Given right triangle ABCABC with altitude \overline{BD}BDdrawn to hypotenuse ACAC. If AC=12AC=12 and DC=4,DC=4, what is the length of \overline{BC}BCin simplest radical form?

Answer:

Solution in attached photo.

Step-by-step explanation:

Steps are in attached photo, to do dot product, make sure you are multiplying x-coordinates of one vector with x-coordinates of another vector, and likewise for y-coordinates.

Answer:

[tex]x \approx 31.6\textdegree[/tex]

Step-by-step explanation:

Apply the Law of Sines.

[tex]\dfrac{\sin(A)}{a} = \dfrac{\sin(B)}{b}[/tex]

↓ plug in given side lengths and angle measures

[tex]\dfrac{\sin(74\textdegree)}{22} = \dfrac{\sin(x)}{12}[/tex]

↓ multiply both sides by 12

[tex]\dfrac{12\sin(74\textdegree)}{22} = \sin(x)[/tex]

↓ take the inverse sine of both sides

[tex]\sin^{-1}\left(\dfrac{6\sin(74\textdegree)}{11}\right) = x[/tex]

↓ evaluate using a calculator and round to tenths place

[tex]x \approx 31.6\textdegree[/tex]

The value of c for which the solution satisfies the initial condition y(2)= 9 is c = 9 - e^{-2}.

Let us solve this problem step by step.

For the given equation: y' + 2y = e^{-x}

We have the solution as: y = ce^{-2x} + e^{-x}

We have to find the value of c for which the solution satisfies the initial condition y(2)= 9.

Substituting x = 2 in the above equation, we get:

9 = c + e^{-2}.

Therefore, we can solve for c as:

c = 9 - e^{-2}

Hence, the value of c for which the solution satisfies the initial condition y(2)= 9 is c = 9 - e^{-2}.

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The number of Mia's marbles that show some blue colour would be = 214.3 marbles

A fraction of a number is part of the whole number which is represented as a numerator and a denominator.

The total number of marbles = 250

The fraction of the marble that are red = 2/3

The fraction of the marble that are blue = 6/7

The fraction of the marble that are yellow = 3/4

The quantity of marbles that are blue

= 6/7× 250

= 1500/7

= 214.3 marbles

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The Mean is 350 and Median is 383.3

The first step in calculating the mean of grouped data is to establish the midpoint (also known as a class mark) of each interval or class. The frequencies of the relevant classes must then be multiplied by these midpoints. The mean value is determined by dividing the total number of values by the sum of the products.

Given:

 CI                  F        x        d       Fd        cf

100 - 200 2 150     -2       4   2  

200 - 300 2 250     -1       -2   4  

300 - 400 3 350      0       0   7  

400 - 500 6 450       1       6   13

                     n= 13                              

Mean = 350+ 0/13 x 100

Mean = 350+0

Mean = 350

Hence, the Mean is 350 and Median is 383.3

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The range of the function shown on the graph is, y ≥ 2.

The equation of line in point-slope form passing through the points

(x₁ , y₁) and (x₂, y₂) with slope m is defined as;

⇒ y - y₁ = m (x - x₁)

Where, m = (y₂ - y₁) / (x₂ - x₁)

We have to given that;

The graph of function is shown in figure.

Now, We know that;

The range is the set of possible outputs values, which are shown on the y - axis.

Hence, By the graph;

The range of function is,

⇒ y ≥ 2

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Answer:

Step-by-step explanation:

210

The probability that both of them wear glasses is 35%.

The probability that both students wear glasses is equal to the product of the probability of the first and the second student wearing glasses

If 59% of all students on campus wear glasses, then the probability of a single student wearing glasses is 0.59.

The probability that both students wear glasses is:

0.59 * 0.59 = 0.3481

Approximate

0.59 * 0.59 = 0.35

So, the probability that both students wear glasses is approximately 0.35 or 35%.

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The marks of the pupil who was recently added is 3.

Given is that overall mark for english test for 15 pupils is 82. If one more pupil being added, the overall mark become 85.

For 15 peoples, the overall mark for english is 82. When one more person is added, the overall mark become 85. Then, the marks of the pupil who recently added was -

85 - 82

3

Therefore, the marks of the pupil who was recently added is 3.

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The linear equations in slope-intercept form are  y=2x+3 ,  y=-1x-2 , y=-3x+4 and y = 3x .

What is linear equation ?

Linear equation can be defined as equation in which highest degree is one.

Given ,

We have to write linear equations in slope intercept form.

So,

An equation in the slope-intercept form is written as

y = mx + b

Where m is the slope of the line and b is the y-intercept. You can use this equation to write an equation if you know the slope and the y-intercept.

So, from given graphs we can say that

y - 3 = 2x

y=2x+3

y + 2 = -x

y=-1x-2

y - 4 = -3x

y=-3x+4

y - 0 = 3x

y=3x+0

y = 3x

Therefore, The linear equations in slope-intercept form are  y=2x+3 ,  y=-1x-2 , y=-3x+4 and y = 3x .

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Answer:

15,910

Step-by-step explanation:

(390*26)+5770=15910

On solving the provided question, we cans ay that in the equation 390t + 5770, In the year 2021, the total average credit card debt for a U.S.

household will be 5,910 dollars.

A mathematical equation is a formula that joins two statements and uses the equal symbol (=) to indicate equality. A mathematical statement that establishes the equality of two mathematical expressions is known as an equation in algebra. For instance, in the equation 3x + 5 = 14, the equal sign places the variables 3x + 5 and 14 apart. The relationship between the two sentences on either side of a letter is described by a mathematical formula. Often, there is only one variable, which also serves as the symbol. for instance, 2x – 4 = 2.

in the equation 390t + 5770

t = 26

In the year 2021, the total average credit card debt for a U.S.

household will be 5,910 dollars.

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Answer:

you have to do this by formula in the copy try this by l×b or b×h

The solution set of the linear equation 2x - t.6.t-12 = 0 can be represented parametrically as x = t + 6.

A linear equation is an algebraic equation of the form y=mx+b. m is the slope and b is the y-intercept. The above is sometimes called a "linear equation in two variables" where y and x are variables.

The equation 2x - t.6.t-12 = 0 is a linear equation in the variable x and can be solved by isolating x to one side of the equation. To do this, first distribute t on the right side of the equation, giving 2x - 6t^2 + 12 = 0. Then add 6t^2 - 12 to both sides of the equation to get 2x = 6t^2 - 12. Finally, divide both sides of the equation by 2 to get x = 3t^2 - 6. This is a parametric representation of the solution set of linear equations. x is a function of parameter t. The solution is given by x = 3t^2 - 6.

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The mean of the score is 3.4 and the standard deviation is 3.2.

Scores of students = 5, 4, 4, 3 and 1

When determining the mean of a group of numbers, add up all the numbers and divide by the total number in the group.

Calculating the mean -

= Sum of total values / Total values

= 5 + 4 + 4 + 3 + 1 / 5

= 17/5

= 3.4

By averaging the squared deviations of each integer from the mean, you may get the variance before calculating the standard deviation.

Calculating the Standard deviation -

σ = √ ∑ (xi - u)²/n

σ² = (1² + (3.4 - 1)² + (3.4 - 4)² + (3.4 - 4)²+ (3.4 - 5)²) / 5

= 3.2

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The inequalities that are equivalent to the inequality $x<3$ are:

$x<3-b$

$x-b<3$

Note that $x-b-3.0$ and $x-b-3<0$ are not equivalent to $x<3$

and -3 is not an inequality.

To find the intersection point of Runways AB and CD, we need to find the point where the two lines intersect. The two lines can be described as follows:

What exactly is a math graph?

A math graph is a visual representation of mathematical data or concepts. It typically uses Cartesian coordinates (x, y) to plot points, lines, and shapes that help to illustrate mathematical relationships. Graphs can be used to represent many types of mathematical information, including functions, equations, data sets, and geometric shapes. They are commonly used in fields such as science, engineering, economics, and statistics to analyze and understand complex information. Graphs are also an important tool for conveying mathematical ideas and results in a clear and understandable way.

Line AB: y = mx + b, where m = (9-(-24))/(6-(-5)) = 33/11 and b = -24 - (33/11)(-5) = -24 + 165/11 = 141/11

Line CD: 2x + y - 11 = 0

Now we can substitute the equation of the line AB into the equation of the line CD:

2x + (33/11)x + 141/11 -11 = 0

Combining like terms we get:

(33/11+2)x + (141/11 - 11) = 0

35/11x + 130/11 = 0

35/11x = -130/11

x = -130/35 = -26/7

Now we can use this value of x to find the corresponding y value using the equation of the line AB

y = (33/11)x + 141/11 = (33/11)(-26/7) + 141/11 = -86/11

So the point of intersection of the two lines is (-26/7, -86/11)

So the intersection point of Runways AB and CD is (-26/7, -86/11)

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The solution is Option A.

The value of the missing length parallel to the side LN of the triangle is 44

If two triangles' corresponding angles are congruent and their corresponding sides are proportional, they are said to be similar triangles. In other words, similar triangles have the same shape but may or may not be the same size. The triangles are congruent if their corresponding sides are also of identical length.

Corresponding sides of similar triangles are in the same ratio. The ratio of area of similar triangles is the same as the ratio of the square of any pair of their corresponding sides

Given data ,

Let the first triangle be represented as ΔLMN

Let the second triangle be represented ΔVUN

The triangles LMN and VUN are similar

The measure of side LV = x

The measure of side LN = 11 + x

So , the corresponding sides of the similar triangles are in ratio

MU / UN = LV / VN

24 / 8 = x / 11

On simplifying the equation , we get

3 = x / 11

Multiply by 11 on both sides of the equation , we get

x = 33

Now , the measure of side LN = x + 11

The measure of side LN = 33 + 11 = 44

Hence , the measure of side LN is 44

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The solution to 3x + 2(4 + 6x) = 113 is x=7

3x + 2(4 + 6x) = 113

We will multiply the 2 with the brackets

3x+8+12x=13

adding the coefficients os x together

15x+8=113

adding -8 on both the sides

15x=105

dividing both the sides by 15

x=105/15

after division we get

x=7

To check our solution

we first put value of x in our original equation

3x7 + 2(4 + 6x7) = 113

we simplify the equation

21+2(4+42)=113

According to BODMAS we first solve the bracket

21+2(46)=113

now we solve the multiplication part

21+92=113

solving this

113=113

∴ LHS=RHS

Hence the solution is correct.

Therefore, the solution to 3x + 2(4 + 6x) = 113 is x=7

BODMAS is an acronym used to remember the order of precedence for operations. It stands for Brackets of Order(exponents) Division, multiplication, Addition, Subtraction

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Therefore , the solution of the given problem of expression comes out to be 4x -2y is the sum of given expression.

Multiplying, dividing, adding, and subtracting are necessary math operations. When combined, the following expression would appear: A mathematical operator, some data, and an equation Values, variables, and functions make up a statement of fact's constituent parts, such as additions, deductions, multiplications, divisions, etc. It is possible to contrast and compare words and phrases.

Here,

Given :

expressions are : 3x,x,2y and y-5y

To find the sum of the expression ,

We add them:

=> 3x+ x+2y+y-5y

=> 4x+3y-5y

=>4x -2y

Therefore , the solution of the given problem of expression comes out to be 4x -2y is the sum of given expression.

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To calculate y, we use the sum() function with an argument of x < 5 to count the number of entries in x that are strictly less than 5. To calculate z, we use the sum() function with an argument of (x > -5) & (x < 5) to count the number of entries in x that are both greater than -5 and less than 5.

y = sum(x < 5)

z = sum((x > -5) & (x < 5))

y = sum(x < 5)

 = sum(TRUE, FALSE, FALSE, TRUE, FALSE, FALSE, TRUE, FALSE, TRUE, TRUE, TRUE, FALSE)

 = 5

z = sum((x > -5) & (x < 5))

 = sum(TRUE, FALSE, FALSE, TRUE, FALSE, FALSE, TRUE, FALSE, TRUE, FALSE, TRUE, FALSE)

 = 4

The first variable, y, is the number of entries of x that are strictly less than 5. To calculate y, we can use the sum() function with an argument of x < 5. This will count the number of entries in x that are less than 5. For example, if

x <- c(-5, 7, 9, 3, 14, 9, 5, 10, 2, -2, 0, 12), then y will be equal to 6 because there are 6 entries in x that are strictly less than 5.

The second variable, z, is the number of entries of x that are strictly between -5 and 5. To calculate z, we can use the sum() function with an argument of (x > -5) & (x < 5). This will count the number of entries in x that are both greater than -5 and less than 5. For our example, z will be equal to 4 because there are 4 entries in x that are strictly between -5 and 5.

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The numeric value of the function f(x) = 4x² + 2x + 2 at x = x + 2 is given as follows:

f(x + 2) = 4x² + 20x + 22.

To find the numeric value of a function or of an expression, we replace each instance of the variable in the function or in the expression by the value at which we want to find the numeric value.

The function for this problem is defined as follows:

f(x) = 4x² + 2x + 2

To obtain the numeric value at x = x + 2, we replace the two instances of x by x + 2, hence:

f(x + 2) = 4(x + 2)²  + 2(x + 2) + 2

f(x + 2) = 4(x² + 4x + 4) + 4x + 4 + 2

f(x + 2) = 4x² + 16x + 16 + 4x + 6

f(x + 2) = 4x² + 20x + 22.

The problem asks for the numeric value of the function f(x) = 4x² + 2x + 2 at x = x + 2.

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The extent of a square prism is given by V = l × w × h, so that you can quantity without the hollow would be 18×22×77=30,492cm³

Determine the most intensity to which the container can be packed with water so that when the water freezes, the ice does not move above the top of the container.?

The surface region is:

A = 2 × (l × w + w × h + h × l)

= 2×(18×22+22×77+77×18)

=4862cm³

Now cut a cylindrical hollow through the middle … however in what route??? because the orientation of the hole isn't given, the valuable axis of the cylindrical hole might be parallel to any individual of the three most important axes of the square prism!

[Note: a ‘round’ hole in any other direction – not parallel to a major axis – would have either elliptical ends, or (worse) cut through vertices or even corners.]

Therefore, The extent of a square prism is given by V = l × w × h, so that you can quantity without the hollow would be 18×22×77=30,492cm³

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Answer: x = 57°

Step-by-step explanation:

63 + x + 2x + 183 - x = 360 (exterior ∡s of a polygon adds up to 360°)

                  x + 2x - x = 360 - 63 - 183

                             2x = 114

                                x = 57°

The value of x in the given diagram is 57° .

We know that the sum of internal angles of a triangle is equal to 180°.

But here we are given the external angles of the triangle:

Angle 1 : 63 + x

Angle 2 : 2x

Angle 3 : 183 - x

Now , changing the given angles into internal angles , we get:

Angle 1 : 180 - ( 63 + x ) = 117 - x

Angle 2 : 180 -2x

Angle 3: 180 -(183-x) = x-3

Now,

(117-x) + (180 -2x) + (x-3) =180

114 - 2x = 180

x = 114/2=57

Therefore, The value of x in the given diagram is 57° .

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The ratio of the sides of the triangle is the same one angle therefore, the two of the given triangles are similar triangles.

What are similar triangles?

If the respective angles are congruent and the corresponding sides are proportional, two triangles are said to be similar. To put it another way, comparable triangles are similar in shape but not necessarily in size. In addition, the triangles are congruent if their corresponding sides are of identical length.

As it we know that for the two triangles to be similar triangles, the ratio of their corresponding sides should be the same.

DC/RD = DB/SD

∠R = ∠C = 68°

So, ΔDCB ≈ ΔRSD by Side Angel Side property.

Hence, we can see that the ratio of the sides of the triangle is the same one angle therefore, the two of the given triangles are similar triangles.

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The explicit formula of the arithmetic sequence in this problem is given as follows:

[tex]a_n = 40 - 7n[/tex]

An arithmetic sequence is a sequence of values in which the difference between consecutive terms is constant and is called common difference d.

The explicit formula of an arithmetic sequence is given as follows:

[tex]a_n = a_1 + (n - 1)d[/tex]

From the recursive formula, each term is the previous term subtracted by 7, hence the common difference is given as follows:

d = 7.

The first term of the sequence is given as follows:

[tex]a_1 = 33[/tex]

Hence the explicit formula is obtained as follows:

[tex]a_n = a_1 + (n - 1)d[/tex]

[tex]a_n = 33 - 7(n - 1)[/tex]

[tex]a_n = 40 - 7n[/tex]

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The missing side of problems using sine and cosine is 9.

The formula for calculating the triangle's side is: c = [a2 + b2 - 2ab cos C] where the triangle's sides are a, b, and c.

According to the Cosine Rule, the square of any side of a given triangle is equal to the sum of the squares of the other sides minus twice the product of the other two sides multiplied by the cosine of the angle that separates them. Law of cosines and Cosine Formula are other names for the cosine rule.

a2 = b2 + c2 – 2bc cos ∠x

b2 = a2 + c2 – 2ac cos ∠y

c2 = a2 + b2 – 2ab cos ∠z

6^2 + 7^2 = X

x = 9^2

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The required, mean for the specific data set will also be greater than the median.

Statistics is the study of mathematics that deals with relations between comprehensive data.

Here,
The relative values of the mean and median are typical of data that have a significant skew to the left, If the data set significant skew to the left implies that the mean of the data set is greater than the median of the data set.

Thus, the required, mean for the specific data set will also be greater than the median.

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Answer:

  f(x) = (x +6)(x -5)  or  f(x) = x² +x -30

Step-by-step explanation:

You want a quadratic function whose zeros are -6 and +5.

If p is a zero of the polynomial f(x), then (x-p) is a factor. The two given zeros meant that the factors are ...

  f(x) = (x -(-6))·(x -5)

  f(x) = (x +6)(x -5) . . . . . . . . simplify a bit

This can be put in standard form by multiplying the factors:

  f(x) = x(x -5) +6(x -5) = x² -5x +6x -30

  f(x) = x² +x -30 . . . . . . . . simplified quadratic function

The quadratic function of the zeros is f(x) = [tex]x^2+x-30[/tex].

What is quadratic function?

A polynomial function that has one or more variables and a variable having a maximum exponent of two is said to be quadratic. A polynomial of degree 2 is referred to as such because the greatest degree term in a quadratic function is of second degree. There must be at least one second-degree term in a quadratic function. It is a mathematical function.

Here the zeros of the function is -6 and 5.

We know that if zeros of the function is a then the factor is (x-a).

Then,

=> f(x) = (x-(-6))(x-5)

=> f(x) = (x+6)(x-5)

=> f(x) = [tex]x^2+6x-5x-30[/tex]

=> f(x) = [tex]x^2+x-30[/tex]

Hence the quadradic function is f(x) = [tex]x^2+x-30[/tex].

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The number of games played is 6 and the number of rides is 2.

Here it is given that the number of games played was 3 times the number of rides. So, by this, we get the value of both is the ratio.

That is $15,

number of games played: number of rides = 3: 1

Let x be the common ratio. So, we have

Number of games played = 3x, and number of rides = x

It is given that the cost of each game is $1.50 and cost of each ride is $3 and the total cost is $15.

By this, we get an equation,

1.50 × 3x + 3 × x = 15

4.5x + 3x = 15

7.5x = 15

x = 2

Therefore the number of games= 3x = 3×2 = 6 and the number of rides = x = 2.

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