Prove the following trigonometric identities
csc x - sin x - cosx cos x

The two equations which can be used to form a system of equations for the scenario are X + Y = 12 and 3.50X + 4.50Y = 50

To solve this problem, we need to form a system of equations. Let X be the amount of chocolate and Y be the amount of almonds. The first equation we can form is based on the total amount of snack mix that Penny needs, which is 12 ounces:

X + Y = 12

The second equation we can form is based on the cost of the ingredients. We know that chocolate costs $3.50 per ounce and almonds cost $4.50 per ounce. If X is the amount of chocolate and Y is the amount of almonds, then the total cost of the snack mix will be:

3.50X + 4.50Y = 50

This equation represents the fact that Penny has $50 to spend on the snack mix. Now we have a system of two equations that we can use to solve for X and Y. We can use substitution or elimination to solve the system and find the values of X and Y that satisfy both equations.

Once we have those values, we can check that they add up to 12 and that the total cost is $50. This system of equations allows us to calculate the amount of chocolate and almonds Penny needs to make the snack mix within her budget.

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Answer: x intercepts = (-1.5,0) and (-1,0)

Step-by-step explanation: Graphed it in desmos :)

The equation of the line passing through the given points is y = 3x-1.

Given that is a line passing through two points (0, 2) and (-1, -1) we need to find the equation of the line using them,

We know that the equation of a line passing through points (x₁, y₁) and (x₂, y₂) is =

y-y₁ = y₂-y₁ / x₂-x₁ (x-x₁)

Here (x₁, y₁) and (x₂, y₂) are (0, 2) and (-1, -1),

Therefore, the required equation is =

y+1 = -1-2/-1 (x-0)

y+1 = 3x

y = 3x-1

Hence, the equation of the line passing through the given points is y = 3x-1.

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We can read 91 books in a year if you increase your reading speed so that each page takes you 30 seconds less than it did before.

Now If each page now takes 30 seconds less to read than before,

then you will save 30*200 = 6000 seconds (or 100 minutes) on each book.

So, the time it will take you to read a 200-page book will be

20 minutes - 100 minutes = -80 minutes,

which means you will finish a 200-page book in 80 minutes (or 1 hour and 20 minutes).

In a year, there are 365 days. If you read for 20 minutes per day, then you will read for a total of

365 * 20 = 7300 minutes (or 121.67 hours) in a year.

Since you can finish a 200-page book in 80 minutes, you can read 7300/80 = 91.25 books in a year.

However, since you cannot read a fraction of a book, the maximum number of 200-page books you can read in a year is 91 books.

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Complete question is :

If you increase your reading speed so that each page takes you 30 seconds less than it did before,

and you begin reading 20 minutes per day,

How many 200 page books can you now read in a year?

Answer:

Here we try the method of trial and error to find out if the equations have a common answer as zero

Step-by-step explanation:

Now,

(a) x²-4=0

(b) x²=-4

(c) 3x²+12=0

(d) 4x²=16

(e) 2(x-2)2=0

Here if we check the first equation i.e x²-4=0

Equating - 2²-4=4-4

=0

2²-4= 4-4

=0

So option (a) is true

Now x²=-4

Substituting, - 2²=4 & 2²=4

So here we get 4≠-4

Therefore, (b) is not true

Now 3x²+12=0

3(-2)²+12= 3(4)+12

=12+12= 24

3(2)²+12= 12+12

= 24

4x²=16

Substituting, 4(-2)²= 4(4)= 16

4(2)²= 4(4)= 16

So option (d) is also true

Now, 2(x-2)2=0

Substituting, 2(-2-2)2= 2(-4)2

4(-4)= - 16

2(2-2)2= 2(0)2

=4(0)=0

Here when we put x=-2, we get - 16

when we put x=2, we get 0

So the following equation is true only for x=2 and not x=-2

I hope this helps ;)

The mean absolute deviation (MAD) of the data in the pictograph is equals to the one basketball.

We have a data in the pictograph. In mathematics, a pictograph is a pictorial representation of data using images, icons. It is also known as a pictogram. We have a pictograph, in which the vertical axis is labeled Baskets made and the horizontal axis is labeled Student. Here, one basketball picture equals two baskets. Mean absolute deviation (MAD) is a statistical measure of the average absolute distance between each data value and the mean of a data set. It is a parameter or statistic that measures the spread, or variation, in data.

Mean is defined as the sum of data values divided by number of values.

Sum of data values = 4 + 4×2 + 3×2 + 5×2

= 28

So, mean = 28/4 = 7

Now, | 4 - 7| + |8 - 7| + |6 -7 | + | 10 - 7|

= 3 + 1 + 1 + 3 = 8

So,, mean absolute deviation (MAD) of the data = 8/7 = 1.1 ~ 1. Hence, required value is 1.

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Complete question :

Find the mean absolute deviation (MAD) of the data in the pictograph below. Baskets

The key says one basketball picture equals two baskets. The key says one basketball picture equals two baskets. A picture graph labeled Baskets each student made. The vertical axis is labeled Baskets made. The horizontal axis is labeled Student. The names from left to right on the horizontal axis are Reynaldo, Marcelle, Allie, and Fernando. There are two basketball pictures above Reynaldo. There are four basketball pictures above Marcelle. There are three basketball pictures above Allie. There are five basketball pictures above Fernando

I believe credit cards tend to be the entry point for establishing credit for so many consumers because they provide an easy and accessible way for individuals to begin building their credit history. Credit card companies report to credit bureaus on a regular basis, which helps establish a credit score and credit history.

Additionally, credit cards offer a convenient way for individuals to make purchases and build their credit at the same time. However, it is important for individuals to use their credit cards responsibly and make timely payments in order to maintain good credit standing.

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1. For the sequence : 35, 29, 23, 17, ?; the next term is 11

2. For the sequence : 1, 2, 5, 10, ?; the next term is 17

From the question, we are to calculate the next term in each of the given sequence

From the given sequence,

35, 29, 23, 17, ?.

To determine the next term, we will determine the common difference

Common difference = Second term - First term

Common difference = 29 - 35

Common difference = -6

Thus,

To determine the next term, we will add the common difference to the last term

That is,

17 + - 6 = 17 - 6

= 11

The next term is 11

For the sequence 1, 2, 5, 10, ?.

Common difference = successive odd numbers

To get the second term, we will add to the first term the first natural odd number

To get the third term, we will add to the second term the second natural odd number

And so on.

In the given sequence, we are to determine the 5th term

Thus,

We will add to the fourth term, the fourth natural odd number

That is,

10 + 7 = 17

Hence, the next term is 17

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The value of coordinate M and K are,

M = (25, 0)

K = (2b + 4c - 25, 2d)

Given that;

In KLMO,

OM = 25

L = (2b + 4c , 4d)

Hence, We can formulate;

The value of coordinate of M is,

M = (25, 0)

Since, M lies on x - axis.

Let the coordinate of K is,

K = (x, y)

Hence, Midpoint of LO is same as midpoint of KM.

Midpoint of LO is,

(0 + 2b + 4c / 2, 4d/2)

(b + 2c, 2d)

Midpoint of KM is

(x + 25/2, y + 0/2)

(x + 25/2 , y/2)

By comparing,

x + 25/2 = b + 2c

x + 25 = 2b + 4c

x = 2b + 4c - 25

y/2 = 2d

y = 2d

Thus, the coordinate of K is,

K = (2b + 4c - 25, 2d)

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Answer:

83 and 99 The sequence is going in +16 so the answer is 83 and 99 sorry if I am a bad explainer I’m new to this app :(

Step-by-step explanation:

The area of the sidewalk in square meters is 69π.

To find the area of the sidewalk, we need to subtract the area of the flower bed from the area of the outer circle formed by the sidewalk.

First, we need to find the area of the flower bed. We know that the diameter of the flower bed is 20 m, so the radius is half of that, which is 10 m. We can use the formula for the area of a circle, which is A = πr^2, where A is the area and r is the radius.

Area of flower bed = π(10m)^2 = 100π square meters

Next, we need to find the area of the outer circle formed by the sidewalk. Since the sidewalk is 3 m wide, the radius of the outer circle will be 10 + 3 = 13 m (10 m for the flower bed radius plus 3 m for the width of the sidewalk).

Area of outer circle = π(13m)^2 = 169π square meters

Finally, we can find the area of the sidewalk by subtracting the area of the flower bed from the area of the outer circle:

Area of sidewalk = Area of outer circle - Area of flower bed
Area of sidewalk = (169π) - (100π)
Area of sidewalk = 69π square meters

Therefore, the area of the sidewalk in square meters is 69π, or approximately 216.6 square meters (if we use 3.14 for π).

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Answer:

Step-by-step explanation:

3x+y = -6

y=-3x-6

The Y-intercept is -6, therefore there is one point (0,-6)

Now go 3 down and one right, so there is your second point (1,-9)

Answer:

(-13)^3

Step-by-step explanation:

Exponents can be used for repeated multiplication.

In this case, the number "negative 13" is repeated several times, all connected with multiplication.

There are a total of three "negative 13"s being multiplied together ("negative 13" appears three times on the page).

To rewrite using exponents, we would write one of the following:

(-13)^3

[tex](-13)^3[/tex]

The location of the point Kristin will add to represent the 13 bottles of water sold at 39 degrees Fahrenheit is the bottom left of the scatter plot.

A scatter plot represents the relationship between two variables. In this case, the temperature (independent variable) is plotted along the x-axis, while the number of bottles of water sold (dependent variable) is plotted along the y-axis. As the temperature increases, it is expected that more bottles of water would be sold.

The bottom left area of the scatter plot is where lower values of both temperature and the number of bottles sold would be found. Since 39 degrees Fahrenheit is relatively low and 13 bottles of water is a lower quantity, the point representing this data will be in the bottom left quadrant of the scatter plot.

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Complete question:

Kristin made a scatter plot to represent the relationship between the temperature and the number of bottles of water sold at her concession stand at a soccer tournament. Which is a description of the location of the point Kristin will add to represent the 13 bottles of water that were sold when the temperature was 39 degrees Fahrenheit? Select one:

O  Top left of the scatter plot

O  Bottom right of the scatter plot

O  Top right of the scatter plot

O  Bottom left of the scatter plotâ

First, using the Pythagorean theorem, we get the hypotenuse = 12.

sin = opposite/hypotenuse = [tex]\frac{6\sqrt{3} }{12} = \frac{\sqrt{3} }{2}[/tex]

cos = adjacent/hypotenuse = [tex]\frac{6}{12} =\frac{1}{2}[/tex]

tan = opposite/adjacent = [tex]\frac{6\sqrt{3} }{6} = \sqrt{3}[/tex]

csc = hypotenuse/opposite = [tex]\frac{12}{6\sqrt{3} } =\frac{2}{\sqrt{3} }[/tex]

sec = hypotenuse/adjacent = [tex]\frac{12}{6} =2[/tex]

cot = adjacent/opposite = [tex]\frac{6}{6\sqrt{3} } = \frac{1}{\sqrt{3} }[/tex]

2.25 liters of solution in all is there in the test tubes. So, the correct option is 2.25 l.

The total volume of solution in the 15 test tubes can be calculated by multiplying the volume of one test tube by the number of test tubes:

Total volume = 15 test tubes × 150 milliliters/test tube

Total volume = 2,250 milliliters

However, the question asks for the answer in liters, so we need to convert milliliters to liters by dividing by 1,000:

Total volume = 2,250 milliliters ÷ 1,000

Total volume = 2.25 liters

Therefore, there are 2.25 liters of solution in all the test tubes.

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The percentage is 9.0% of the senior class are either members of the Key Club, enrolled in AP Physics, or both.

We need to find the percentage of seniors who are either members of the Key Club, enrolled in AP Physics, or both. We can use the formula:

Total percentage = Key Club percentage + AP Physics percentage - Both percentage

Step 1: Identify the given percentages
Key Club percentage = 2.3%
AP Physics percentage = 8.6%
Both percentage = 1.9%

Step 2: Apply the formula
Total percentage = 2.3% + 8.6% - 1.9%

Step 3: Calculate the result
Total percentage = 9.0%

So, 9.0% of the senior class are either members of the Key Club, enrolled in AP Physics, or both.

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The percentage of the family's income that is spent on food is 25%.

Firstly, noting down the family's monthly spend on food ($550) and their total monthly income ($2,200).

Next, dividing the amount spent on food by the total income to find the ratio of the spend to income: $550 / $2,200.

Now, calculating this division: 550 ÷ 2,200 = 0.25.

Finally, finding the percentage, multiply the ratio (0.25) by 100: 0.25 x 100 = 25%.

So, the family spends 25% of their monthly income on food.

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The system of equations to describe the earnings by Avery and Janelle would be:

Avery's earnings: y = 120x + 300

Janelle's earnings: y = 24x + 60s

The problem provides two scenarios with different methods for earning money. Avery earns a fixed amount of $120 each week, in addition to a one-time bonus of $300. To represent this situation as an equation, we can use the formula:

y = 120x + 300

where y is Avery's total earnings, x is the number of weeks she works, and 300 is the one-time bonus she receives.

For Janelle, her earnings consist of a fixed weekly rate of $24 plus a variable amount based on the number of students she teaches.

We can represent Janelle's earnings as an equation using the formula:

y = 24x + 60s

where y is Janelle's total earnings, x is the number of weeks she works, and s is the number of students she teaches.

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The  correct way to solve this equation for m in terms of y is  m = b - y/x

It is important to note that subject of formula is the variable that is made to stand alone in an equation.

It is described as the variable that is being worked out in an equation.

The subject of formula in an equation is worked to stand alone on on end of the equality sign.

Example of equations

x = y - 2

The variable 'x' is the subject of formula

From the information given, we have that;

y= mx+b

collect the terms

mx = b - y

Divide by the coefficient

m = b - y/x

The equation for m is m = b - y/x

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The complete question:

The equation y=mx+b is used to express the equation of a line. Which solution is a correct way to solve this equation for m in terms of y?

The height of the tower is: 274.7m

To find the height of the tower, we will use the tangent trigonometric function.

The tangent function relates the angle of elevation to the ratio of the opposite side (height of the tower) to the adjacent side (distance from the observer to the tower).

In this case, the angle of elevation is 70°, and the distance from the observer to the tower is 100 meters.

The formula we will use is:

tan(θ) = opposite / adjacent

tan(70°) = height / 100m

To calculate the height, we will rearrange the formula:

height = 100m * tan(70°)

Using a calculator, we find that tan(70°) ≈ 2.747.

Therefore, the height of the tower is: 274.7m

height ≈ 100m * 2.747 ≈ 274.7m

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The question in English is:

A person observes a tower from a distance of 100m with an elevation angle of 70, with which trigonometric function would you obtain the height of the tower? Calculate the height of the tower

The solution of the exponents is shown below.

Exponents are mathematical shorthand for multiplying a number by itself a certain number

We have that;

5^n = 1

5^n = 5^0

n = 0

2) 2^-7/2^n = 2^2

2^-7 - n = 2^2

-7 - n = 2

-n = 2 + 7

n = -9

3) 6^5 * 6^n = 6^1

6^ 5 + n = 6^1

5 + n = 1

n = 1 - 5

n = -4

4) (8^n)^7 = 8^21

8^7n = 8^21

7n = 21

n = 3

5) 4^n = (1/4)

4^n = 4^-1

n = -1

In each of the cases, we have applied the laws of the exponents as we know them.

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The equation of the forms are matched as;

2⁻⁷/2ⁿ = 2², n = -9

6⁵ * 6ⁿ = 6. n = -4

(8ⁿ)⁷ = 8²¹, n = 3

4ⁿ = 1/4 , n = -1

Index forms are simply described as mathematical forms that are used to represent numbers of variables that are too large or small.

To multiply index forms, you need to add the exponents of the same bases.

To divide index forms, you need to subtract the exponents of the same bases.

From the information given, we have that;

2⁻⁷/2ⁿ = 2²

cross multiply the values

2⁻⁷ = 2²⁺ⁿ

Then,

-7 = 2 + n

n = -9

6⁵ * 6ⁿ = 6

take the exponents

5 + n= 1

n =- 4

(8ⁿ)⁷ = 8²¹

We have;

7n = 21

Make 'n' the subject

n = 3

4ⁿ = 1/4

4ⁿ = 4⁻¹

n =-1

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The surface area of the rectangular prism is 4960 cm².

The rectangular prism can be divided into six rectangular faces, with opposite faces having the same area. To find the surface area, we need to calculate the area of each face and add them up.

The net shows three rectangles with dimensions of 62 cm x 21 cm, 62 cm x 16 cm, and 21 cm x 16 cm.

Therefore, the surface area of the rectangular prism is:

Area of the first rectangle = 62 cm x 21 cm = 1302 cm²

Area of the second rectangle = 62 cm x 16 cm = 992 cm²

Area of the third rectangle = 21 cm x 16 cm = 336 cm²

Total surface area = 2(Area of first rectangle) + 2(Area of second rectangle) + 2(Area of third rectangle)

= 2(1302) + 2(992) + 2(336)

= 2604 + 1984 + 672

= 4960 cm²

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The value of 'a' remains the same, 'h' is equal to -b/(2a), and 'h' and 'k' cannot both equal zero.

When rewriting a quadratic expression of the form y = ax^2 + bx + c into the vertex form y = a(x - h)^2 + k, the following must be true:

1. The value of 'a' remains the same in both expressions, as it represents the parabola's vertical stretch or compression.

2. 'h' is equal to -b/(2a), which is derived from completing the square to transform the standard form into the vertex form.

3. 'k' and 'c' do not necessarily have the same value. 'k' is the value of the quadratic function when 'x' equals 'h', which can be found by substituting 'h' back into the original equation and solving for 'y'.

4. 'h' and 'k' cannot both equal zero, unless the vertex of the parabola is at the origin (0,0).

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The function has a horizontal asymptote at y = 3. Other types of discontinuities can also occur in rational functions

A polynomial is a mathematical expression that consists of variables and coefficients, which are combined using arithmetic operations such as addition, subtraction, multiplication, and non-negative integer exponents.

A rational function can have a discontinuity at any point where the denominator of the function becomes zero since division by zero is undefined. These points are called "vertical asymptotes."

For example, consider the rational function f(x) = (x² - 1) / (x - 1). The denominator becomes zero when x = 1, which causes a vertical asymptote at x = 1. At x = 1, the function approaches positive infinity from the left-hand side and negative infinity from the right-hand side. This creates a "hole" or a "removable discontinuity" in the graph of the function.

Another type of discontinuity that can occur in a rational function is a "horizontal asymptote." This occurs when the degree of the numerator is less than the degree of the denominator. In this case, the function approaches a horizontal line (the horizontal asymptote) as x approaches infinity or negative infinity.

For example, consider the rational function f(x) = (3x² - 2x + 1) / (x² + 1). As x approaches infinity or negative infinity, the function approaches the horizontal line y = 3.

Therefore, the function has a horizontal asymptote at y = 3.

Other types of discontinuities can also occur in rational functions, such as "slant asymptotes" or "oscillating behavior," but these are less common and typically require more advanced techniques to identify.

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The formula for calculating the amount of money in an account with continuous compounding is:

[tex]A = Pe^{(rt)}[/tex]

where A is the amount of money in the account, P is the principal (initial deposit), e is the mathematical constant e (approximately equal to 2.71828), r is the interest rate (expressed as a decimal), and t is the time (in years).

Plugging in the given values, we get:

A =[tex]40000 * e^{(0.025 * 8)[/tex]

Using a calculator, we find that [tex]e^{(0.025 * 8)[/tex] is approximately 1.2214, so:

A = 40000 * 1.2214 = $48,856.12

Therefore, the amount of money in the account after eight years with continuous compounding is $48,856.12.

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There is a positive correlation and as such as time passes, spending increases.

The descriptions of the graph from the question are given as

From the above statements, we can make the following summary

As the year increase, the spending also increase

The above summary is about the correlation of the graph

And it means that there is a positive correlation and as such as time passes, spending increases.

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The work done by the force field F along the semicircle and the line segment is 32/3.

The work done by a force field F along a curve C from point A to point B is given by the line integral:

W = ∫ F dot dr

where dot represents the dot product and dr is the differential displacement vector along the curve C.

Let's divide the curve C into two parts: the semicircle from P to Q, denoted by C1, and the line segment from Q to P, denoted by C2.

For C1, the curve can be parameterized as x = 2cos(t) and y = 2sin(t) for t in [0, pi]. The differential displacement vector is then given by:

dr = (-2sin(t) dt)i + (2cos(t) dt)j

The force field F(x,y) = x^2i - ryj, so we have:

F(x,y) = (2cos^2(t))i - (2rsin(t))j

The dot product F dot dr is then:

F dot dr = (2cos^2(t))(-2sin(t) dt) + (2rsin(t))(2cos(t) dt)

= -4cos^2(t)sin(t) dt + 4rcos(t)sin(t) dt

= 4sin(t)cos(t)(r - cos(t)) dt

Therefore, the work done along C1 is:

W1 = ∫ C1 F dot dr

= ∫[0, pi] 4sin(t)cos(t)(r - cos(t)) dt

This integral can be evaluated using the substitution u = cos(t), du = -sin(t) dt:

W1 = -∫[1, -1] 4u(r - u) du

= 4r∫[1, -1] u du - 4∫[1, -1] u^2 du

= 0

Hence, the work done along C1 is 0.

For C2, the curve is simply the line segment from Q(-2,0) to P(2,0), which is parallel to the x-axis. Therefore, the differential displacement vector is given by:

dr = dx i

where i is the unit vector in the x-direction. The force field is the same as before, F(x,y) = x^2i - ryj. Along C2, y = 0, so the force field reduces to:

F(x,y) = x^2i

The dot product F dot dr is then:

F dot dr = x^2 dx

Therefore, the work done along C2 is:

W2 = ∫ C2 F dot dr

= ∫[-2, 2] x^2 dx

= 32/3

Hence, the work done along C2 is 32/3.

The total work done along the curve C is the sum of the work done along C1 and C2:

W = W1 + W2 = 0 + 32/3 = 32/3

Therefore, the work done by the force field F along the semicircle and the line segment is 32/3.

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