Write and solve an equation using the constant of proportionality to answer each question.

The ratio between the number of children (c) on a field trip and number of teachers (t) on the trips is 14/3. There are 70 children on the field trip.how many teachers are on the trip?

Answer:

X || Z

Step-by-step explanation:

We are not told about any lines being perpendicular, so we cannot conclude any lines to be perpendicular.

Theorem:

If two lines are parallel to the same line, then the two lines are parallel to each other.

X is paralle to Y; Z is parallel to Y.

Line X and Z are parallel to the same line, Y, so lines X and Z are parallel.

Answer: X || Z

In mathematics, if lines X and Y are parallel, and lines Y and Z are parallel, then lines X and Z are also parallel. This is known as the transitive property.

In mathematics, when we say one line is parallel to another, we mean they are moving in the same direction and they will never intersect. In the case of X being parallel to Y and Y being parallel to Z, according to the transitive property in mathematics, it follows that X would be parallel to Z. To visualize this, imagine three straight lines on a piece of paper. If Line X and Line Y never meet and continue in the same direction, and Line Y and Line Z also follow the same rule, then logically, Line X and Line Z must also be continuing in the same direction and never intersecting, hence they are essentially parallel to each other.

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Answer:

57 children

54 adults

Step-by-step explanation:

Let's call x the number of children admitted and call z the number of adults admitted.

Then we know that:

[tex]x + z = 111[/tex]

We also know that:

[tex]3.25x + 4z = 401.25[/tex]

We want to find the value of x and z. Then we solve the system of equations:

-Multiplay the first equation by -4 and add it to the second equation:

[tex]-4x - 4z = -444[/tex]

[tex]3.25x + 4z = 401.25[/tex]

----------------------------------

[tex]-0.75x = -42.75[/tex]

[tex]x =\frac{-42.75}{-0.75}\\\\x=57[/tex]

Now we substitute the value of x in the first equation and solve for the variable z

[tex]57 + z = 111[/tex]

[tex]z = 111-57[/tex]

[tex]z = 54[/tex]

Answer:

Number of children=57

Number of adults=54

Step-by-step explanation:

We can start by forming simultaneous equations from the information provided.

Let the number of children be x and adults be y, then the the sum of the amount collected from both children and adults=3.25x+4y=401.25

The total number of people in attendance x+y=111

Let us solve these equations simultaneously.

3.25x+4y=401.25

x+y=111

Using substitution method.

y=111-x

3.25x+4(111-x)=401.25

3.25x+444-4x=401.25

-0.75cx=-42.75

x=57

Number of children=57

Adults=111-57

=54

Answer:

C) 1287 cubic feet

Step-by-step explanation:

To find the volume of a rectangular prism the formula is length time width times height. That means area=9*11*13 which is 1287 or C.

Answer:

[tex]\large\boxed{\dfrac{x^2-1}{x^2-x}=\dfrac{x+1}{x}=1+\dfrac{1}{x}}[/tex]

Step-by-step explanation:

[tex]\dfrac{x^2-1}{x^2-x}=\dfrac{x^2-1^2}{x(x-1)}\qquad\text{use}\ a^2-b^2=(a-b)(a+b)\\\\=\dfrac{(x-1)(x+1)}{x(x-1)}\qquad\text{cancel}\ (x-1)\\\\=\dfrac{x+1}{x}=\dfrac{x}{x}+\dfrac{1}{x}=1+\dfrac{1}{x}[/tex]

Answer:

its b on edge 2021

Step-by-step explanation:

Answer:

7, 3 and 0

Step-by-step explanation:

10, 3, 5, 7

Because the numbers are differ by prime numbers less than 10, i.e, the difference between the numbers are 7,5 and next will be 3.

8, 5, 4, 3.

The difference between the numbers are 3, 4 and similarly it will be differ by 5 which means next will be no. 3.

12, 6, 3, 0.

The numbers are differ by 6, 9 and next will be differ by 12 resulting the next no. 0.

Answer:

[tex]\frac{\sqrt{6}+\sqrt{2}}{4}[/tex]

Step-by-step explanation:

I'm going to write 105 as a sum of numbers on the unit circle.

If I do that, I must use the sum identity for sine.

[tex]\sin(105)=\sin(60+45)[/tex]

[tex]\sin(60)\cos(45)+\sin(45)\cos(60)[/tex]

Plug in the values for sin(60),cos(45), sin(45),cos(60)

[tex]\frac{\sqrt{3}}{2}\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\frac{1}{2}[/tex]

[tex]\frac{\sqrt{3}\sqrt{2}+\sqrt{2}}{4}[/tex]

[tex]\frac{\sqrt{6}+\sqrt{2}}{4}[/tex]

Sin 105 degrees is equivalent to (√6 - √2) / 4.

The exact value of sin 105 degrees can be determined using trigonometric identities. Knowing that sin (90 + θ) = cos θ, we can rewrite sin 105 degrees as sin (90 + 15) degrees.

Applying the identity, sin (90 + 15) degrees equals cos 15 degrees.

Utilizing the trigonometric values of common angles, cos 15 degrees can be expressed as (√6 - √2) / 4.

This value is derived from trigonometric relationships, providing an exact representation of sin 105 degrees without resorting to decimal approximations.

[tex]5x + 3x -4 = 10\\8x=14\\x=\dfrac{14}{8}=\dfrac{7}{4}[/tex]

one

Answer:0ne

Step-by-step explanation:

Answer:

Oops I went too far.

The other point is (9,4).

The average rate of change is 1/5.

Step-by-step explanation:

So I think your function is [tex]f(x)=\sqrt{x}+1[/tex]. Please correct me if I'm wrong.

You want to find the slope of the line going through your curve at the points (4,f(4)) and (9,f(9)).

All f(4) means is the y-coordinate that corresponds to x=4 and f(9) means the y-coordinate that corresponds to x=9.

So if [tex]f(x)=\sqrt{x}+1[/tex], then

[tex]f(4)=\sqrt{4}+1=2+1=3[/tex] and

[tex]f(9)=\sqrt{9}+1=3+1=4[/tex].

So your question now is find the slope of the line going through (4,3) and (9,4).

You can use the formula [tex]\frac{y_2-y_1}{x_2-x_1}[/tex] but I really like to just line up the points vertically and subtract then put 2nd difference over 1st difference. Like this:

( 9  ,   4)

-( 4  ,   3)

-------------

 5         1

So the slope is 1/5.

The average rate of change of the function f on the interval [4,9] is 1/5.

A function describes the relationship between related variables.

Given that:

[tex]f(x) = \sqrt x + 1,\ 4 \le x \le 9[/tex]

The average rate of change (m) is calculated as:

[tex]m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}[/tex]

[tex]m = \frac{f(9) - f(4)}{9 - 4}\\[/tex]

So, we have:

[tex]m = \frac{f(9) - f(4)}{5}[/tex]

Calculate f(4)

[tex]f(x) = \sqrt x + 1[/tex]

[tex]f(4) = \sqrt 4 + 1[/tex]

[tex]f(4) = 2 + 1[/tex]

[tex]f(4) = 3[/tex]

Calculate f(9)

[tex]f(x) = \sqrt x + 1[/tex]

[tex]f(9) = \sqrt 9 + 1[/tex]

[tex]f(9) = 3 + 1[/tex]

[tex]f(9) = 4[/tex]

So, we have:

[tex]m = \frac{f(9) - f(4)}{5}[/tex]

[tex]m = \frac{4 - 3}{5}[/tex]

[tex]m = \frac{1}{5}[/tex]

Recall that:

[tex]f(4) = 3[/tex] --- this represents the coordinate of the start interval

[tex]f(9) = 4[/tex] --- this represents the coordinate of the end interval

Hence, the coordinates of the end interval is (9,4)

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Answer:

Option C and E are correct.

Step-by-step explanation:

We need to solve the following equation

(2x+3)^2=10

taking square root on both sides

[tex]\sqrt{(2x+3)^2}=\sqrt{10}\\2x+3=\pm\sqrt{10}[/tex]

Now solving:

[tex]2x+3=\sqrt{10} \,\,and\,\,2x+3=-\sqrt{10}\\2x=\sqrt{10}-3 \,\,and\,\,2x=-\sqrt{10}-3\\x=\frac{ \sqrt{10}-3}{2} \,\,and\,\,x=\frac{-\sqrt{10}-3}{2}[/tex]

So, Option C and E are correct.

Answer: E .√10 - 3 / 2 or

c. -√10 - 3 / 2

Step-by-step explanation:

(2x + 3)^2 = 10

take the square root of bothside

√(2x + 3)^2 = ±√10

2x + 3 = ±√10

subtract 3 from bothside

2x = ±√10 - 3

Divide bothside by 2

x = ±√10 - 3 / 2

Either x = √10 - 3 / 2 or

x = -√10 - 3 / 2

Answer:

16x³ + 9x² + 9x + 13

Step-by-step explanation:

Given

6x³ + 8x² - 2x + 4 and 10x³ + x² + 11x + 9

Sum the 2 expressions by adding like terms, that is

= (6x³ + 10x³) + (8x² + x²) + (- 2x + 11x) + (4 + 9)

= 16x³ + 9x² + 9x + 13

Answer:

220 glasses.

Step-by-step explanation:

Quantity of fruit juice = 13 liters 200 ml

Number of glasses of each capacilty 60ml can be filled = ?

The quantity of fruit juice has 2 units. one is liter and the other is milliliter.

So we will convert liter into milliliter.

We have the quantity 13 liters 200 ml:

We know that:

1 liter = 1000 ml

Hence we have 13 liters so 13 will be multiplied by 1000 to convert it into milliliter.

13 * 1000 = 13000 ml

Now we have 13000 ml 200ml

Notice that we have two milliliters, so we will add both the quantities to make it one.

(13000+200)ml = 13200ml

Total quantity of fruit juice = 13200ml

Now divide the total quantity by the capacity of 60ml

=13200ml/60ml

= 220 glasses

It means that 220 glasses can be filled....

5(x - 2) - 3x + 7.

First use the distributive property:

5(x-2) = 5x-10

Now you have:

5x - 10 - 3x +7

Now combine like terms to get:

2x - 3

Answer:

2x -3

Step-by-step explanation:

5(x - 2) - 3x + 7

Distribute the 5

5x -10 -3x+7

Combine like terms

5x-3x   -10 +7

2x -3

Answer:

A.  s = 3/10 m.

Step-by-step explanation:

s = km  where k is a constant , s = the length of the string and m = the mass of the object.

Substituting m = 30 and s = 9:

9 = k * 30

k =  9/30 = 3/10

So the equation is s = 3/10 m.

Megan can have 7 different values of A for the number of 25c coins she possesses.

Given:

Megan has 25c coins and 10c coins with a total value of $1.95.

To find:

Number of different values of A (number of 25c coins) Megan can have.

Step-by-step solution:

Therefore, Megan can have 7 different values of A for the number of 25c coins she possesses.

Answer:

Step-by-step explanation:

Use FOIL: (a + b)(c + d) = ac + ad + bc + bd

(x - 4)(x + 3) = (x)(x) + (x)(3) + (-4)(x) + (-4)(3)

= x² + 3x - 4x - 12          combine like terms

= x² + (3x - 4x) - 12

= x² - x - 12

Answer:

Step-by-step explanation:

You have a dangling bracket. I'm not sure what to make of it. I will solve it as

(x - 4)(x + 3) if this is not correct, could you leave me a note.

x^2 + 3x - 4x - 12

x^2 - x - 12

If you meant

(x - 4(x + 3)) then it would be solved as

x - 4x - 12     combine the xs

-3x - 12

dangling brackets are to math what dangling modifiers are to English.

Running up a tree, I saw a squirrel.

If you mean anything but that you were running up a tree, the sentence is incorrect.

Answer:

254 and 180

Step-by-step explanation:

First of all we will define mode

"A mode is the most frequent value in the data"

In order to find mode the data is observed and the data value with most number of occurrence is called mode.

A data set can have more than one modes.

So in the given data,

The repeated data values are:

254 = two times

180 = two times

So the modes are 254 and 180 ..

Answer:

Mode of staff wages =   £180

, and  £254

Step-by-step explanation:

Points to remember

Mode of a data set is the most repeating item in the given data set.

To find the mode of staff wages

The given data set is,

£254, £254, £310, £276, £116, £90, £312, £180, £180, £536, £350, £243, £221, £165, £239, £700

Ascending order of data set

£90,  £116,  £165, £180, £180,  £221, £239, £243, £254, ,£254, £276,  £310,  £312, £350,  £536,  £700

Most repeating data = £180 , and  £254

Mode of staff wages =   £180 , and  £254

Answer:

The foci are located at [tex](0,\pm \sqrt{5})[/tex]

Step-by-step explanation:

The standard equation of an ellipse with a vertical major axis is [tex]\frac{x^2}{b^2}+\frac{y^2}{a^2}=1[/tex] where [tex]a^2\:>\:b^2[/tex].

The given equation is [tex]9x^2+4y^2=36[/tex].

To obtain the standard form, we must divide through by 36.

[tex]\frac{9x^2}{36}+\frac{4y^2}{36}=\frac{36}{36}[/tex]

We simplify by canceling out the common factors to obtain;

[tex]\frac{x^2}{4} +\frac{y^2}{9}=1[/tex]

By comparing this equation to

[tex]\frac{x^2}{b^2}+\frac{y^2}{a^2}=1[/tex]

We have [tex]a^2=9,b^2=4[/tex].

To find the foci, we use the relation: [tex]a^2-b^2=c^2[/tex]

This implies that:

[tex]9-4=c^2[/tex]

[tex]c^2=5[/tex]

[tex]c=\pm\sqrt{5}[/tex]

The foci are located at [tex](0,\pm c)[/tex]

Therefore the foci are [tex](0,\pm \sqrt{5})[/tex]

Or

[tex](0,-\sqrt{5})[/tex] and [tex](0,\sqrt{5})[/tex]

Answer: 1

Step-by-step explanation:

A formula for the slope of a line is rise/run.

First let’s find the ‘rise’

To find the rise, we must look at the difference between the elevation of the points. We can find the rise by subtracting y1 from y2. 7-3=4, so the rise is 4

Run is the same concept. We must subtract x2 - x1.

2-(-2)=2+2=4

So the rise is 4 and the run is 4. There for rise/run = 4/4 =1

The slope of the line is 1

Answer:

[tex]1.17*10^{10}[/tex]

Step-by-step explanation:

wee know that

To multiply two numbers in scientific notation, multiply their coefficients and add their exponents

In this problem we have

[tex](2.6*10^{1})*(4.5*10^{8})=(2.6*4.5)*10^{1+8}=11.7*10^{9}=1.17*10^{10}[/tex]

Answer:

The correct answer is 6-4i

Answer:

6 - 4i .

Step-by-step explanation:

I will assume that (10 - 51) is ( 10 - 5i) since the other number is a complex number.

You add the real parts and the imaginary parts separately.

(-4 + i) + (10 - 5i)

= (-4 + 10) + (i - 5i)

= 6 - 4i .

Answer:

3i[tex]\sqrt{5}[/tex]

Step-by-step explanation:

Using the rule of radicals

[tex]\sqrt{a}[/tex] × [tex]\sqrt{b}[/tex] ⇔ [tex]\sqrt{ab}[/tex]

and [tex]\sqrt{-1}[/tex] = i

Given

[tex]\sqrt{-45}[/tex]

= [tex]\sqrt{9(5)(-1)}[/tex]

= [tex]\sqrt{9}[/tex] × [tex]\sqrt{5}[/tex] × [tex]\sqrt{-1}[/tex]

= 3 × [tex]\sqrt{5}[/tex] × i

= 3i[tex]\sqrt{5}[/tex]

The probability of guessing a 3-digit pin correctly on the first attempt, when repetition of digits is allowed, is 1 out of 1000, or 0.001.

The subject under scrutiny is related to the concept of probability in mathematics. To solve this, we need to consider that there are 10 possible digits (0 to 9) on the keypad for each of the 3 input spots in the pin. Since a digit can be repeated, each spot has 10 possibilities. The total number of possible pin combinations is thus 10*10*10 = 1000.

The probability of guessing the pin correctly on the first attempt would be 1 (since there's only one correct pin) divided by the total number of possibilities, which is 1000. Therefore, the probability of this happening is 1/1000 or 0.001.

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The probability of a correct guess on the first try is 1/729.

In this scenario, the thief must randomly guess the correct three-digit pin code from a 9-key keypad where repetition of digits is allowed. To determine the probability of a correct guess on the first try, we need to calculate the ratio of favorable outcomes to total outcomes.

There are 9 possible digits to choose from, and repetition is allowed. Therefore, the number of outcomes is 9 raised to the power of 3 (since the thief needs to guess a three-digit pin code). This gives us 729 total outcomes. There is only one favorable outcome in this case, which is the correct pin code. Therefore, the probability of a correct guess on the first try is 1/729.

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Answer:

asa only because the two triangles are facing each other like a reflection making them congruent

ΔWZV and ΔWZY are congruent by AAS congruency

Considering    ΔWZV         and        ΔWZY

                      ∠WVZ            =            ∠WYZ  (given)

                        ∠WZV           =           ∠WZY   ( both angles are 90° since it is given that WZ is perpendicular to VY)

                           WZ is common side

So  ΔWZV and ΔWZY are congruent (AAS congruency)

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Answer:

2y = 3x

Step-by-step explanation:

general formula for a straight line is given as

y = mx + b

Given 2 points (2,3) and (4,6), we can use the attached formula to calculate the slope, m of the line which connects the 2 points

m = (6-3) / (4-2) = 3/2

hence the general formula becomes

y = (3/2) x + b

Substitute  one of the given points into this equation to find b.

We pick point (2,3)

3 = (3/2) (2) + b

b = 0

hence the equation is

y = (3/2)x

2y = 3x  (answer)

Answer:

the angle Ф is 63.9 degrees.

Step-by-step explanation:

A right triangle describes this situation.  The height of the triangle is 45 ft and the base is 22 ft.  The ratio height / base is the tangent of the angle of elevation and is tan Ф.  We want to find the angle Ф:

                opp

tan Ф = ---------- = 45 / 22 = 2.045.  

               adj

Using the inverse tangent function on a calculator, we find that the angle Ф is 63.9 degrees.

Answer:

First draw two axes x and y. Then mark all points for which x=4, this is a vertical line. Do the same for the other sides and you will find a square with side length 8.

For this case we have that by definition, the slope of a line is given by:

[tex]m = \frac {y_ {2} -y_ {1}} {x_ {2} -x_ {1}}[/tex]

We have the following points:

[tex](x_ {1}, y_ {1}) :( 3,2)\\(x_ {2}, y_ {2}): (- 2,3)[/tex]

Substituting we have:

[tex]m = \frac {3-2} {- 2-3} = \frac {1} {- 5} = - \frac {1} {5}[/tex]

Finally, the slope is:

[tex]m = - \frac {1} {5}[/tex]

Answer:

[tex]m = - \frac {1} {5}[/tex]

Answer:

67.972 ÷ 10 = 6.7972

679.72 ÷ 10 = 67.972

6797.2 ÷ 10 = 679.72

6797200 ÷ 10 = 679720

Area of a full circle is PI x r^2

Area = 3.14 x 8^2 = 200.96

Area of a sector is area *  central angle / full circle

Area = 200.96 x 5PI/3 / 2PI = 160PI/3

Answer is C.

Answer:[tex]\frac{160\pi }{3} units^2[/tex]

Step-by-step explanation:

Given

Angle subtended by sector is[tex]\frac{5\pi }{3}[/tex]

radius =8 units

[tex]Area of sector=\frac{\theta }{2\pi }\pi r^2 [/tex]

[tex]A=\frac{20\times 8\pi }{3}[/tex]

[tex]A=\frac{160\pi }{3}[/tex]

(4 - 7n) – (20+5)

Simplify:

(4 -7n) - 25

Remove parenthesis:

4 - 7n -25

Combine like terms:

4-25 = -21

Now you have:  -21-7n

The term with the variable is usually rearranged to be in front, so it becomes:  -7n-21