{10} , we\ get\ 5 = 6[/tex]

[tex]\frac{1}{2} = \frac{3}{5}\ we\ get\ 5 = 6[/tex]

[tex]\frac{3}{4} =\frac{9}{10} \ we\ get\ 5 =6[/tex]

So, In every 5 hour 6 m water height increase in tank.

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The first 5 elements of set A are: {-1, -2, 1, -3, 2}.

What is the set of elements?

The objects in a set are called the elements (or members ) of the set; the elements are said to belong to the set (or to be in the set), and the set is said to contain the elements. Usually, the elements of a set are other mathematical objects, such as numbers, variables, or geometric points.

Set A is defined as:

A = {x | x = k(-1)ⁿ where k ∈ Z* and n ∈ N}

To find the first 5 elements of A, we can substitute different values of k and n:

For k = 1 and n = 1, x = 1(-1)¹ = -1

For k = 2 and n = 1, x = 2(-1)¹ = -2

For k = 1 and n = 2, x = 1(-1)² = 1

For k = 3 and n = 1, x = 3(-1)¹ = -3

For k = 2 and n = 2, x = 2(-1)² = 2

Therefore, the first 5 elements of set A are: {-1, -2, 1, -3, 2}.

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The ratio of the sides of the small box and the medium box is 1:3

The ratio of the area of the bases of the small box and the medium box is 1:9.

The ratio of the volumes of the small box and the medium box is 1:27.

The volume of a medium box is 5832 cubic inches.

What is a cube?

The solid shape of a cube has six square faces. Because every square face has the same side length, each face is the same size. A cube has 8 vertices and 12 edges. Each vertex is the intersection of three cube edges.

Given that a cube-shaped small packaging box has a volume of 216 cubic inches.

Assume the length of sides is a.

The volume of a cube is side³.

The volume of the small box is a³.

Thus,

a³ = 216

a³ = 6³

a = 6

The length of the small box is 6.

The area of the base of small box is 6×6  = 36 square inches.

The dimensions of a medium box are tripled in length, width, and height.

Thus the length of the medium box is 3×6 = 18 inches.

The area of the base of the medium box is 18×18 = 324 square inches.

The volume of cube medium box is 18×18×18 = 5832 cubic inches.

The ratio of the sides of small box and medium box is

6:18 = 1:3

The ratio of the area of bases of small box and the medium box is

36:324 = 1:9

The ratio of the volumes of small box and the medium box is

216:5834 = 1:27

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The peaches cost $2.50 per pound.

Conversion rate is the percentage of website visitors or potential customers who take a desired action, such as making a purchase, signing up for a newsletter, or filling out a form. In other words, it is the rate at which website visitors are converted into customers or subscribers.

To find the cost per pound of peaches, we need to divide the total cost by the weight in pounds.

Cost of 3.1 pounds of peaches = $7.75

Cost of 1 pound of peaches = $7.75 / 3.1

Cost of 1 pound of peaches = $2.50 (rounded to the nearest cent)

Therefore, the peaches cost $2.50 per pound.

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The value of XS in the perpendicular lines is 16 units.

The line VU is perpendicular to the line  XS. Perpendicular lines are defined as two lines that meet or intersect each other at right angles.

Therefore, perpendicular bisector is a line that divides a given line segment exactly into two halves forming 90 degrees angle at the intersection point.

Hence,

b + 5 = 3b - 1

b - 3b = -1 - 5

-2b = -6

divide both sides by -2

b = -6 / -2

b = 3

Therefore, let's find XS as follows:

XS = b + 5 + 3b - 1

XS = 4b + 4

XS = 4(3) + 4

XS = 12 + 4

XS = 16 units

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The vector components and direction of the river and motorboat are;

(a) 12·i

(b) 12·i + 12·√3·j

(c) 24·i + 12·√3·j

(d) 31.7 mi/h

A vector is a quantity or measurement that indicates a magnitude and a direction of the measurement.

The direction the river flows = East

The speed of the river = 12 mi/h

The direction the boater heads = 60° from the shore

Speed of the motor boar = 24 mi/h

The unit vector in the east direction = i

Unit vector in the north direction = j

(a) The velocity of the river, which is 12 mi/h east expressed in vector form is therefore;

(b) The velocity of the motorboat relative to the river, in vector component form is therefore;

Component of the velocity in the east direction = 24 × cos(60°)·i = 12·i

Component in the north direction = 24 × sin(60°)·j = 12·√3·j

The component form of the velocity of the boat is therefore;

(c) The true velocity of the boat is the vector sum of the velocity of the river and the velocity of the boat, therefore;

True velocity of the boat = 12·i + 12·i + 12·√3·j = 24·i + 12·√3·j

(d) The true speed is the magnitude of the velocity of the boat which can be found ads follows;

The (true) direction of the motorboat, θ, can be obtained from the arctangent of the ratio of the velocity component in the northern direction to the velocity in the eastern direction as follows;

The velocity component of the motorboat in the northern direction = 12·√3

The component of the velocity in the eastern direction = 24

Therefore;

tan(θ) = 12·√3/(24)

θ = arctan(12*√3/(24)) ≈ 40.9°, which is about East 40.9° North

The direction from the north, therefore is found as follows;

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The sides of the dilated triangle are parallel to the sides of the original triangle, but their lengths are multiplied by the scale factor k.

A triangle is a closed two-dimensional figure with three straight sides and three angles. It is one of the basic shapes in geometry and is used in many different areas of mathematics, science, and everyday life. A triangle is defined by its three vertices (or corners), which are the points where the sides of the triangle intersect. The three sides of the triangle are the line segments connecting the vertices. The three angles of the triangle are the angles formed by the intersection of two sides.

Here,

To dilate a triangle with a center at the origin, we need to multiply the coordinates of each vertex by the scale factor k. However, in this case, the center of dilation is not at the origin, but at the point Irgun. To perform the dilation, we can follow these steps:

Plot the original triangle with vertices (2,0), (4,-4), and (6,2).

Draw a line segment from the center of dilation Irgun to each vertex of the triangle.

Measure the distance from Irgun to each vertex, and multiply it by the scale factor k = 1.5 to find the new distance.

Use the new distances to construct the dilated triangle by drawing line segments from Irgun to points that are 1.5 times farther away from Irgun than the corresponding vertices of the original triangle.

The resulting dilation of the triangle with center at Irgun and scale factor k = 1.5 is shown below:

In this image, the original triangle is shown in black, and the dilated triangle is shown in blue. The center of dilation Irgun is marked with a red dot. Note that the new vertices of the dilated triangle are (0.5,0), (3,-6), and (7.5,3), which are 1.5 times farther away from Irgun than the corresponding vertices of the original triangle.

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The length would decrease at first to AB - BC then it would increase to the maximum of B + BC

From the diagram we have that two triangles are formed when we have AB = 3 cm and when we have BC = 1.5cm

This would happen when at point AC we have C moving in a counterclockwise direction around A

As AC goes towards A, then we would have AC decrease at first to AB - BC. Then it would increase. The maximum length would be AB + BC

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Answer:

It decreases

Step-by-step explanation:

Answering the question, we may state that So the equilibrium price of equation tea is $2.10 per unit.

A mathematical equation is a process that links two statements and indicates equality using the equals sign (=). A mathematical statement that proves the equality of two mathematical expressions is known as an equation in algebra. For instance, the equal sign separates the numbers 3x + 5 and 14 in the equation 3x + 5 = 14. A mathematical formula may be used to understand the link between the two sentences that are written on opposite sides of a letter. Frequently, the logo and the particular software are identical. like in 2x - 4 = 2, for example.  

To find the equilibrium price of tea, we need to set the quantity demanded equal to the quantity supplied:

[tex]34P + 53 = -25P + 177\\59P + 53 = 177\\59P = 124\\P = 2.10[/tex]

So the equilibrium price of tea is $2.10 per unit.

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Answer: 565.49ft^3/ 565.49ft cubed

Hopefully it is correct!

Answer:

First statement is true.

Second statement is false--there are some 3d shapes that unit cubes cannot be used to find the volume of. Examples are cylinders and cones.

Since opposite angles are equal, this means that GHJK is a parallelogram, because opposite sides are parallel.

Additionally, each pair of opposite sides of a parallelogram are congruent (of equal length), and the opposite angles of a parallelogram are also congruent (of equal measure). These properties make parallelograms useful in various fields such as engineering, architecture, and geometry.

There are different types of parallelograms, including rectangles, rhombuses, and squares, which have additional properties such as perpendicular sides, equal sides, or right angles. A rectangle is a type of parallelogram with four right angles, while a rhombus has four sides of equal length, and a square has both equal sides and right angles.

To prove that GHJK is a parallelogram, we need to show that opposite sides are parallel.

Since GK is parallel to HJ, we have:

∠LGK + ∠KHL = 180° (because GK // HJ)

But we are given that:

∠LGK = ∠LKL (given)

Therefore:

∠LKL + ∠KHL = 180°

This means that ∠KHL is a supplementary angle to ∠LKL.

Similarly, we can show that ∠KJH is a supplementary angle to ∠JKL:

∠JKL + ∠KJL = 180° (because GK // HJ)

But we are given that:

∠LGK = ∠LKL

Therefore:

∠LGK + ∠KJL = 180°

This means that ∠KJL is a supplementary angle to ∠LGK.

Now, we can use the fact that the opposite angles of a parallelogram are equal to show that GHJK is a parallelogram:

∠LKL = ∠JKL (alternate interior angles)

∠KHL = ∠KJH (alternate interior angles)

Since opposite angles are equal, this means that GHJK is a parallelogram, because opposite sides are parallel.

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To make a regular polygon with 96 sides we can take a circle as a starting point and divide it into 96 equal sides.

To create a regular polygon with 96 sides, we must take into account that all its sides must be equal, so we can take a circumference as a starting point. In this case, the circumference has 360°, so we must divide this into 96 parts and the result would be the distance between the edges of the polygon.

In this case, we must make a circle with a protractor and mark every 3.75°, when we have completed this procedure in the entire circumference we will have a regular polygon with 96 sides.

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Answer:

A'(4,-3) B'(6,11) C'(-7,-2)

Step-by-step explanation:

X-axis is (x,y) --> (x,-y)

A(4,3) --> (4,-3)

B(6,-11) --> (6,11)

C(-7,2) --> (-7,-2)

♡♡♡

The exponential growth of bacteria gives the growth percentage to be 4.05%.

What is an exponential function?

The formula for an exponential function is [tex]f(x) = a^x[/tex], where x is a variable and a is a constant that serves as the function's base and must be bigger than 0.

Let N(t) be the number of bacteria in the population at time t.

According to the continuous exponential growth model, we have -

[tex]N(t) = N0 \times e^{(rt)}[/tex]

where N0 is the initial number of bacteria, r is the hourly growth rate parameter, and t is the time elapsed in hours.

We know that a sample of 1800 bacteria reached the size of 2204 bacteria in five hours, so we can use this information to find the hourly growth rate parameter.

First, we can set up a ratio of the final size to the initial size -

[tex]\frac{2204}{1800} = e^{5r}[/tex]

Simplifying, we get -

[tex]e^{5r} = 1.2244[/tex]

Taking the natural logarithm of both sides, we get -

5r = ln(1.2244)

Solving for r, we get -

r = ln(1.2244) / 5

On further simplifying, we get -

r = 0.20245 / 5

r ≈ 0.04049

To express the growth rate as a percentage, we can multiply by 100 -

r percentage = 100 × r ≈ 4.049

Rounding to the nearest hundredth, we get -

r percentage ≈ 4.05%

Therefore, the hourly growth rate parameter is approximately 4.05%.

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From the given scatter plot,  point L is an outlier. The association is negative linear association.  The line is not the best fit accurate. Graph shows a negative linear association. As the number of weeks increases, the number of tickets sold decreases

What is scatter plot?

A scatter plot is a type of data visualization that displays the relationship between two numerical variables. It is a graph with points plotted on it that represent the values of two different variables.

1. In a scatter plot, an outlier is a data point that lies far away from the other data points in the plot, and is not in line with the overall pattern or trend that the other data points display.

2. The scatter plot for this data is likely to show a negative linear association between the number of people in a group and the zoo ticket price per person. As the number of people in a group increases, the zoo ticket price per person decreases.

The rate of change of Zoo Ticket Price with respect to the the number of persons:

[tex]m=14.50-1500/2-1=-0.5[/tex]

3. No because the line should touch every point.  When we draw a line passing through the origin, it passes through the data points (2,3), (4,3), and (4,5), but it does not pass through the other data points (6,5), (6,6), and (8,7).

This indicates that the line of best fit may not accurately represent the overall trend of the data.

4. The given data points are (1,5), (2,4), (3,3), (4,2), (5,1), and (6,0). When we plot these points on a graph with x-axis representing variable x and y-axis representing variable y, we get a downward sloping straight line passing through all the points.

This pattern suggests a negative linear association between the variables x and y. As the value of x increases, the value of y decreases at a constant rate.

Therefore, the graph shows a negative linear association between x and y.

5. The answer is option (a) "As the number of weeks increases, the number of tickets sold decreases".

As per the given data points and the line passing through (1,100) and (6,35), we can observe that the number of tickets sold decreases as the number of weeks a movie has been in theaters increases. Therefore, the statement "As the number of weeks increases, the number of tickets sold decreases" best describes the relationship between the number of weeks a movie has been in theaters and the number of tickets sold.

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Answer:

the charge for opening up the account was $155.

Step-by-step explanation:

To find the charge for opening up the account, we need to subtract the interest from the total amount borrowed.

The total amount repaid at the end of the loan period is the sum of the amount borrowed and the interest charged. In this case, the total amount repaid is:

$2,500 + $155 = $2,655

The interest charged on the loan is the difference between the total amount repaid and the amount borrowed. Therefore, the amount charged for opening up the account is:

$2,655 - $2,500 = $155

So the charge for opening up the account was $155.

The surface area of the shape which was formed when four identical cubes were joined, is 882 yd ².

The surface area of a cube can be found by the formula :

= Number of faces x Length of a side ²

When four identical cubes were joined, this leads to the creation of 18 faces which means that the surface area of the cube would then be :

= 18 x 7 x 7

= 18 x 49

= 882 yd ²

In conclusion, the area is 882 yd ².

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The dimensions of the rectangle are, length = 73 cm and width = 17 cm.

Finding the values of the variables employed in a system of equations entails solving the set of equations. By keeping the equations balanced on both sides, we compute the values of the unknown variables. Finding the value of the variable that makes the condition of all the provided equations true is the primary goal when solving an equation system. A given system of equations may have a variety of solutions.

Let us suppose the width of the rectangle = w.

Given that, length is 5 more than four times its width.

Thus,

l = 5 + 4w

Now, the perimeter of the rectangle is given as:

P = 2(l + w)

Substitute the value of P = 180:

180 = 2l + 2w

The system of equations is:

l = 5 + 4w

180 = 2l + 2w

Substitute the value of l from equation 1 into equation 2:

180 = 2(5 + 4w) + 2w

180 = 10 + 8w + 2w

180 - 10 = 10w

170 = 10w

w = 17

Substitute the value of w = 17 in the first equation:

l = 5 + 4w

l = 5 + 4(17)

l = 73

Hence, the dimensions of the rectangle are, length = 73 cm and width = 17 cm.

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