The length of a side of an equilateral triangle is 10 cm. Find the height of the triangle

Answer:

\(y=-\frac{4}{5}x-\frac{21}{5}\)

Step-by-step explanation:

The fastest way is to use point-slope form with \(m=-\frac{4}{5}\) and \((x_1,y_1)=(-4,-1)\):

\(y-y_1=m(x-x_1)\\y-(-1)=-\frac{4}{5}(x-(-4))\\y+1=-\frac{4}{5}(x+4)\\y+1=-\frac{4}{5}x-\frac{16}{5}\\y=-\frac{4}{5}x-\frac{21}{5}\)

To graph the line passing through (-4,-1) with slope m = -4/5, we can use the slope-intercept form of the equation of a line, which is:

y = mx + b

where m is the slope and b is the y-intercept.

Substituting m = -4/5, x = -4, and y = -1, we can solve for b:

-1 = (-4/5)(-4) + b

-1 = 3.2 + b

b = -4.2

Therefore, the equation of the line is:

y = (-4/5)x - 4.2

To graph the line, we can plot the given point (-4,-1) and then use the slope to find additional points. Since the slope is negative, the line will slope downwards from left to right. We can find the y-intercept by setting x = 0 in the equation:

y = (-4/5)x - 4.2

y = (-4/5)(0) - 4.2

y = -4.2

So the y-intercept is (0,-4.2).

Using this point and the given point (-4,-1), we can draw a straight line passing through both points.

Here is a rough sketch of the graph:

     |

     |

     |     *

     |    /

     |   /

     |  /

-----*--------

     |

     |

     |

     |

     |

The point (-4,-1) is marked with an asterisk (*), and the y-intercept (0,-4.2) is marked with a dash. The line passing through these two points is the graph of the equation y = (-4/5)x - 4.2.

One simple kind of transformation involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a vertical shift, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function.

The production schedule for maximum profit is; X = 3 and Y = 6 with a maximum profit of $960

We are told that  two products X and Y, has a total production capacity of 9 tones per day, X and Y requiring the same production capacity

The firm has a permanent contract to supply at least 2 tones of X and at least 3 tones of Y per day to another company.

Let product A be x and product B be y. Therefore we have the following inequalities and constraints as;

x + y ≤ 9

x ≥ 2

y ≥ 3

Now, we are told that each tonne of A requires 20 machine hours of production time and each tonne of B requires 50 machine hours of production time. The daily maximum possible number of machine hours is 360. Thus, we have;

20x + 50y ≤ 360

Wea re told that all the firm's output can be sold and the profit made is $80 per tonne of A and $120 per tonne of B. Thus, we have the inequality as;

Z = 80x + 120y  maximize

The solution from the graph attached is;

x = 3, y = 6

Thus, the maximum profit is;

Z = 80(3) + 120(6)

Z = 960

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Answer:

12.5

Step-by-step explanation:

In a rectangle, the two diagonals are equal length, so MK = JL = 2x + 9

Note MK = 2 * MN, so

2x + 9 = 2 * (3x + 1)

expand:

2x + 9 = 6x + 2

Solve:

9 - 2 = 6x - 2x

4x = 7

x = 7/4

MK = JL = 2x + 9 = 2 * 7/4 + 9 = 7/2 + 9 = 3.5 + 9 = 12.5

Answer:

Step-by-step explanation:

S.I =720×6×5/100×365

=21600/36500

=0.59178

So balance will be 720+0.59178

=720.59178

Given function :-

\(\bf \implies g(x) = \dfrac{3}{x}+7\)

\(\bf \implies y = \dfrac{3}{x}+7\)

\(\bf \implies x = \dfrac{3}{y}+7\)

\(\bf\implies \dfrac{3}{y}= x -7 \\\\\bf\implies y =\dfrac{3}{x-7} \)

\(\implies\boxed{\red{\bf f^{-1}(x) = \dfrac{3}{x-7} }}\)

Answer:

  (a)  twenty-eight ninths, square root of nineteen, cube root of eighty-eight

Step-by-step explanation:

When ordering a list of numbers by hand, it is convenient to convert them to the same form. Decimal equivalents are easily found using a calculator.

The attachment shows the ordering, least to greatest, to be ...

  \(\dfrac{28}{9}.\ \sqrt{19},\ \sqrt[3]{88}\)

__

Additional comment

We know that √19 > √16 = 4, and ∛88 > ∛64 = 4, so the fraction 28/9 will be the smallest. That leaves us to compare √19 and ∛88, both of which are near the same value between 4 and 5.

One way to do the comparison is to convert these to values that need to have the same root:

  √19 = 19^(1/2) = 19^(3/6) = sixthroot(19³)

  ∛88 = 88^(1/3) = 88^(2/6) = sixthroot(88²)

The roots will have the same ordering as 19³ and 88².

Of course, these values can be found easily using a calculator, as can the original roots. By hand, we might compute them as ...

  19³ = (20 -1)³ = 20³ -3(20²) +3(20) -1 = 8000 -1200 +60 -1 = 6859

  88² = (90 -2)² = 90² -2(2)(90) +2² = 8100 -360 +4 = 7744

Then the ordering is ...

  28/9 < 19³ < 88²   ⇒   28/9 < √19 < ∛88

Answer:

the ordering is

28/9 < 19³ < 88²   ⇒   28/9 < √19 < ∛88

Step-by-step explanation:

Answer:Y=x3

2nd answer: Just add it all up and divide by how many things you had to add. For an example: (2+2+2+2)/4.

Step-by-step explanation:

Answer:

UT = TW + WU = 27 + 27 = 54

WT = 27

ST = UT = 54

arc(XT) =  angle XZT = 52 degrees

arc(ST) = angle SZT = 104

arc(US) = arc(ST) + arc(UT) = 104 + 104 = 208

I know it was super drawn out and there were probably quicker ways to do it, but I was justusing methods that came to mind rather than going for efficiency.  let me know if any part is confusing.  

Step-by-step explanation:

Let's start by just building on what we know

Since we know sv and vz can be made into the side of two triangles, SZV and TZV, we can use some similar triangle rules.

In SZV and TZV we know SZ = TZ since they are both radii, ZV = ZV because they are the same line and angle ZVS = angle ZVT since they are supplementary and one is 90 degrees.  This allows us to solve and see if they are congruent or not.  To do so we use the normal trig functions

sin(ZTV) = ZV/ZT = ZV/12

sin(ZSV) = ZV/SV = ZV/12

So the angles could be equal, but just to be sure lets look at all possible answers.  with sine an angle x gives the same result as 180-x.  Since both angles are acute thugh we know this can't be true so both angles are equal.    Now we can say both triangles are congruent by AAS.  Since they are congruent we can then confirm SV = VT.

In general you can say if a radii intersects a chord at a right angle then the intersection is a bisection.  So now we know VT = 27

We can use similar methods to show UW = WT and if you made two triangles ZWT and ZWU you will see they are congruent.  and since arc UT is 104 degrees that means angle UZT is 104 degrees.  Since ZW bisects UT and the two triangles are congruent ZW also bisects angle UZT so UZW = TZW = 52 degrees.

Now we have 4 triangles, SZV, TZV, ZWT and ZWU and we know SZV and TZV are congruent and ZWT and ZWU are congruent.  Now, if we can make one from each pair congruent we would knwo all four are congruent.  Since TZV and ZWT share a side  that's a good place to start.  Looking at these two  triangles we know those two sides are equal, and they both have a right angle.  Also we know VZ = ZW fromt he instructions, so this is again a chance to check with trig.

Let's look at angle ZTW and ZTV.

sin(ZTW) = ZW/ZT = 27/ZT

sin(ZTV) = VZ/ZT = 27/ZT

Using the same reasoning as before we can say that the two angles are equal, so the two triangles are congruent.  Now we can say a lot.

angle UZT = angle SZT = 104 degrees

angle SZV = angle VZT = angle TZW = angle WZU = 52

SV = VT = TW = WU = 27

Let's start solving.

UT = TW + WU = 27 + 27 = 54

WT = 27

ST = UT = 54

arc(XT) =  angle XZT = 52 degrees

arc(ST) = angle SZT = 104

arc(US) = arc(ST) + arc(UT) = 104 + 104 = 208

Answer:

There are 20 nickels and 12 quarters.

Step-by-step explanation:

20 nickels = 1 dollar.

12 quarters = 3 dollars.

$1 + $3 = $4

There will be 20 nickels and 12 quarters in the pocket of Seo-jun.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that Seo-jun has 32 coins in his pockets. The coins consist of only nickels and quarters. The total value of the coins is $4.00.

The number of nickels and the quarters will be calculated as,

20 nickels = 1 dollar.

12 quarters = 3 dollars.

$1 + $3 = $4

Therefore, there will be 20 nickels and 12 quarters in the pocket of Seo-jun.

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Answer:

It is not a integer, it is a rational number

Answer: No, this isn't. An integer is any whole number, positive or negative. Having extraneous digits to the right of the decimal other than zero, radicals or any fractions are excluded from being considered an integer. Integers, for example, are 2, 85, -923, etc. Hope this helps!

Step-by-step explanation:

Let a=\(4x^{4} -20x^{2} -3x^{2} +15\) . So the resulting polynomial after factorization is a= \((4x^{2} -3)(x^{2} -5)\).

A polynomial is an expression that only uses the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. Variables are also known as indeterminates.

In an equation such as the quadratic equation, cubic equation, etc., a polynomial function is a function that only uses non-negative integer powers or only positive integer exponents of a variable. For instance, the polynomial 2x+5 has an exponent of 1.

Let

\(a=4x^{4}-20x^{2} -3x^{2} +15\\=4x^{4} -3x^{2} -20x^{2} +15\\=x^{2} (4x^{2} -3)-5(4x^{2} -3)\\=(4x^{2} -3)(x^{2} -5)\)

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The function of the power series f(x) = 6ln(1+x) = ∑=1 , ∞ 6(-1)⁽ⁿ⁻¹⁾xⁿ/n.

And a_n = 6(-1)⁽ⁿ⁻¹⁾xⁿ/n.

The distance between each number in a geometric series is known as the common ratio. The ratio between two consecutive numbers, or a number

divided by the number before it in the sequence, is known as the common ratio since it is the same for all numbers or common.

To find the power series for f(x), we first need to find the power series for g(x) = 6/(1+x):

g(x) = 6/(1+x) = 6(1 - x + x² - x³ +...) (geometric series with common ratio -x)

Next, we integrate term by term:

∫ g(x) dx = ∫ 6(1-x+x²-x³+...) dx

= 6(x - x²/2 + x³/3 - x⁴/4 + ...) + C

where C is the constant of integration.

Since we're only interested in finding the coefficients of the power series, we can ignore the constant term.

Thus, the power series for f(x) is:

f(x) = 6ln(1+x) = 6(x - x²/2 + x³/3 - x⁴/4 + ...) = ∑=1 , ∞ 6(-1)⁽ⁿ⁻¹⁾xⁿ/n

where a_n = 6(-1)⁽ⁿ⁻¹⁾xⁿ/n.

Therefore, f(x) = 6ln(1+x) = ∑=1 , ∞ 6(-1)⁽ⁿ⁻¹⁾xⁿ/n.

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Answer:

13

Step-by-step explanation:

4 ÷ 2 + (2 × 3 + 5) = 2 + (6 + 5)

                             = 2 + 11

                             = 13

Evaluate the multiplication and divisions first before evaluating the additions and subtractions in an equation. Brackets also have a priority of beign evaluated first.

Answer:

18

Step-by-step explanation:

4/2=2

2×(3+5)=16

16+2=18

The equation for the relationship between the number of cans of paint, x, and the total weight of the cans, y, in slope-intercept form is y = 6x + 0. This equation can be derived from the formula for the slope-intercept form of a line, y = mx + b, where m is the slope of the line and b is the y-intercept. In this case, since the weight of each can of paint is 6 pounds and there is no y-intercept (the line passes through the origin), the slope of the line is 6 and the y-intercept is 0.

A table of possible values for the number of cans of paint, x, and the total weight of the cans, y, based on this equation is shown below:

x (number of cans of paint) y (total weight of cans)

1 6

2 12

3 18

4 24

5 30

As the table shows, for each additional can of paint, the total weight of the cans increases by 6 pounds. This is because each can of paint weighs 6 pounds, and the weight of the cans is equal to the number of cans multiplied by the weight of each can.

Answer: y = 6x

Step-by-step explanation:

       For each can of paint (x) 6 pounds is added. This is represented as;

         y = 6x

       For the table of possible values, we will plug a value in for x, and then evaluate the function for y. See attached.

y = 6(1) = 6

y = 6(2) = 12

y = 6(3) = 18

Answer:

X/4=3/6

Start off by cross multiplying the tops and bottoms, so X/4=3/6 turns into 6X=12. Now divide by 6 on both sides.

6X=12      X=2

Answer:

2

Step-by-step explanation:

Cross Multiple and Divide

X. 3

----. = ----

4. 6

So what you want to do is multiply diagonally:

(6)(X) = 6X

(3)(4) = 12

Now that we have our two answers, we put an equal sign between them.

6X = 12

Now to get X alone. Since it's being multiplied by 6, you want to do the opposite and divide.

So remember to divide on both sides to get rid of the 6.

6X divided by 6 is just X.

12 divided by 6 is 2

We got our answers, put the equal sign in-between

X = 2

The length KL is 8 units

The value of x is undefined

The length KL is 12 units

From the question, we have the following parameters that can be used in our computation:

The rhombus

Also, we have

DK = 8

A rhombus is a quadrilateral with all sides equal.

So, we have

KL = 8

Here, we have

SKAL = 2x - 8

There is no point S on the rhombus

This means that

x = undefined

Here, we have

DM = 5y + 2 and DK = 3y + 6

A rhombus is a quadrilateral with all sides equal.

So, we have

5y + 2 = 3y + 6

Evaluate

2y = 4

Divide

y = 2

So, we have

KL = 5 * 2 + 2

KL = 12

Hence, the length KL is 12 units

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Answer:

Step-by-step explanation:

Either way works. It comes to the same thing.

(2x-5)(x+1)=9

F: 2x*x = 2x^2

O: 2x*1 = 2x

I: -5 * x = -5x

L: -5*1 = -5

2x^2 + 2x - 5x - 5 = 9    Subtract 9 from both sides

2x^2 - 3x - 14 =0            Divide by 2 (it makes life easier).

Factoring you get

(2x - 7)(x + 2)

2x - 7 = 0

2x = 7

x = 3.5

x + 2 = 0

x = -2

The x intercepts = (3.5,0) and (-2,0)

I've put a graph up for the given equation.

Answer:

1/2 = 4/8 so  7/8- 4/8 = 3/8

Step-by-step explanation:

Answer:

3/8

Step-by-step explanation:

find greatest common factor wich will be 8, u need to multiply 2 by another number to get the denominator of 8

8÷2=4 so 2×4=8

multiply 1/2 by 4 to get a denominator of 8 and numerator of 4

now u have 7/8 -4/8 =3/8

The only matrix that is certainly symmetric is a)\((a^2 - b^2).\)

We know that a matrix A is symmetric if it is equal to its transpose, that is \(A = A^T.\)

a) \((a^2 - b^2)\)

We can write the transpose of this matrix as

\((a^2 - b^2)^T = (a^2)^T - (b^2)^T = a^2 - b^2\), which is equal to the original matrix. Therefore, this matrix is symmetric.

b) (A+B)(A-B)

Expanding this expression, we get \((A+B)(A-B) = A^2 - AB + BA - B^2.\)

Taking the transpose of this, we get \((A^2)^T - (AB)^T + (BA)^T - (B^2)^T = A^2 - BA + AB - B^2.\)

Since AB and BA may not be equal, we cannot say for certain that this matrix is symmetric.

c) ABA

Taking the transpose of this matrix, we get\((ABA)^T = A^T B^T A^T\). Since \(A^T and B^T\) may not commute, we cannot say for certain that this matrix is symmetric.

d) ABAB

Taking the transpose of this matrix, we get \((ABAB)^T = B^T A^T B^T A^T\). Again, since \(A^T and B^T\) may not commute, we cannot say for certain that this matrix is symmetric.

Therefore, the only matrix that is certainly symmetric is a) \((a^2 - b^2).\)

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Answer:

12

Step-by-step explanation:

The value of x = 16.5.

In Plane Geometry, a figure which is formed by two rays or lines that shares a common endpoint is called an angle. The two rays are called the sides of an angle, and the common endpoint is called the vertex.

We know that, a ray is a line that continues on forever inn one direction with a point at it's start.

Here, given that,

m∠8=(4x+7)

m∠2=107

from that given figure,

we get,

m∠2=m∠4=107

and, m∠4 + m∠8 =180

m∠8=73

(4x+7)=73

4x=66

x=16.5

hence, The value of x = 16.5.

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Answer:

See explanation

Step-by-step explanation:

sin(α + β) = sin α cos β + sin β cos α

sin(α - β) = sin α cos β - sin β cos α

\(\frac{1}{2}\)[ sin(α + β) + sin(α - β)] = \(\frac{1}{2}\)[sin α cos β + sin β cos α + sin α cos β - sin β cos α]

                               = \(\frac{1}{2}\)[2 sinα cos β]

                               = sin α cos β = \(\frac{1}{2}\)[ sin(α + β) + sin(α - β)]

Answer:  19/21 or 0.90 rounded

Step-by-step explanation  

First step: subtract 2/3 from 3 1/5 which equals 2 8/15

Second step is divide 2 8/15 by 2 4/5 which equals 19/21

Answer: The answer is y=10.

Step-by-step explanation:

4y + 10 = -10 + 6y First subtract 4y from both sides.

10 = -10 + 2y Then, add 10 to both sides.

20 = 2y Finally, divide both sides by 2.

y = 10.

Answer:

\((\frac{1}{2}+\frac{\sqrt{3}}{2}i)^5=\frac{1}{2}-\frac{\sqrt{3}}{2}i\)

Step-by-step explanation:

Convert 1/2 + i√3/2 to rectangular form

\(\displaystyle z=a+bi=\frac{1}{2}+\frac{\sqrt{3}}{2}i\\\\r=\sqrt{a^2+b^2}=\sqrt{\biggr(\frac{1}{2}\biggr)^2+\biggr(\frac{\sqrt{3}}{2}\biggr)^2}=\sqrt{\frac{1}{4}+\frac{3}{4}}=\sqrt{1}=1\\\\\theta=\tan^{-1}\biggr(\frac{b}{a}\biggr)=\tan^{-1}\biggr(\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}\biggr)=\tan^{-1}(\sqrt{3})=\frac{\pi}{3}\\\\z=\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\)

Use DeMoivre's Theorem

\(\displaystyle z^n=r^n(\cos(n\theta)+i\sin(n\theta))\\\\z^5=1^5\biggr(\cos\biggr(\frac{5\pi}{3}\biggr)+i\sin\biggr(\frac{5\pi}{3}\biggr)\biggr)\\\\z^5=\frac{1}{2}-\frac{\sqrt{3}}{2}i\)

The midpoint of segment YZ is (-a, -2b).

Given the coordinates of the rhombus WXYZ:

W(0, 4b)

X(2a, 0)

Y(0, -4b)

Z(-2a, 0)

The midpoint formula is given by:

Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)

Substituting the coordinates of Y and Z:

Midpoint of YZ = ((0 + (-2a)) / 2, (-4b + 0) / 2)

= (-a, -2b)

Therefore, the midpoint of segment YZ is (-a, -2b).

To demonstrate that the segments joining the midpoints of the rhombus are perpendicular, we need to prove that the slopes of these segments are negative reciprocals of each other.

Let's consider the segments joining the midpoints:

Segment joining the midpoints of WX and YZ:

Midpoint of WX: ((0 + 2a) / 2, (4b + 0) / 2) = (a, 2b)

Midpoint of YZ: (-a, -2b)

Slope of WX = (2b - 4b) / (a - 0) = -2b / a

Slope of YZ = (-2b - (-4b)) / (-a - 0) = 2b / a

The slopes of WX and YZ are negative reciprocals of each other, indicating that these segments are perpendicular.

We have shown that the segments joining the midpoints of a rhombus are perpendicular to each other and have equal lengths. Therefore, these segments form a rectangle.

Additionally, the midpoint of segment YZ is (-a, -2b).

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Answer:

x=2

Step-by-step explanation:

2 times 2 = 4. 4 times 2 =8. 8+4=12. 12-4=8

With the help of probability we can say that, DUM={GG,BB,GB,BG).

Probability is the concept that describes the likelihood of an event occurring.

In real life, we frequently have to make predictions about how things will turn out.

We may be aware of the result of an occurrence or not.

When this occurs, we state that there is a possibility that the event will occur.

In general, probability has many excellent applications in games, commerce, and this newly growing area of artificial intelligence

The chance of an event can be calculated using the probability formula by only dividing the favourable number of possibilities by the total number of potential outcomes.

hence, DUM={GG,BB,GB,BG)

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Answer:

D

Step-by-step explanation:

m^ (-3 · -4) = m^12

n ^ (-2 · -4) = n^8

m^12 n^8