Find a unit vector with positive first component that is orthogonal to the plane passing through the pointsP=(4,5,−7),Q=(13,14,2), andR=(13,14,8)

Answer:

see below

Step-by-step explanation:

We need to solve the below two equations by graphing. We will plot the graph of the given equations and the point of intersection of line would be the solution to the equation.

[tex] 1. \begin{cases} y = -\dfrac{1}{2}x+2\\\\ y = -3x - 3 \end{cases}[/tex]

Firstly let's make a table;

[tex]\boxed{\begin{array}{c|c|c} x & 0 & 1 \\ y & 2 & 1.5 \end{array}}[/tex]

Now plot the points , (0,2) and (1,1.5) .

Similarly make another table for second equation [tex] y = -3x - 3[/tex] as ,

[tex]\boxed{\begin{array}{c|c|c} x & 0 & 1 \\ y & -3 & -6\end{array}}[/tex]

Now draw the lines joining the points. Hence from the graph the line intersects at (-2,3)

So the solution to equations are ,

_________________________________

Similarly we will solve the second set of equations as ,

[tex] 2. \begin{cases} x-4y=12\\\\ 7x-4y=12 \end{cases}[/tex]

make a table for plotting equation [tex] x-4y=-12[/tex]

[tex]\boxed{\begin{array}{c|c|c} x & 12 & 16 \\ y & 0 & 1\end{array}}[/tex]

make a table for plotting equation [tex] 7x-4y=12[/tex]

[tex]\boxed{\begin{array}{c|c|c} x & 0 & 2 \\ y & -3 & 0.5\end{array}}[/tex]

Now we see that the equations intersect at point (4,4) So the solution to the given equation is ,

________________________________

[tex] 3. \begin{cases} -8x-4y = 12 \dots (1)\\\\ x=y\dots(2)\end{cases}[/tex]

substitute value of equation 2 in 1

[tex]\longrightarrow -8x - 4x = 12 \\ [/tex]

simplify,

[tex]\longrightarrow -12x = 12[/tex]

Divide both sides by 12 ,

[tex]\longrightarrow\underline{\underline{ x = -1}} [/tex]

Substitute this value in equation 2 as ,

[tex]\longrightarrow x = y \\ [/tex]

[tex]\longrightarrow\underline{\underline{ y = -1}} [/tex]

Hence our required solution is ,

___________________________________

[tex] 4. \begin{cases} 2x + 4y = -10 \dots (1)\\\\ -7x -y = 22(2)\end{cases}[/tex]

Taking equation 2 ,

[tex]\longrightarrow y = -7x -22 [/tex]

substitute this value in equation 1,

[tex]\longrightarrow 2x + 4(-7x-22)=-10\\[/tex]

[tex]\longrightarrow 2x -28x -88 = -10\\ [/tex]

[tex]\longrightarrow -26x = 78\\[/tex]

divide both sides by 26,

[tex]\longrightarrow\underline{\underline{ x = -3}} [/tex]

Again, substitute this value in equation 2 , as ;

[tex]\longrightarrow y = -7(-3) -22\\ [/tex]

[tex]\longrightarrow y = 21 -22\\ [/tex]

[tex]\longrightarrow\underline{\underline{ y = -1}} [/tex]

hence our required solution is ,

And we are done!

Answer:

h = F/4g

Step-by-step explanation:

Answer:

21°

Step-by-step explanation:

For this right-angled triangle:

∠M = angle in question

Side LM = Hypotenuse

With respect to ∠M:

Side LN = Opposite side

Side MN = Adjacent side

Trigonometric function applied:

sin∠M = [tex]\frac{opposite}{hypotenuse}[/tex]

           = [tex]\frac{8}{22}[/tex]

 ∠M = [tex]sin^{-1} (\frac{8}{22})[/tex]

          = 21.32°

  ∠M  = 21° (Rounded to the nearest degree)

Answer:

∠M = 21°

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{9.4 cm}\underline{Trigonometric ratios} \\\\$\sf \sin(\theta)=\dfrac{O}{H}\quad\cos(\theta)=\dfrac{A}{H}\quad\tan(\theta)=\dfrac{O}{A}$\\\\where:\\ \phantom{ww}$\bullet$ $\theta$ is the angle. \\ \phantom{ww}$\bullet$ $\sf O$ is the side opposite the angle. \\\phantom{ww}$\bullet$ $\sf A$ is the side adjacent the angle. \\\phantom{ww}$\bullet$ $\sf H$ is the hypotenuse (the side opposite the right angle). \\\end{minipage}}[/tex]

From inspection of the given right triangle:

Therefore, substitute the given values into the sine ratio and solve for ∠M:

[tex]\implies \sin(M)=\dfrac{8}{22}[/tex]

[tex]\implies M=\arcsin \left(\dfrac{8}{22}\right)[/tex]

[tex]\implies M=21.323686...^{\circ}[/tex]

[tex]\implies M=21^{\circ}\sf\;(nearest\;degree)[/tex]

Therefore, the measure of ∠M to the nearest degree is 21°.

The level of scale of measurements is the ratio scale for the data of the average weight of newborn babies in ounces.

A ratio scale is a numerical scale with a genuine zero and uniform distances between adjacent points. A zero on a ratio scale, in contrast to an interval scale, indicates complete lack of the variable being measured. Examples of ratio scales include length, area, and population.

As, There are four scales of measurement: Nominal, Ordinal, Interval, and Ratio.

First, The categories in this scale are given names, hence the label "nominal." There is no innate hierarchy among the categories. Simply said, it is impossible to say that one category is superior to or better than another.

Ordinal scale: It is possible to rationally organise the various categories in a meaningful order. The distinction between the categories is not "meaningful," nevertheless.

Interval scale: Doubling has no significance, but values (not categories) can be arranged and have substantial differences. The lack of a "absolute zero" is the reason of this.

Scale of ratios: The values can be arranged in a way that makes a meaningful difference, and doubling also makes a difference. "Absolute zero" exists.

As a result, the ratio scale for the level of measurements is used to determine the average weight of newborn babies.

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The average rate of change of the function in the interval [ 4 , 9 ] is given by the equation m = -67

The average amount that a function changed per unit during that time period is the rate of change of that function. It is calculated using the slope of the line connecting the interval's endpoints on the graph of the function.

Rate of change = f ( b ) - f ( a ) / ( b - a )

Given data ,

Let the equation of line be represented as A

Now , the value of A is

Let the first point be P ( 4 , -84 )

Let the second point be Q ( 9 , -419 )

Now , the slope of the line is m = ( y₂ - y₁ ) / ( x₂ - x₁ )

Substituting the values in the equation , we get

The rate of change of the function = slope

Slope m = ( -419 - ( -84 ) ) / ( 9 - 4 )

On simplifying the equation , we get

Slope m = ( -419 + 84 ) / 5

Slope m = -335/5

Slope m = -67

Hence , the rate of change of function is m = -67

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Answer:

12

Step-by-step explanation:

6 - 3/4x + 1/3 = 1/2x + 5

6 - 9/12x + 4/12 = 6/12x + 5  If I multiply all the way through by 12 I will eliminate the fraction

72 - 9x + 4 = 6x+ 60

The coordinates of the vertices of the image include the following: (-1, 3), (-1, -1), (5, -1), and (7, 3).

In this exercise, we would have to dilate (enlarge) the coordinates of the pre-image by using a scale factor of 2 centered at the point (1, 5) by using this mathematical expression:

(x, y)  →  (k(x - a) + a, k(y - b) + b)

For coordinate 1, we have;

Coordinate 1 = (2, 4)  →  (2(2 - 1) + 1, 2(4 - 5) + 5)

Coordinate 1 = (2, 4)  →  (2(-1) + 1, 2(-1) + 5)

Coordinate 1' = (-1, 3).

For coordinate 2, we have;

Coordinate 2 = (2, 2)  →  (2(2 - 1) + 1, 2(2 - 5) + 5)

Coordinate 2 = (2, 2)  →  (2(-1) + 1, 2(-3) + 5)

Coordinate 2' = (-1, -1).

For coordinate 3, we have;

Coordinate 3 = (3, 2)  →  (2(3 - 1) + 1, 2(2 - 5) + 5)

Coordinate 3 = (3, 2)  →  (2(2) + 1, 2(-3) + 5)

Coordinate 3' = (5, -1).

For coordinate 4, we have;

Coordinate 1 = (4, 4)  →  (2(4 - 1) + 1, 2(4 - 5) + 5)

Coordinate 1 = (4, 4)  →  (2(3) + 1, 2(-1) + 5)

Coordinate 1' = (7, 3).

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The value of the equation is A = 50 ( n ) , where n is the number of employees

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the equation be represented as A

Now , the value of A is

Substituting the values in the equation , we get

Let the number of employees be n

So , A = product of 50 and the number of employees

A = 50 ( n )   be equation (1)

Therefore , the value of A is 50n

Hence , the equation is A = 50n

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The confidence interval: 59.5 to 61.5; margin of error: ± 1.0. Option B is the correct answer.

The confidence interval (CI) is a range of values that, with a particular level of confidence, is likely to include a population value. A population mean that sits between an upper and lower interval is frequently stated as a percentage.

The confidence interval is given as:

x' ± 1.96 s/√n

Substituting the value of x' = 60.5, s = 5, and n = 100.

60.5 ± 1.96 (5/ √√100)

60.5 ± 1.96 (5/10)

60.5 ± 0.98

Hence, the confidence interval: 59.5 to 61.5; margin of error: ± 1.0. Option B is the correct answer.

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Answer:

They would need 35 cups of rice.

Step-by-step explanation:

4 x 5 = 20

7 x 5 = 35

Answer:

11.3

Step-by-step explanation:

13.3 - 2 = 11.3

the missing species in each nuclear equation are:

1.  ¹⁰₄Be ---> ¹⁰₅B + ⁰₋₁β

2. ³⁴₁₄Be ---> ³⁴₁₅P + ⁰₋₁β

3. ¹⁹²₇₈Pt -----> ¹⁹⁰₇₆Os + ⁴₂α

4.  ²⁸₁₂Mg ---> ²⁸₁₃Al + ⁰₋₁β

Nuclear equations represent the reactants and products in radioactive decay, nuclear fission, or nuclear fusion. Instead of chemical equations where it shows the different number of elements are conserved in a reaction, in a nuclear reaction the atomic mass and proton number are conserved.

1. In the first equation, the Beryllium-10 isotope undergoes beta decay, emitting a beta-particle to form the boron-10 isotope. The balanced nuclear equation is given below:

¹⁰₄Be ---> ¹⁰₅B + ⁰₋₁β

2. In this reaction, the silicon-34 isotope undergoes beta decay, emitting a beta-particle to form the phosphorus-34 isotope. The balanced nuclear equation is given below:

³⁴₁₄Be ---> ³⁴₁₅P + ⁰₋₁β

3. In this equation, the platinum-192 isotope undergoes alpha-particle decay emitting an alpha-particle to form the osmium-190 isotope. The balanced nuclear equation is given below:

¹⁹²₇₈Pt -----> ¹⁹⁰₇₆Os + ⁴₂α

4. In this equation, the magnesium-28 isotope undergoes beta decay, emitting a beta-particle to form the aluminum-28 isotope. The balanced nuclear equation is given below:

²⁸₁₂Mg ---> ²⁸₁₃Al + ⁰₋₁β

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The value of the given expression is 64ab.

Expressions are mathematical statements which consist of two or more terms and terms are connected to each other using mathematical operators like addition, multiplication, subtraction and so on.

Given a mathematical expression which includes the basic operations of multiplication and subtraction.

There are two terms for the expression, (-2a × 4b) and (-12a × 6b).

-2a × 4b = -8ab

-12a × 6b = -72ab

(-2a × 4b) - (-12a × 6b) = (-8ab) - (-72ab)

Two negatives become positive.

(-8ab) - (-72ab) = -8ab + 72ab

Both are like terms since the variables a and b are same. Just subtract it.

-8ab + 72ab = 64ab

Hence the solution of the difference of the expression is 64ab.

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Answer:

2

Step-by-step explanation:

mangoes left=10-5-2-1=2

On solving the provided question, we can say that the equation of inequality will be as follows 150x + 100y [tex]\leq[/tex] 300

An inequality in mathematics is a relationship between two expressions or values that is not equal. Thus, imbalance leads to inequality. An inequality creates the link between two values that are not equal in mathematics. Egality is distinct from inequality. When two values are not equal, most commonly use the not equal sign (). Different inequalities are used to contrast values, no matter how little or large. Many simple inequalities may be resolved by modifying the two sides until the variables are all that remain. But a number of things contribute to inequality: Negative values on both sides are divided or added. Trade off the left and right.

A patient is not allowed to have more than 300 milligrams of cholesterol per day from a diet of eggs and meat.

Each egg provides 150 milligrams of cholesterol.

Each once of meat contains 100 milligrams of cholesterol

so, the equation of inequality will be as follows

150x + 100y [tex]\leq[/tex] 300

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The missing side lengths are given as follows:

The three trigonometric ratios are defined as follows:

Considering the tangent of 30º, we can obtain the length y, as follows:

tan(30º) = y/12

y = 12 x tangent of 30 degrees

y = 6.93.

Applying the Pythagorean Theorem, we can obtain the length x, as follows:

x² = 6.93² + 12²

x = sqrt(6.93² + 12²)

x = 13.86.

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the system is inconsistent Infinitely many solutions.

What is matrix?

The plural version of the word matrix is a matrix, which refers to the arrangements of numbers, variables, symbols, or phrases in a rectangular table with varying numbers of rows and columns. These arrays have a rectangular shape, and several operations like addition, multiplication, and transposition are specified for them. The components of the matrix are referred to as its entries or numbers. Vertical and horizontal entries in matrices are referred to as columns and rows, respectively. A matrix with m rows and n columns will contain m n entries. The uppercase letter 'A' stands for "matrix," Aij.

The given matrix is [tex]\left[\begin{array}{ccccc}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\0&0&0&0&0\end{array}\right][/tex]

This is a linearly dependent matrix a row is 0.

Hence, the system is inconsistent Infinitely many solutions.

Hence the element s₂₃ of the given matrix is -2

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The solution will lie on the number line from (-∞ , 0)

An inequality is the representation of an algebraic statement with two non-equal values. One of the two variables can be larger than, more than or equal to, smaller than, or equal to another value by using a sign of inequality.

One of the following options is available for resolving an inequality: Add the same amount to each side, deduct the same amount from each side, multiply each side by the same positive number, or split each side in half. If you multiply or divide each side by a negative value, the inequality sign must be reversed.

Given,

(x-0)(x-6)<0

⇒ x < 0 and x < 6

the solution will lie on the number line from -∞ to 0.

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To maximize the store's profit, Hector should aim to sell as many lemons as possible while paying the lowest price possible for them. Since Hector estimates that he can sell a maximum of -2000p + 7540 lemons at price p, and Walt can supply at most 5000 lemons, Hector should buy all of the 5000 lemons from Walt.

To determine the price at which Hector should buy the lemons from Walt, we need to consider the average cost of the lemons. If Hector buys 5000 lemons at $1.60 per lemon, he will spend a total of 5000 x $1.60 = $8000 on lemons.

If he sells all 5000 lemons at the price p, his revenue will be (-2000p + 7540) x 5000 = -10,000,000p + 37,700,000.

His profit will be the difference between revenue and cost, which is given by:

Profit = -10,000,000p + 37,700,000 - $8000

To maximize profit, we need to take the derivative of the profit function with respect to p and set it equal to zero:

dProfit/dp = -10,000,000 = 0

Solving for p, we get:

p = $1.00

Therefore, Hector should buy 5000 lemons from Walt at $1.00 per lemon to maximize the store's profit.

The formula cosec (A + B) = sinA cosB - cosA sinB ÷ (sinA + sinB)(sinA - sinB) can be proven using the trigonometric identities.

Starting with the definition of cosecant:

cosec (A + B) = 1/sin (A + B)

Using the sum-to-product identity:

sin (A + B) = sinA cosB + cosA sinB

Substituting this expression into the definition of cosecant:

cosec (A + B) = 1 / (sinA cosB + cosA sinB)

Using the difference-to-product identity:

sinA - sinB = 2sin((A - B)/2)cos((A + B)/2)

Dividing both numerator and denominator by 2sin((A - B)/2)cos((A + B)/2), we get:

cosec (A + B) = (sinA cosB - cosA sinB) ÷ (sinA + sinB)(sinA - sinB)

Therefore, the formula cosec (A + B) = sinA cosB - cosA sinB ÷ (sinA + sinB)(sinA - sinB) has been proven.

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Answer:

6(f-2)

Step-by-step explanation:

is a factor of 6f & 12

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1.

to get factors

write 1 to 10

to see if any of the factor numbers can be divided by any of the numbers between 1 thru 10 or sometimes even up to 20

2.

write

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

so 6 & 12 can be divided by

1 2 3 4 6 12

Answer:

564 inches is equivalent to 47 yards and 0 feet.

Malia can't buy 75 water bottles and 100 pairs of socks because it is not a solution to the system.

Here we have a graph of inequalities, and we just need to see if the pair (75, 100) is a solution.

The solutions of the system are all the points in the intersection of the two shaded areas (in this case are all the points in the green area)

Now, clearly the point (75, 100) is not a solution

Even if you evaluate that in the first inequality:

x + y ≥ 200

we get:

75 + 100 ≥ 200

175 ≥ 200

which is false.

So no, Malia can't buy 75 water bottles and 100 pairs of socks.

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Only Euna successfully constructed an equilateral triangle inscribed in a a circle.

An equilateral triangle's sides are congruent, which indicates that they are all the same length. The inner angles are all equal (60°), as well. Equilateral is a compound word made up of the words "equi," which means equal, and "lateral," which means side. Thus, an equilateral triangle is one that has three sides of equal length. An equilateral triangle in geometry is a triangle whose three sides are equal in length. An equilateral triangle is also equiangular in the conventional Euclidean geometry, meaning that each of the three internal angles is 60 degrees and that all three are congruent with one another.

So, Euna successfully constructed an equilateral triangle inscribed in a a circle.

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For the given scenario the mean changes from 80.42 to 82.

In maths, the average value in a set of numbers is the middle value, calculated by dividing the total of all the values by the number of values. When we need to find the average of a set of data, we add up all the values and then divide this total by the number of values.

Given that, the average high temperatures in degrees for a city are 58, 61, 71, 77, 91, 100, 105, 102, 95, 82, 66, 57.

Now, average = (58+61+71+77+91+100+105+102+95+82+66+57)/12

= 965/12

= 80.42

If a value of 101° is added to the data, then the new average is

(965+101)/(12+1)

= 1066/13

= 82

Therefore, for the given scenario the mean changes from 80.42 to 82.

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Answer:

Step-by-step explanation:

tan L = 8/15

answer: 39.5

Step-by-step explanation:

arrange numbers in ascending order....

5,22,31,38,41,55,75,69,80,83,89,91,96,97,100

then find median of the arranged list of numbers

which is 69

after that find the median between the arranged numbers starting from 5 to the median which we found as 69

which will be 38 and 41

So we add those two numbers together and then divide them by 2 .....which will give u 39.5

Before solving for fractions, you must know what you're doing. This will list information you should know to solve fraction problems.

A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters

[tex]\frac{5}{12} + \frac{1}{4}[/tex] is our equation, so let's solve this step by step,

Common Denominator

As you can see, the fractions have unlike denominators. An unlike denominator is when 2 or more fractions have different denominators. One of our denominators is 12 and the other is 4, so this shows we have unlike denominators. The way we get the denominators to be the same is multiplying. Its a quicker method to easily make them the same. We can also skip count until the numbers meet:

12: 12,

4: 4, 8, 12

The fraction denominator can be 12. We can also prove that it is 12 by showing the Least Common Denominator, or LCD. LCD (5/12, 1/4) = 12.

[tex]\frac{5}{12}[/tex] × [tex]\frac{1}{1}[/tex] = [tex]\frac{5}{12}[/tex] . We multiply this by [tex]\frac{1}{1}[/tex] because the denominator is already 12.

[tex]\frac{1}{4}[/tex] × [tex]\frac{3}{3}[/tex] = [tex]\frac{3}{12}[/tex].  We multiply this by [tex]\frac{3}{3}[/tex]  so we can make the denominator 12.

Adding the fractions

Now that we have our equation, [tex]\frac{5}{12}[/tex]  + [tex]\frac{3}{12}[/tex], we can easily add the fractions.

We do not add the denominators, however, we do add the numerators. We can express this as [tex]\frac{5+3}{12}[/tex]. 5 + 3 = 8. So, the numerator is 8. This means our answer is [tex]\frac{8}{12}[/tex].

Reduce

As your question says the sum must be in simplest form, you must reduce the answer to get it in simplest form. To reduce, you must divide. [tex]\frac{8}{12}[/tex] can be reduced, or divided by 4 to get it to simplest form. [tex]\frac{8}{12}[/tex] ÷ [tex]\frac{4}{4}[/tex] = [tex]\frac{2}{3}[/tex].

Answer

Simplest form: [tex]\frac{2}{3}[/tex]

Not in its simplest form: [tex]\frac{8}{12}[/tex]