given constants $a,$ $b,$ and $c,$ let $\alpha$ and $\beta$ be solutions to the equations \[a \cos \theta b \sin \theta

Answer:

Area_shaded

= 340.9 cm^2

Step-by-step explanation:

Find the area of the circle, then subtract the area of the triangle.

The 25cm side of the triangle is a hypotenuse of the right triangle. It is also the diameter of the circle.

The radius of the circle is 1/2 the diameter. So the radius is 1/2•25, or 12.5cm

We need the radius to find the area of the circle.

Area_circle

= pi•r^2

= pi•12.5^2

= 490.9 cm^2

The two legs of the triangle can serve as our base and height to find the area of the triangle.

Area_triangle

= 1/2b•h

= 1/2(20)(15)

= 150 cm^2

Subtract.

490.9 - 150

= 340.9cm^2

The area of the shaded region is 340.9cm^2.

The length of the second side, x, is 12.

What is Pythagoras' theorem?

In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs of the triangle). This theorem is usually represented by the equation: a^2 + b^2 = c^2, where c is the length of the hypotenuse, and a and b are the lengths of the legs.

In this case, the first side has a length of 9, the second side has a length of x, and the third side has a length of 15, which is the hypotenuse. We can use the Pythagorean theorem to solve for x:

x^2 + 9^2 = 15^2

x^2 = 15^2 - 9^2

x^2 = 225 - 81

x^2 = 144

x =√144

x = 12

Hence, the length of the second side, x, is 12.

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If an open box is to be constructed from a square piece of sheet metal by removing a square of side 3 feet from each corner and turning up the edges. the dimensions of the sheet​ metal is 16 feet.

Let x represent the dimension of the sheet metal

So,

Volume =3×(x-2×3)×(x-2×3) = 300

Volume =3×(x-6)×(x-6) = 300

(x -6)² = 300/3 = 100

x-6 = √100 =10

x = 10 +6

x = 16 feet

Therefore the dimensions of the sheet​ metal is 16 feet

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Expressing the binomial coefficients in terms of factorials and simplifying algebraically. we have proved that  

[tex]$\left(\begin{array}{l}n \\ r\end{array}\right) = \dfrac{n-r+1}{r} \cdot \left(\begin{array}{c}n \\ r-1\end{array}\right)[/tex]

The binomial coefficient [tex]\left(\begin{array}{l}n \\ k\end{array}\right)[/tex] is the number of ways of picking k unordered outcomes from n possibilities, also known as a combination or combinatorial number. The symbols [tex]_nC_k[/tex] and [tex]\left(\begin{array}{l}n \\ k\end{array}\right)[/tex] are used to denote a binomial coefficient, and are sometimes read as "n choose k."

[tex]$$In general, $\left(\begin{array}{l}n \\ r\end{array}\right) \neq\left(\begin{array}{c}n \\ r-1\end{array}\right)$To solve (a), simply expand LHS.$$\left(\begin{array}{l}n \\r\end{array}\right)=\frac{n !}{r ! \cdot(n-r) !}=\frac{n !}{r \cdot(r-1) ! \cdot(n-r) !}$$Multiply numerator and denominator by $(n-r+1)$, then,$$\left(\begin{array}{l}n \\r\end{array}\right)=\frac{n-r+1}{r} \cdot \frac{n !}{(r-1) ! \cdot(n-r+1) !}=\frac{n-r+1}{r} \cdot\left(\begin{array}{c}n \\r-1\end{array}\right)$$[/tex]

Thus, Expressing the binomial coefficients in terms of factorials and simplifying algebraically. we have proved that  

[tex]$\left(\begin{array}{l}n \\ r\end{array}\right) = \dfrac{n-r+1}{r} \cdot \left(\begin{array}{c}n \\ r-1\end{array}\right)[/tex]

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Full question:

Expressing the binomial coefficients in terms of factorials and simplifying algebraically, show that:

(a) [tex]$\left(\begin{array}{l}n \\ r\end{array}\right) = \dfrac{n-r+1}{r} \cdot \left(\begin{array}{c}n \\ r-1\end{array}\right)[/tex]

Triangle LMN is similar to triangle OPQ. The measure of side OP is 12.6

Triangle LMN is similar to triangle OPQ.

Similar forms or figures have corresponding sides that are proportionate to one another. The ratio of their matching sides would be identical because LMN OPQ. Thus:

MN/PO = LN/OQ

LN = 31

OQ = 8

MN = 49

RO = ?

Put the values in.

31 / 8 = 49 / PO

multiply by cross

31 * PO  =  49 * 8

31  * PO =  392

PO = 392 / 31

PO = 12.6

Triangle LMN is similar to triangle OPQ. The measure of side OP is 12.6

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Answer:

N103. 33

Step-by-step explanation:

let the cost of bottle of water be represented by B

Let the cost of wine be represented by W

3×W + 4×B =N450

3W+4B=450....(1)

one bottle of water(B)Costs N35

3W+ 4(N35)= N450

3W + 140=450

collect the.like terms

3W= 450-140

3W = 310

divide through by 3

W= N103.33

It will cost the company $321,400 to produce 14,300 items.

An arithmetic sequence has a constant difference between each consecutive pair of terms. This is similar to the linear functions that have the form y=mx+b. A geometric sequence has a constant ratio between each pair of consecutive terms. In its simplest form in algebra, the definition of an equation is a mathematical statement that shows that two mathematical expressions are equal.

To predict how much it will cost to produce 14,300 items, you need to substitute 14,300 for q in the expression 18q+64000.

So, 18q + 64000 becomes 18(14300)+64000 = 257400+64000 = 321400

Therefore, it will cost the company $321,400 to produce 14,300 items.

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Answer:

The inequality can be solved by isolating the variable x on one side of the inequality. To do this, we can first add 4x to both sides of the inequality:

-2 ≤ 4x + 3 ≤ 4

Next, we can subtract 3 from both sides of the inequality:

-5 ≤ 4x ≤ 1

Finally, we can divide both sides of the inequality by 4 to solve for x:

x ≥ -5/4 and x ≤ 1/4

So the solution of the compound inequality is x ≥ -5/4 and x ≤ 1/4

We can also express this solution in interval notation: [-5/4, 1/4]

Note that the compound inequality has two separate inequality signs, indicating that the solution is the set of all x-values that make both inequalities true.

Step-by-step explanation:

The volume of the large container is given as follows:

113.04 in³.

The ratio of the volumes is of:

B) 2.25

The volume of a cylinder of radius r and height h is given by the equation presented as follows:

V = πr²h.

The diameter of the smaller cylinder is of:

4 inches.

Hence the diameter and radius of the larger container is of:

As the height of the cylinder is of 4 inches, the volume is given as follows:

V = 3.14 x 3² x 4

V = 113.04 in³.

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The measures of the angles are (a) 180-C, and 180-A (b) 180 (c) 180 and supplementary.

You should understand that  cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle

For better understanding note the following

(a) m<A = 180 - m<C

m<C = 180 - m<A

(b)  m<A + m<C = 180 .......... (opposite angles are supplementary)

(c) m<A + m <C = 180 (Sum of opposite angles = 180⁰)

(d) <A and <C are supplementary angles

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To find the highest common factor (HCF) of two polynomials, we can use the Euclidean algorithm or the method of factoring the polynomials and comparing their common factors.

The first polynomial is x² - 2x - y² + 4y-3 and the second polynomial is x² + 2x - y² + 8y-15

x² - y² is common in both polynomials, let's factor it out.

x² - y² = (x-y)(x+y)

so the first polynomial can be written as (x-y)(x+y) - 2x + 4y - 3

and the second polynomial can be written as (x-y)(x+y) + 2x + 8y - 15

So the HCF of these two polynomials is (x-y)(x+y)

It is important to note that the HCF of polynomials is not unique, there can be more than one factorization.

The coordinates given in the question are not provided. Without the coordinates it's impossible to graph the line and determine which statement is true.

It is important to have the coordinates to graph the line, and then we can determine if the line is a proportional relationship by looking at the slope of the line, if it is a constant value then it represent a proportional relationship.

The statement A and B are incorrect, as passing through the origin does not determine whether the relationship is proportional or not.

C and D are correct, whether the line passes through the origin or not, it can be determined if it represents a proportional relationship by looking at the slope of the line.

Please provide the coordinates in order to proceed with the question.

Answer:

Amaka passes hers 2/3

Step-by-step explanation:

she just will

The distance between two points on a line segment .

Learn how to use the distance formula to calculate the distance between two points, which is an application of the Pythagorean theorem. To calculate the distance between any two points, we may rewrite the Pythagorean theorem as d=((x 2-x 1)2+(y 2-y 1)2). Sal Khan and the CK-12 Foundation created it.

A line segment is delimited by two separate points on a line in geometry. A line segment is a section of a line that links two locations. A line has no endpoints and stretches in both directions indefinitely, but a line segment has two defined or definite endpoints.

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The total number of ordered triples $(a,b,c)$ satisfying the conditions [tex]$0\le a \le b \le c \le 10$[/tex]  is 10275.

The total number of ordered triples $(a,b,c)$ satisfying the conditions [tex]$0\le a \le b \le c \le 10$[/tex]  is given by

[tex]$n= \sum_{a=0}^{10}\sum_{b=a}^{10}\sum_{c=b}^{10} 1$$= \sum_{a=0}^{10}\sum_{b=a}^{10} (c-b+1)$$= \sum_{a=0}^{10}\sum_{b=a}^{10} c - \sum_{a=0}^{10}\sum_{b=a}^{10} b + \sum_{a=0}^{10}\sum_{b=a}^{10} 1$$= \sum_{a=0}^{10}\sum_{b=a}^{10} 10 - \frac{10(11-a)(a+1)}{2} + \sum_{a=0}^{10} (b-a+1)$$= \sum_{a=0}^{10} 105 - \frac{10(11-a)(a+1)}{2} + \frac{10(a+1)}{2}$$= \sum_{a=0}^{10} \frac{185 + 10a^2}{2}$$= \frac{(185 + 10\times 100) \times 10}{2}$ $= \frac{2055 \times 10}{2}$ $= 10275$[/tex]

Therefore, the total number of ordered triples $(a,b,c)$ satisfying the conditions [tex]$0\le a \le b \le c \le 10$[/tex]  is 10275.

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F g - x hence equals -f x. -g x. It follows from this that f g - x = f g x. Given that f and g are two odd functions, it follows that f g is an even function. As a result, f g is not an unusual function.

If the equation for every x and x in the domain of f holds, then a function f is odd. F(x)=F(x) or F(x)=F(x) An odd function has a graph that, geometrically speaking, is rotationally symmetric with respect to the origin, meaning that the graph is unaffected by a 180° rotation of the origin.

If f - x = f x, then a function f x is said to be even. These functions have symmetric properties around the y-axis.

F - x Equals F x if two functions f and g are two even functions.

g - x = g x

Now, the definition of the product function f g is as follows:

[tex]$f g x=f x \cdot g x$[/tex]

Analyze the fg at -x product function.

[tex]$f g-x=f-x \cdot g-x$[/tex]

The two functions are currently equal functions. Utilize the knowledge that f - x = f and g - x = g.

Consequently, f g - x = f x. g x It follows from this that f g - x = f g x. This suggests that f g, where f and g are two even functions, is an even function.

If f - x = -f x, then a function f x is said to be odd. These functions have symmetry at their origin.

If f and g are two odd functions, then f - x = -f x and g - x = -g x, respectively.

Now, the definition of the product function f g is as follows:

[tex]$f g x=f x \cdot g x$[/tex]

Evaluate the product function f g at -x

[tex]$f g-x=f-x \cdot g-x$[/tex]

The two functions are now peculiar functions. Utilize the knowledge that f-x = -f-x and g-x = -g-x.

F g - x hence equals -f x. -g x. It follows from this that f g - x = f g x. Given that f and g are two odd functions, it follows that f g is an even function. As a result, f g is not an unusual function.

Assume that g is odd and f is even. G - x = -g x and f - x = f x, respectively.

The following is a definition of the product function:

[tex]f g-x & =f-x \cdot g-x \\[/tex]

[tex]& =f x \cdot-g x \\[/tex]

[tex]& =-f x \cdot g x \\[/tex]

[tex]& =-f g x[/tex]

Because f is even and g is odd, f g is an odd function.

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A correct stacked bar chart would show store location on the horizontal axis, and the percentage of time spent on each task on the vertical axis. Each store location would be represented as a separate bar, with each bar displaying the percentage of time spent on each task as stacked sections within the bar.

A stacked bar chart is a type of graph that is used to visualize data. It is a bar chart that displays the relative contribution of different data points to the whole. It is particularly useful for comparing the contribution of different data points across different categories. For example, a stacked bar chart can be used to show the percentage of time spent on different tasks at different store locations. The horizontal axis would represent the store locations, while the vertical axis would represent the percentage of time spent on each task. Each store location would then be represented as a separate bar, with each bar displaying the percentage of time spent on each task as stacked sections within the bar. The colors used to represent each task can be used to make the chart more visually appealing, and help viewers differentiate between the tasks. The stacked bar chart is a useful tool for analyzing data over multiple categories, and can be used to quickly identify trends and patterns.

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The complete question is

which of the following shows a correct stacked bar chart with store location on the horizontal axis and percentage of time spent on each task on the vertical axis Store Location Task

1. The antiderivative would be x.

2. The antiderivative would be sec(θ)tan(θ)−5e^(θ^3).

3. The antiderivative would be 1+2x.

What is the antiderivative of the function?

An antiderivative of a function is a function whose derivative is equal to the original function. It is also known as an indefinite integral of a function with respect to its variable. The process of finding an antiderivative is called integration.

1. The most general antiderivative of the function f(x)=x is F(x) = (1/2)x^2 + C, where C is the constant of the antiderivative. To check this, we can differentiate F(x) using the power rule for derivatives and see if we get back f(x) = x: d/dx [(1/2)x^2 + C] = d/dx (1/2)x^2 + d/dx C = x

2. The most general antiderivative of the function r(θ)=sec(θ)tan(θ)−5e^(θ^3) is R(θ)=∫sec(θ)tan(θ)−5e^(θ^3) dθ = -ln(cos(θ))+5e^(θ^3)/3+C, where C is the constant of the antiderivative. To check this, we can differentiate R(θ) using the rules of integration by substitution, and see if we get back r(θ) = sec(θ)tan(θ)−5e^(θ^3): d/dθ [-ln(cos(θ))+5e^(θ^3)/3+C] = sec(θ)tan(θ)−5e^(θ^3)

3. The most general antiderivative of the function f(x)=1+2x is F(x) = x+x^2 + C, where C is the constant of the antiderivative. To check this, we can differentiate F(x) using the power rule for derivatives and see if we get back f(x) = 1+2x: d/dx (x+x^2 + C) = d/dx x+d/dx x^2 + d/dx C = 1+2x

Hence,

1. The antiderivative would be x.

2. The antiderivative would be sec(θ)tan(θ)−5e^(θ^3).

3. The antiderivative would be 1+2x.

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The probability of rolling an odd number on a fair six-sided die is P(odd) = (1/6) + (1/6) + (1/6) = 3/6 = 1/2. This is because there are three odd numbers (1, 3, and 5) out of the six possible outcomes, and they are each equally likely to occur.

The probability of rolling a number divisible by 3 on a fair six-sided die is P (divisible by 3) = (1/6) + (1/6) = 2/6 = 1/3. This is because there are two numbers (3 and 6) out of the six possible outcomes that are divisible by 3, and they are each equally likely to occur.

It's important to note that in both of these examples, we are assuming that the die is fair and unbiased, which means that each of the six possible outcomes has an equal chance of occurring. If the die is not fair, the probabilities of each outcome could be different.

Complete question:

More probability Aa Aa An event's probability measures the likelihood of its occurrence. The probability that an event X occurs is between 0 and 1, inclusive, and is denoted by P(X). If it is impossible for X to occur, then P(X) Rolling a standard six-sided die, the probability of rolling a 7 is zero, P(7)-0. If X is the set of all possible outcomes in the experiment, then X is certain to occur and P(X)1 In the die example, X could be that an integer is rolled, which must occur, and thus P(integer)-1 Otherwise, X is possible but uncertain, and 0<P(X) 1. In the die example, the probabilities of each of the six outcomes are equally likely (so long as the die is fairly balanced), so P(1)-P(2)-P(3)-P(4) = P(5)-P(6)- 1/3. If possible, you want to identify all the possible outcomes, called the If you are rolling a 1/6). fair die, your outcomes range from 1 to 6, with equal probability (P The probability of rolling an odd number is the probability of rolling a number divisible by 3 is

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Kasey's claim that her model is a good fit for the data is best supported by a correlation coefficient of r = 0.91. The correct answer would be an option (B).

A correlation coefficient (r) is a number between -1 and 1 that measures the strength and direction of a linear relationship between two variables.

A coefficient of r = 1 indicates a perfect positive linear relationship, meaning as one variable increases, the other variable also increases.

A coefficient of r = -1 indicates a perfect negative linear relationship, meaning as one variable increases, the other variable decreases.

A coefficient of r = 0 indicates no linear relationship between the variables.

In this case, a coefficient of r = 0.91 is close to 1 which means that there is a strong positive linear relationship between the data and the model. On the other hand, the other coefficients -0.25, 0.01, and -0.79 are not close to 1, which means that there is no or weak linear relationship between data and the model.

Hence, the correct answer would be an option (B).

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The angle CFD is 67.6 degrees.

An angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.

Given that ∠CFE=177 degrees.

We have to find angle CFD.

∠CFD+∠EFD=117

4z+74+9z+64=117

13z+138=117

13z=21

z=-1.6

Now substitute in ∠CFD =4(-1.6)+74

∠CFD =-6.4+74

=67.6 degrees

Hence, the angle CFD is 67.6 degrees.

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Answer:

400

Step-by-step explanation:

[tex]100 + 200 + 300 + 400 + 500 + 600 + 700 = 2800 [/tex]

There are 7 numbers ( No. of cakes in a month)

Therefore, we should divide it by sum of the cakes

[tex]2800 \div 7 = 400[/tex]

SO MEAN NUMBER IS 400//

Three skeins are the most Sharon can afford to purchase because a number of skeins have to be a whole number.

For her sister's birthday present, Sharon has a budget of no more than $25. Sharon is therefore limited to spending no more than $25. She has paid $14.99 for a knitting machine for her. She wants to get her sister some yarn skeins to go with it as well. Cost per skein is $2.75. Let y be the number of yarn skeins.

The necessary inequality is:

14.99 + 2.75y ≤ 25

2.75y ≤ 25 - 14.99

2.75y ≤ 10.01

y ≤ 10.01 / 2.75

y ≤ 3.64

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The polygon was translated 5 units left and 9 units up using the rule for (x, y) ⇒ (x - 5, y + 9)

Translation is the movement of a point in the coordinate plane either up, left, down or right. Translation is a type of rigid transformation because it preserves both the shape and size of the figure.

Other types of rigid transformations are reflection and rotation.

The points D(5, -10) and T(-9, -6) were translated to give D'(0,-1), T'(-14,3). This means that the translation was 5 units left and 9 units up.

The rule for this translation is (x, y) ⇒ (x - 5, y + 9)

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The bisection method up to five iterations and finding the root to 2 decimal places for the following would be 2.19

Bisection technique

By continuously reducing the interval in which the root is located, the bisection method roughly approximations the root of a function. The procedure first evaluates the function at the interval midway before swapping out the interval's end with a new one that has the same sign.

The beginning range, in this case, is [1, 3], and f(1) > 0 and f(3) 0. (1+3)/2 = 2 represents the interval's midpoint. The interval endpoint (0) that had f(x) > 0 is replaced with the midpoint (2) when we evaluate f(2) and discover that f(2) > 0. The new midpoint is x = 5/2, and the new interval is [2, 3]. The first iteration is now complete.

Iterations

2nd iteration: f(5/2) < 0 ⇒ interval is [2, 5/2]; midpoint is 9/4

3rd iteration: f(9/4) < 0 ⇒ interval is [2, 9/4]; midpoint is 17/8

4th iteration: f(17/8) > 0 ⇒ interval is [17/8, 9/4]; midpoint is 35/16 ≈ 2.19

5th iteration: f(35/16) < 0 ⇒ interval is [17/8, 35/16]; midpoint is 69/32 ≈ 2.16

The approximate root of 2.16 is revealed by the fifth iteration, but that is not an option for the solution. We think this is the response you're searching for because the fourth iteration yields a root that is around 2.19.

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The full question:

What is the root of the following equation using the bisection method correct to three places of decimal f(x) =x3-3x-5?

Answer:

[tex]y=-\dfrac{1}{30}x+12[/tex]

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{6.3 cm}\underline{Slope-intercept form of a linear equation}\\\\$y=mx+b$\\\\where:\\ \phantom{ww}$\bullet$ $m$ is the slope. \\ \phantom{ww}$\bullet$ $b$ is the $y$-intercept.\\\end{minipage}}[/tex]

The y-intercept is the y-value of the point at which the line crosses the y-axis.

Therefore, from inspection of the given graph, the y-intercept is 12:

The slope of a line is the change in y over the change in x.

From inspection of the given graph, the change in y is -3 and the change in x is 90.  Therefore, the slope is:

Substitute the found values of m and b into the slope-intercept formula to create an equation of the line:

Answer:

[tex]\frac{17}{8}[/tex]

Step-by-step explanation:

[tex]2\frac{1}{8} = (2*8)+1 =17= \frac{17}{8}[/tex]

Answer:

17/8

Step-by-step explanation:

The total length of paper needed to cover all the tables is equal to 10 tables multiplied by 104 inches per table, which is equal to 1040 inches.

According to the Pythagorean theorem, the hypotenuse's square length is equal to the sum of the squares of the two legs' lengths in a right triangle. The formula for this is a2 + b2 = c2, where a and b represent the lengths of the two legs, respectively, and c represents the length of the hypotenuse.

The theory used in this question is the Pythagorean theorem.

Number of tables:

Let's say there are 10 tables.

Length of each table plus 8 inches:

Each table is 8 feet long, which is equal to 96 inches. The extra 8 inches makes it 104 inches per table.

Total length of paper needed:

The total length of paper needed to cover all the tables is equal to 10 tables multiplied by 104 inches per table, which is equal to 1040 inches.

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Answer:

Step-by-step explanation:

Given : ∠1 ≅ ∠2

To prove : p║q

Solution :

In the figure attached,

               Statements                 Reasons

1). m∠1 ≅ m∠2                      1). Given

2) m∠1 ≅ m∠3                      2). Vertical angles

3) m∠2 ≅ m∠3                     3). Transitive property of congruence of the                                                     angles

4). Line p║q                         4). Corr