Geometry
Write a paragraph proof for the following:
Given: ∠3 and ∠2 are complementary; m∠1 + m∠2 = 90
Prove: ∠3 is congruent to ∠1
Plan: First show that ∠1 and ∠2 are complementary. Then show that ∠3 is congruent to ∠1 because they are complementary to the same angle 2.

If two sides of a triangle have lengths 13.2 and 21.5, then the third side length be less than 34.7.

When a triangle's two sides are given lengths of 13.2 and 21.5, we must determine what the third side's length should be.

The sum of the lengths of any two sides of a triangle can always be more than the length of the third side.

The length difference between any two triangle sides will never be greater than the length of the triangle's third side.

13.2 and 21.5 add up to 34.7, and their difference equals 8.3.

The above condition must not be violated, and the length of the third side must be greater than 8.3 and less than 34.7.

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Check the picture below.

[tex]cos(Y )=\cfrac{\stackrel{adjacent}{6}}{\underset{hypotenuse}{15}}\implies cos(Y)=\cfrac{2}{5}\implies cos^{-1}[cos(Y)]=cos^{-1}\left( \cfrac{2}{5}\right) \\\\\\ Y=cos^{-1}\left( \cfrac{2}{5}\right) \implies Y\approx 66.4^o[/tex]

Make sure your calculator is in Degree mode.

Answer:

66.4

Step-by-step explanation:

Hope this helped!

Answer:

Step-by-step explanation:

The answer is C. 5x + 15 because the 5 outside the parentheses () is supposed to multiply by what's inside the parentheses. So multiply 5 and x and 5 and 3 (5 * X = 5x) (5 * 3 = 15).

The p-value is a probability and it should be between 0 and 1.

The p-value of a hypothesis test is the probability of obtaining a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true.

In this case, the null hypothesis is that the population proportion of students who went to the movie theatre last weekend is equal to 0.2 and the alternative hypothesis is that the population proportion is greater than 0.2.

Given that the test statistic is 2.25, the p-value would be the probability of getting a test statistic as extreme or more extreme than 2.25 assuming the null hypothesis is true. Without more information, we can't give you an exact p-value, but you can use a one-sided test and find the p-value using a z-table or using software that can calculate the p-value.

It is also important to note that the p-value is a probability and it should be between 0 and 1.

Therefore,  The p-value is a probability and it should be between 0 and 1.

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In given triangle ABC the length if BC is 7.1 .

What is trigonometric functions?

A subfield of mathematics called trigonometry is concerned with the study of triangles. It is frequently referred to as "trig" casually. In trigonometry, mathematicians look at how triangles' sides and angles relate to one another.

Here in given triangle contains 90°, So it is right triangle.

Now using tangent function then

=> tan B = [tex]\frac{opposite}{adjacent}[/tex] then

=> tan 38° = [tex]\frac{AC}{BC}[/tex]

=> BC = tan 38° × AC

=> BC = tan 38° × 9.1

=> BC = 7.1

Hence the length of the side BC is 7.1.

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The length of BC in the triangle ABC is 7.109.

Trigonometry is a branch of mathematics concerned with the study of triangles. It is commonly referred to as "trig" in a casual manner. Trigonometry examines how the sides and angles of triangles relate to one another.

Because the given triangle contains 90°, it is a right triangle.

Now, using the tangent function,

The tangent function is one of trigonometry's six primary functions.

Tan A = Opposite Side/Adjacent Side is the Tangent Formula. In terms of sine and cosine, tangent may be represented as: Tan A = Sin

then,

= tan B = opposite side / adjacent side

= tan 38° = AC/BC

= BC = 38° tan x AC

= BC = tan 38° × 9.1

= BC = 7.109

As a result, the length of the side BC is 7.109.

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Answer: y=-33.25 or -33.3

Step-by-step explanation:

2x=88

x=44

the triangle has 180 total angle

4y+(5x+5)+2x=180

substituting the numbers

4y+(5(44)+5)+2(44)=180

4y+(220+5)+88=180

4y+225+88=180

4y+313=180

(-313)

4y=-133

(divide by 4)

y=-33.25 or -33.3

3:7 are the odds in favor of choosing a book of poetry

given that

The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a statement is true. An event's probability is a number between 0 and 1, where, roughly speaking, 0 denotes the event's impossibility and 1 denotes certainty. [note 1] [1] [2] The likelihood that an event will occur increases with its probability. A straightforward illustration is tossing a fair (impartial) coin. The chance of both outcomes ("heads" and "tails") is equal because the coin is fair, "heads" is more likely than "tails," there are no other conceivable outcomes, and the likelihood of either outcome is half

the probability of randomly picking out a book of poetry from a bookshelf is 3/10

now, we need to find what are the odds in favor of choosing a book of poetry

P(poerty) = 3/10

P(non-poerty) = 1 - 3/10 = 7/10

number of odds = P(poerty) /P(non-poerty)

                          = 3/10/7/10

                          = 3/7

                          = 3:7

3:7 are the odds in favor of choosing a book of poetry

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The numbers are 4.5 and 13.5.

The ratio of two numbers = 1:3

Let the numbers be x and 3x.

After increasing each number by 9, the numbers will be x + 9 and 3x + 9.

Now, According to the question, after increasing each number by 9 the ratio becomes 3:5

Hence,         [tex]\frac{x + 9}{3x + 9}[/tex]  = [tex]\frac{3}{5}[/tex]

                  5x + 45 = 9x + 27

                  9x - 5x =  45 - 27

                     4x     = 18

                       x     = 18 / 4

                              = 4.5

if x = 4.5, then 3x = 4.5 * 3 = 13.5

So, the numbers are 4.5 and 13.5.

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The provided integral may be expressed as follows using the following statement as a guide: 128/3 - 31 [tex]\sqrt{17}[/tex] /3

An integral in mathematics is either a number that depicts the area underneath the graph of a function over a specific period of time or a about before, the reversed of which includes the initial result (indefinite integral). To complete the whole, an essential component is required. The term crucial is a close synonym in this context. Integrals of functional and equations are a concept in mathematics. The word "aal" is derived from Middle English, which is derived from Latin Word integralis "thinking up unity," from Latin "with, entire."

To find [tex]$\int_0^1 \frac{r^3}{\sqrt{16+r^2}} d r$[/tex]

Put [tex]$16+r^2=t^2$[/tex]

[tex]$2 r d r=2 t d t$[/tex]

[tex]$r d r=d t$[/tex]

[tex]$r=0 \Rightarrow t=4$[/tex]

[tex]$r=1 \Rightarrow t=\sqrt{17}$[/tex]

[tex]$\int_0^1 \frac{r^3}{\sqrt{16+r^2}} d r=\int_4^{\sqrt{17}} \frac{t^2-16}{t} t d t$[/tex]

[tex]$=\left[\frac{t^3}{3}-16 t\right]_4^{\sqrt{7}}$[/tex]

[tex]$=\frac{(\sqrt{17})^3}{3}-16(\sqrt{17})-\frac{(4)^3}{3}+64$[/tex]

[tex]$=\frac{128}{3}-\frac{31 \sqrt{17}}{3}$[/tex]

[tex]$\int_0^1 \frac{r^3}{\sqrt{16+r^2}} d r=\frac{128}{3}-\frac{31 \sqrt{17}}{3}$[/tex]

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$45.38 and $46.42 is the  interval can the branch manager be 95% certain that the sample mean will fall within

Using the Central Limit Theorem, the branch manager can be 95% certain that the sample mean will fall within $1.034 of the mean.

What does the Central Limit Theorem state?

It states that the sampling distribution of sample means of size n has standard deviation .

By the Empirical Rule, 95% of the sample means fall within 2 standard errors of the mean.

In this problem, we have that the standard deviation and the sample size are given as follows:

Hence the standard error is given by:

[tex]s = \frac{10.34}{\sqrt{400}} = 0.517.

Two standard errors is represented by:

2 x 0.517 = $1.034.

Hence, the branch manager can be 95% certain that the sample mean will fall within $1.034 of the mean.

$45.38 and $46.42 is the  interval can the branch manager be 95% certain that the sample mean will fall within

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Answer:

  see attached

Step-by-step explanation:

You want to fill in the blanks to complete the description of writing a number in scientific notation.

A number written in scientific notation can be described as a coefficient times a power of ten (2nd attachment). Different terminology is used by different authors to describe these parts.

Sometimes the entire coefficient is also called the "mantissa." Sometimes, the term "mantissa" is used to refer to the fractional part of the coefficient, the digits to the right of the decimal point. Your curriculum calls the coefficient the "first factor."

The power of ten is chosen so that the first factor is always a number greater than or equal to 1, but less than 10.

The power of 10 has an exponent, also called the "characteristic". Your curriculum calls the combination of base (10) and exponent the "second factor." The exponent is equal to the exponent of the place value multiplier of the most-significant digit when the number is written in standard form (3rd attachment).

The filled-in description is shown in the first attachment. You will notice that its wording helps you fill in the blanks. It asks you how many parts there are to the number, then refers to them as the "first part" and the "second part." It asks you what the parts are called, then refers to the first one as the "first factor."

The given example in scientific notation is ...

  [tex]2.54\times10^6[/tex]

The exponent of 6 corresponds to the 6 little left-arrows in the figure in your problem statement. It is the power of 10 that represents "millions," the place-value multiplier of the most-significant digit of the number two million five hundred forty thousand.

The value of (AB)² is 15.

A circle is a two-dimensional figure formed by a set of points that are at a constant or at a fixed distance (radius) from a fixed point (center) on the plane. The fixed point is called the origin or center of the circle and the fixed distance of the points from the origin is called the radius. Two circles can be called congruent if they have the same radius. Equal chords are always equidistant from the center of the circle. The perpendicular bisector of a chord passes through the center of the circle. When two circles intersect, the line connecting the intersecting points will be perpendicular to the line connecting their center points. Tangents drawn at the endpoints of the diameter are parallel to each other.

Equation of first circle:

(x - 2)² + (y + 1)² = 4²

Equation of secnd circle:

(x - 2)² + (y - 5)² = (√10)²

Solve for (x - 2)² :

(x - 2)² + (y + 1)² = 4²   ==>   (x - 2)² = 16 - (y + 1)²

(x - 2)² + (y - 5)² = (√10)²   ==>   (x - 2)² = 10 - (y - 5)²

Then,

16 - (y + 1)² = 10 - (y - 5)²

16 - (y ² + 2y + 1) = 10 - (y ² - 10y + 25)

15 - 2y - y ² = -15 + 10y - y ²

30 - 12y = 0

12y = 30

y = 30/12 = 5/2

Thus, the y coordinate of A and B is 5/2.

Then solve for x :

(x - 2)² = 16 - (5/2 + 1)²

(x - 2)² = 15/4

x - 2 = ± √(15/4) = ±√15/2

x = 2 ± √15/2

Thus, the x coordinates for either A or B are 2 +√15/2 or 2- √15/2.

The intersections points are A = (2 - √15/2, 5/2) and B = (2 + √15/2, 5/2). Thus, the squared distance between them:

(AB)² = [(2 - √15/2) - (2 + √15/2)]² + (5/2 - 5/2)²

(AB)² = (-√15)² + 0²

(AB)² = 15

Thus, the value of (AB)² is 15.

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Answer:

We can set up a system of equations using the information given:

x + 2y = 2830 (equation 1)

2x + y = 2540 (equation 2)

where x is the number of calories in one shake and y is the number of calories in one taco supreme.

We can use either equation to solve for one of the variables and then substitute that into the other equation.

From equation (1), x = 2830 - 2y

And substituting this into equation (2) we get

2(2830-2y) + y = 2540

Now we can solve for y.

5,660 - 4y + y = 2540

-3y = -3120

y = 1040

So, one taco supreme contains 1040 calories.

And substituting this value of y back into equation (1) we can find the number of calories in one shake:

x + 2(1040) = 2830

x = 550

So, one shake contains 550 calories.

Answer:

The bottom most option

Step-by-step explanation:

Lets divide 3 2/5 and 3 1/6

3 2/5 = 17/5 and 3 1/6 = 19/6

17/5/(19/6) = 17/5  x 6/19 = 102/95 = 1 7/95 or the last option.

Pls mark as brainliest for no real reason. Cheers

Yes, a function can have 3 roots. This means that the function has 3 values of x where the y value is equal to 0. A polynomial of degree 3 or higher can have 3 roots.

Yes, a function can have 3 roots. A root of a function is a value of x for which the y value of the function is 0. This means that when a function has 3 roots, it has 3 values of x for which the y value of the function is 0. To have 3 roots, the function must be a polynomial of degree 3 or higher. A polynomial of degree 3 is usually written in the form ax^3 + bx^2 + cx + d = 0. This equation can be solved using the quadratic formula to find the 3 roots, which will be the 3 values of x for which the y value of the function is 0. The roots of the polynomial can also be found graphically by plotting the polynomial and finding where it intersects with the x-axis. This will be the 3 roots of the function.

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The distribution used to obtain the critical value in this situation is the Student's t-distribution.

Critical value is a statistical value used to determine whether a hypothesis test result is statistically significant or not. It is the value that is compared to the test statistic to determine whether the null hypothesis should be rejected or not. The critical value is determined by the level of significance chosen for the study, the sampling distribution of the test statistic, and the degrees of freedom.

The Student's t-distribution is a symmetric probability distribution that is used when the population standard deviation is unknown. It is used in situations where there is a small sample size (less than 30) and the sample is drawn from a normally distributed population. In this case, the owner is interested in developing a 90% confidence interval estimate, so the critical value will be obtained from the Student's t-distribution.

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Answer:

(6x + 15y) / x

Step-by-step explanation:

(2x + 5y) * 3/x =

3 * (2x + 5y) / x =

(3 * 2x + 3 * 5y) / x  ==> distribute 3 to 2x and 5y

(6x + 15y) / x ==> simplify

Stratified sampling is used when the population is heterogeneous.

What is Stratified sampling?

A method of probability sampling (where all members of the population have an equal chance of being included) Population is divided into 'strata' (sub-populations) and random samples are drawn from each. This increases representativeness as a proportion of each population is represented.

The sampling method described in the study is a stratified sampling method.

In stratified sampling, the population is first divided into subgroups or strata based on some relevant characteristic, such as size in this case.

Then, a simple random sample is taken from each stratum.

In this study, the sample is chosen by separating all cars by size and selecting 10 of each size grouping, this is the definition of stratified sampling.

Stratified sampling is useful when the population is heterogeneous and you want to ensure that the sample is representative of specific subgroups within the population.

This method allows for a more precise estimation of population characteristics and can reduce sampling error.

Hence, Stratified sampling is used when the population is heterogeneous.

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The value of c is 72.

What is a quadratic equation?

A quadratic equation is a type of polynomial equation that has the form of ax^2 + bx + c = 0, where x is the variable, and a, b, and c are coefficients. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b and c are constants. The solutions of a quadratic equation are given by the quadratic formula: x = (-b ± √(b^2-4ac)) / 2a

The equation x^2-17x+c=0 is a quadratic equation where a=1, b=-17 and c=c

To find c, we can use the difference of the two solutions of the equation, which is 1.

The two solutions of the equation are given by the quadratic formula

x = (-b ± √(b^2-4ac)) / 2a

x = (-(-17) ± √((-17)^2-41c)) / 2*1

x = (17 ± √(289-4c)) / 2

The difference of the two solutions is (17 + √(289-4c))/2 - (17 - √(289-4c))/2 = 1

we can solve for c:

(17 + √(289-4c))/2 - (17 - √(289-4c))/2 = 1

√(289-4c) = 1

289-4c = 1

4c = 288

c = 72

Hence, the value of c is 72.

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The equation of the least square regression line mated closely with the given data: y = 5.5x + 52.

For the given data:

Option:  y = 5.5x + 52.

Put x = 2,

y = 5.5(2) + 52

y = 63

Put x = 4,

y = 5.5(4) + 52

y = 74

Thus, the equation of the least square regression line mated closely with the given data: y = 5.5x + 52.

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The minimum number of 1 point foul shots that your team must make to win the game is given as follows:

20.

The expression for this problem is defined as follows:

3p + 2f + s.

In which the variables are given as follows:

The opponent made these following shots:

Hence the number of points of the opponent is given as follows:

8 x 3 + 12 x 2 + 11 = 59.

Your team made these following shots:

Hence the number of points of your team is given as follows:

10 x 3 + 8 x 2 + x = 40 + x.

The team wins when:

40 + x > 59

x > 19

Hence the minimum number is of 20, as the number of one point shots is a discrete amount.

Your basketball team made 10 three-pointers, 8 two-point field goals, and x one-point foul shots.

Your opponents made 8 three-pointers, 12 two-point field goals, 11 one-point foul shots.

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The third side of the right angle triangle is x = 15 ft.

What is right angle triangle?

A right triangle is a triangle with one angle at a right angle, meaning that two of its sides are perpendicular. It is also known as a right-angled triangle, an orthogonal triangle, or more commonly as a rectangled triangle. Trigonometry's foundation is the right triangle's relationship between its sides and other angles.

We have to find the measure of third side of given right triangle.

The two sides of right triangle are 9 ft and 12 ft.

Let the side of third side is x.

By using the Pythagoras theorem,

(9)^2 + (12)^2 = x^2

       81 + 144 = x^2

             225 = x^2

                  x = 15

Hence, the third side of the right angle triangle is x = 15 ft.

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The required number of ways to write 9 as sum of 1, 2, 4 is 96 ways.

An arrangement of items in a specific order is referred to as a permutation. Here, the components of sets are arranged in a linear or sequential order. For instance, the set A=1,6 has a permutation of 2, which is 1,6,1.

Two fours and one one Three numbers must be added up. Any one of the three spaces will accept a one before fours are used to fill the remaining slots. There are thus 3 possible arrangements for these. Thus, there are

1 + 8 + 21 + 6 + 20 + 20 + 5 + 12 + 3 = 96 ways

Thus, there are 96 ways to write 9 as sum of 1, 2, 4.

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Answer:

  CD = 3/√2 in

Step-by-step explanation:

You want the measure of altitude CD in isosceles right triangle ACB with the right angle at C.

All of the isosceles right triangles in this figure are similar, and all have sides in the ratio 1 : 1 : √2.

In ∆BCD, side BC is the longest, given as 3 in. That means the other two sides of that triangle are ...

  CD = BD = CB/√2

  CD = 3/√2 in

__

Additional comment

Often, we like to rationalize the denominator of expressions involving roots. That would make the expression for CD be ...

  CD = 3/√2 = (3/2)√2 = 1.5√2

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the ratio of the line segment joining the points (-3, 1) and (-8, 9) is 5:4. The ratio of the line segment joining the points (-3, 1) and (-8, 9) is calculated by subtracting the x-coordinates and y-coordinates of the two points.

5:4

Given two points, (x1, y1) and (x2, y2)

The ratio of the line segment joining the points is:

(x2 - x1) : (y2 - y1)

Therefore, the ratio of the line segment joining the points (-3, 1) and (-8, 9)

The ratio of the line segment joining the points (-3, 1) and (-8, 9) is calculated by subtracting the x-coordinates and y-coordinates of the two points. Specifically, the ratio is given by the expression (x2 - x1) : (y2 - y1). In this case, the x-coordinates are -3 and -8, and the y-coordinates are 1 and 9. By substituting these values into the expression, we get ( -8 - (-3)) : (9 - 1) which simplifies to ( -5 ) : ( 8 ). Therefore, the ratio of the line segment joining the points (-3, 1) and (-8, 9) is 5:4.

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In the given scenario the pairing is natural pairing, hence the samples are dependent because there is a natural pairing between the two samples.

A pairing function in mathematics is a technique to convert two natural numbers into one natural number while maintaining their uniqueness. Insets and rational numbers share the same attributes as natural numbers, which may be shown in set theory using any pairing function.

In the given situation, the pairing between the prior drug sample and later drug sample are conducted in the same men. The results will thus, be dependent as same men will be required for the research to be conducted, and the effect of the drug before will also be taken into consideration.

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There are two types of systems in mathematics: discrete systems, which are composed of distinct, separate elements, and continuous systems, which are composed of elements that are connected and can take on any value over a range.

1. Discrete systems: A system with distinct, separate elements, such as a set of integers or a graph.

2. Continuous systems: A system with elements that are connected and can take on any value over a range, such as a line or a curve.

1. Discrete systems: A discrete system is a system with distinct, separate elements. Examples include a set of integers, a graph, or a network of nodes. Discrete systems are usually studied using discrete mathematics.

2. Continuous systems: A continuous system is a system with elements that are connected and can take on any value over a range. Examples include a line, a curve, a function, or a surface. Continuous systems are usually studied using calculus.

There are two types of systems in mathematics: discrete systems, which are composed of distinct, separate elements, and continuous systems, which are composed of elements that are connected and can take on any value over a range. Discrete systems are studied using discrete mathematics, while continuous systems are studied using calculus.

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Answer:

4ft by 7 ft

Step-by-step explanation:

4*7=28 and 7-4=3 so the dimensions are 4ft by 7ft

The dimensions of the rectangle are 7 ft by 4 ft.

What is the area of the rectangle?

The area of a rectangle is defined as the product of the length and width.

Let x be the length of the rectangle.

The width of the rectangle is 3 less than the length, so the width can be represented as x-3.

The area of a rectangle is the length multiplied by the width, so:

Area = Length * Width

Area = x × (x-3)

Area = x² - 3x

We know that the area of the rectangle is 28 ft², so we can set the equation equal to 28 and solve for x:

x² - 3x = 28

x² - 3x - 28 = 0

Now we can use the quadratic formula or factoring method to find the solutions of x:

x = (3 ± √(3² - 41-28))/(2*1)

x = (3 ± √(9+112))/2

x = (3 ± √121)/2

x = (3 ± 11)/2

x = -4 or 7/2

The dimensions of the rectangle can't be negative, so the only valid solution is x = 7/2, which means the length of the rectangle is 7/2 ft and the width is (7/2 - 3) ft = -1/2 ft.

The width can't be negative, this answer is incorrect. So we have to check the other solution x = 7.

Therefore, the dimensions of the rectangle are 7 ft by 4 ft.

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The number of pencils that he bought will be 21 pencils.

The allocation of weights to the important variables that produce the calculation's optimum is referred to as a direct consequence.

Karani bought 4 pencils and 6 biro pens for $66 and Mary bought 2 pencils and 5 biro pens for $51.

Let 'x' be the cost of pencils and 'y' be the cost of biro pens. Then the equations are given as,

4x + 6y = 66         ...1

2x + 5y = 51

4y + 10y = 102      ...2

From equations 1 and 2, then we have

10y - 6y = 102 - 66

4y = 36

y = 9

Then the value of 'x' is given as,

2x + 5(9) = 51

2x + 45 = 51

2x = 6

x = 3

Ondieki spent $228 to buy the same type of pencils and biro pens. If the number of biro pens he bought was 4 more than the number of pencils.

Let 'm' and 'n' be the number of pencils and biro pens, respectively.

3m + 9n = 288          ...3

n = m + 4                   ...4

From equations 3 and 4, then we have

3m + 9(m + 4) = 288

12m + 36 = 288

12m = 252

m = 21

Then the value of 'n' is given as,

n = 21 + 4

n = 25

The number of pencils that he bought will be 21 pencils.

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X=5times (root2)/2

Step-by-step explanation:

Sin45=x/5

x=5sin45

x=5 times root2/2

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A number or algebraic expression is said to be in its simplest radical form and to be an nth root if it lacks any elements that are perfect nth powers under the radical.

The term "simplest radical form" refers to the most basic representation of a number or algebraic expression. When a number or algebraic expression contains no elements that are perfect nth powers under the radical, it is said to be in its simplest radical form and be an nth root.

When all of a radical expression's perfect squares, cubes, etc. are eliminated from the radical, the radical is in its simplest form. A radical is a phrase containing a root, most frequently a square or cube root.

Simply put, simplifying a radical eliminates the need to find any more square roots, cube roots, fourth roots, etc. when expressing it in its simplest radical form. Additionally, it entails eliminating any radicals from the denominator of a fraction.

Sin45 = x/5

x = 5 sin 45

x = 5 × (√2)/2

Therefore, the value of x be x = 5 × (√2)/2

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