How do you use theorems about triangles to solve problems?

Answer:

The conjecture is that the sum is [tex]30^2+30=930[/tex].

Step-by-step explanation:

I don't see your table... but let's see if we can make a conjecture about the sum of the first 30 positive even numbers.

What is the sum of the first even number? 2=2

What is the sum of the first two even numbers? 2+4=6

What is the sum of the first three even numbers? 2+4+6=12

What is the sum of the first four even numbers? 2+4+6+8=20

What is the sum of the first five even numbers? 2+4+6+8+10=30

What is the sum of the first six even numbers? 2+4+6+8+10+12=42

Alright, let's stop there for a second.

So we have the following sequence of numbers to find a pattern for:

2,6,12,20,30,42,...

Let's look at the common differences:

6-2 ,  12-6 , 20-12 , 30-20, 42-30,...

 4  ,    6     ,  8      ,   10     , 12

No common difference here so let's move on too the second common differences:

6-4  ,  8-6,   10-8,   12-10

 2   ,    2  ,     2  ,     2

So there is a 2nd common difference which means the pattern is a quadratic.

So our expression is of the form [tex]ax^2+bx+c[/tex]

Let's plug in our numbers to come up with a system to solve:

If x=1 , then [tex]ax^2+bx+c=2[/tex]

That is, [tex]a(1)^2+b(1)+c=2[/tex] .

Simplifying this gives: [tex]a+b+c=2[/tex].

If x=2, then [tex]ax^2+bx+c=6[/tex]

That is, [tex]a(2)^2+b(2)+c=6[/tex]

Simplifying this gives: [tex]4a+2b+c=6[/tex].

If x=3, then [tex]ax^2+bx+c=12[/tex]

That is [tex]a(3)^2+b(3)+c=12[/tex]

Simplifying this gives: [tex]9a+3b+c=12[/tex].

So we have this system of equations:

 a+  b+  c=2

4a+2b+  c=6

9a+3b+  c=12

I'm going to set this up as a matrix:

[ 1    1    1    2 ]

[4   2    1    6 ]

[9   3    1    12]

Multiply first row by -4:

[ -4   -4    -4    -8 ]

[  4     2     1     6 ]

[  9     3     1     12]

Add equation 1 to 2:

[ -4    -4    -4     -8]

[   0    -2    -3    -2]

[   9    3      1      12]

Divide first row by -4:

[ 1     1        1       2]

[ 0    -2     -3     -2]

[9      3      1       12]

Multiply top row by -9:

[-9    -9      -9    -18]

[0    -2      -3       -2]

[ 9    3       1         12]

Add equation 3 to 1:

[0    -6      -8     -6]

[0    -2      -3      -2]

[9     3        1        12]

Multiply the second equation by -3:

[ 0    -6     -8     -6]

[0      6      9      6]

[9       3      1     12]

Add equation 1 to 2:

[0      -6    -8    -6]

[0       0     1       0]

[9       3      1       12]

Let's stop there the second row gives us c=0.

So the first row gives us -6b-8c=-6 where c=0 so -6b-8(0)=-6.

Let's solve this:

-6b-8(0)=-6

-6b-0=-6

-6b    =-6

  b    =1

So we have b=1 and c=0 and we haven't used that last equation yet:

9a+3b+c=12

9a+3(1)+0=12

9a+3+0=12

9a+3=12

9a=9

a=1

So your expression for the pattern is [tex]x^2+x+0[/tex] or just [tex]x^2+x[/tex].

Let's test it out for one of our terms in our sequence:

"What is the sum of the first four even numbers? 2+4+6+8=20"

So if we plug in 4 hopefully we get 20.

[tex]4^2+4[/tex]

[tex]16+4[/tex]

[tex]20[/tex]

Looks good!

Now we want to know what happens when you plug in 30.

[tex]30^2+30[/tex]

[tex]900+30[/tex]

[tex]930[/tex]

If you don't like this matrix way, I can think of something else let me.

Step-by-step explanation:

The point-slope form of an equation of a line:

[tex]y-y_1=m(x-x_1)[/tex]

m - slope

We ahve the slope m = 1/2 and the point (-2, 1). Substitute:

[tex]y-1=\dfrac{1}{2}(x-(-2))[/tex]

[tex]y-1=\dfrac{1}{2}(x+2)[/tex] - point-slope form

Covert to the slope-intercept form (y = mx + b):

[tex]y-1=\dfrac{1}{2}(x+2)[/tex]          use the distributive property

[tex]y-1=\dfrac{1}{2}x+1[/tex]          add 1 to both sides

[tex]y=\dfrac{1}{2}x+2[/tex]   - slope-intercept form

Convert to the standard form (Ax + By = C):

[tex]y=\dfrac{1}{2}x+2[/tex]     multiply both sides by 2

[tex]2y=x+4[/tex]        subtract x from both sides

[tex]-x+2y=4[/tex]         change the signs

[tex]x-2y=-4[/tex]   -  standard form

Convert to the general form (Ax+By+C=0):

[tex]x-2y=-4[/tex]        add 4 to both sides

[tex]x-2y+4=0[/tex]   - general form

Answer:

1+cot ²∅=csc² ∅

Step-by-step explanation:

Step-by-step explanation:

Step-by-step explanation:

Ax+By=C

By=C-Ax

y=(C-Ax)/B

To solve the equation Ax + By = C for y, first subtract Ax from both sides to get By = C - Ax. Then, divide by B to isolate y, resulting in y = (C - Ax) / B.

In order to solve the given formula Ax+By=C for y, we need to isolate y in the equation. The steps are as follows:

And that's it! This is the formula solved for 'y'. It shows y in terms of A, B, C, and x.

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Answer:

-2

Step-by-step explanation:

The slope of the line can be found using

m = (y2-y1)/ (x2-x1)

    = (-4-8)/(5--1)

   = (-4-8)/(5+1)

   = -12/6

  = -2

Answer:

The slope is -2.

Step-by-step explanation:

The slope formula is [tex]\Rightarrow \displaystyle \frac{Y_2-Y_1}{X_2-X_1}[/tex].

[tex]\displaystyle \frac{(-4)-8}{5-(-1)}=\frac{-12}{6}=-2[/tex]

[tex]\Large\textnormal{Therefore, the slope is -2, which is our answer.}[/tex]

Hope this helps!

Answer:

50

Step-by-step explanation:

Let the unknown number be n.  Then 0.60n = 30, and n = 30/0.60 = 50.

Answer:

60 divided by 2 =30

100 divided by 2 = 50 so your answer will be c

Step-by-step explanation:

hope your having a good day guys

Answer:

First option, Second option and Fourth option.

Step-by-step explanation:

We need to remember that:

 1) Corresponding angles are located on the same side of the transversal (one interior and the other one exterior). They are congruent. Based on this, we can conclude that:

-Angle 1 and Angle 3 are Corresponding angles. Therefore, they are congruent.

-Angle 2 and Angle 4 are Corresponding angles. Therefore, they are congruent.

2)  Alternate interior angles are between the parallel lines,  and on opposite sides of the transversal. They are congruent.

Based on this, we can conclude that:

Angle 3 and Angle 6 are Alternate interior angles. Therefore, they are congruent.

The domain in this context, which represents the possible values for time from when the soccer ball was kicked until it landed, is the set of all real numbers from 0 to 4.

In the context of this problem, the domain refers to the possible values for time, from when Vanessa kicked the soccer ball until when it landed again. We know from the problem that the ball was in the air for 4 seconds. Therefore, the domain consists of all real numbers from 0 to 4 (both inclusive). Since time cannot be negative in this context, we start the domain at 0, and end at 4 because that's when the ball hit the ground again. Also, time, which is a continuous quantity, can take any value within this period, therefore it's the set of all real numbers between 0 and 4.

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The domain of the function, which represents the soccer ball's flight, is the time interval it was in the air. Therefore, the domain is from 0 to 4 seconds, written as [0,4].

In the problem, Vanessa kicked a soccer ball and it was in the air for 4 seconds before hitting the ground. In this scenario, the height of the soccer ball is considered to be a function of time. As such, the domain of the function, which represents all possible input values for the function, would be the amount of time the ball is in the air. Therefore, the domain for this function would be the interval from 0 to 4 seconds, often written as [0,4].

It's important to understand that in situations involving time, the domain value cannot be negative, as negative time values have no physical meaning. Therefore, the lower limit of the domain is 0, when Vanessa initially kicked the ball. The upper limit is the time the soccer ball spent in the air, or 4 seconds.

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Answer:

y-22=14(x-1)

Step-by-step explanation:

Since we are looking for a line parallel to y=14x-3 then we are looking for a line that has the same slope as y=14x-3.

The slope of y=14x-3 is 14.

So the slope of our line is 14.

So we know our equation is in the form y=14x+b.

We can find b by using a point (x,y) on the line. We know (x,y)=(1,22) is a point on the line we are looking for.

So I'm going to replace x with 1 and y with 22 in the equation y=14x+b giving me

22=14(1)+b

22=14   +b  

Subtract 14 on both sides

8=b

The equation of the line is y=14x+8. That is slope-intercept form.  I will leave this here just in case you are curious of this.

I was do point-slope form now since that is what your question says.

We know the slope is 14 so m=14.

The point-slope form is y-y1=m(x-x1).

We know (x1,y1)=(1,22) so now we substitute.

y-22=14(x-1)

Answer:

C) XZ is your best answer.

Step-by-step explanation:

Note that all the other choices are not line segments:

YZ is an arc segment, and is not your answer.

WY is also an arc segment, and is not your answer.

WZ is not a straight line, which does not make it a "line segment" (a straight line that is terminated on both ends by points).

~

Answer: The answer is C

Answer:

1

Step-by-step explanation:

(8^1)^0

We know that a^b^c = a^ (b*c)

8^1 ^0 = 8^(1*0) = 8^0 =

Anything raised to the 0 power is 1

=1

The answer is option "D. Figure U"

The required simplified solution of the given expression is 8x + 9.

Given that,
To determine the simplified solution of the expression 3(2x+1)+2(x+3).

The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.

Here,
Let,
y = 3(2x+1)+2(x+3)
y = 6x + 3 + 2x + 6
y = 8x + 9

Thus, the required a simplified solution of the given expression is 8x + 9.

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Answer:

Step-by-step explanation:

By looking at the graph I notice an open circle at (-4, -5) which means the function is not evaluable at -4, this restricts the domain.

The Domain of a function represents the set of values for which the function has an output.

The Range of a function is the set containing all the possible values associated with all input.

Domain in interval notation: (-4, 3]. The parenthesis denotes that the interval does not contain the extreme point -4. The brackets are the opposite.

Domain in set builder notation: [tex]$\{x |-4<x \leq 3 \}$[/tex]

Range in interval notation: (-5, -3]

Range in set builder notation:  [tex]$\{y |-5<y \leq 3 \}$[/tex]

Answer:

1) Domain = {x ∈ R | -4 < x ≤ 3}

Step-by-step explanation:

In this case, we have a line segment made up by two points (-4,-5) and (3,-3)

1) To find the Domain, is to find the set of values x may assume for a function.

We can also write it as compound inequality which is a complete form of writing it since inform us the set, the conditions.

{x ∈ R | -4 < x ≤ 3}

Or simply the interval (-4,3]

2) Range or Image is the set of values y may assume once you plug it in valid values for the Domain.

{y ∈ R| -5>y≥-3} or Simply (-5, -3]

Answer:

810.58 dollars per year

Step-by-step explanation:

Ok, so we are given that P=5000, so that makes our function A(t)=5000(1.06)^t .

The average rate of change is really the slope of the line going through (15,y1) and (20,y2).

We can find the corresponding y values by plugging the x's there to A(t)=5000(1.06)^t.

So let's do that:

y1=A(15)=5000(1.06)^(15)=11982.79097

y2=A(20)=5000(1.06)^(20)=16035.67736

Now the slope of a line can be found by using the formula:

(y2-y1)/(x2_x1) or just lining up the points and subtracting vertically and remember y difference goes over x difference.

Like so:

 ( 15  , 11982.79097 )

- ( 20 , 16035.67736)

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

 -5          -4052.886391

So the average rate of change is whatever -4052.886391 divided by -5 is.....

which is approximately 810.58 dollars per year.

Answer:

The average rate of change in dollars per year between years 15 and 20 is:

[tex]m=\$810[/tex]

Step-by-step explanation:

First we calculate the profit of the investment for the year 15

So

[tex]P=5000\\t=15[/tex]

[tex]A(15)=5000(1.06)^{15}[/tex]

[tex]A(15)=11982.79[/tex]

Now we calculate the profit of the investment for the year 20

So

[tex]P=5000\\t=20[/tex]

[tex]A(20)=5000(1.06)^{20}[/tex]

[tex]A(20)=16035.68[/tex]

Now the average rate of change m is defined as:

[tex]m=\frac{A(20) - A(15)}{20-15}[/tex]

Therefore:

[tex]m=\frac{16035.68-11982.79}{20-15}[/tex]

[tex]m=\frac{4052.89}{5}[/tex]

[tex]m=\$810.58[/tex]

Answer:

[tex]308.4579\ inches^2[/tex]

Step-by-step explanation:

We are given the height and radius

[tex]h=19\ inches\\r=5\ inches[/tex]

The formula for lateral area is:

[tex]LA = \pi rl[/tex]

We have to find lateral height first

[tex]l = \sqrt{r^2+h^2}\\  =\sqrt{5^2+19^2}\\ =\sqrt{25+361}\\=\sqrt{386}\\l=19.647\ inches[/tex]

Now,

[tex]LA = \pi rl\\= 3.14*5*19.647\\=308.4579\ inches^2[/tex]

Answer: [tex]308.61\ in^2[/tex]

Step-by-step explanation:

In order to calculate the lateral area of the traffic cone, we can use the following formula:

[tex]LA=\pi rl[/tex]

Where "r" is the radius and "l" is the slant height of the cone.

We need to find the slant height.  Knowing the height and the radius, we can calculate it with the Pythagorean Theorem:

[tex]a^2=b^2+c^2[/tex]

Where "a" is the hypotenuse and "b" and "c" are the legs of the triangle.

For this case, we can say that:

[tex]a=l\\b=19\ in\\c=5\ in[/tex]

Substituting anf solviing for "l", we get:

[tex]l^2=(19\ in)^2+(5\ in)^2\\\\l=\sqrt{(19\ in)^2+(5\ in)^2}\\\\l=19.6468\ in[/tex]

Now we can substitute values into the formula for calculate the lateral area of the traffic cone. This is:

[tex]LA=\pi (5 in)(19.6468\ in)=308.61\ in^2[/tex]

Answer:

B.

Step-by-step explanation:

We are given x is a number that is 7 more than y.

This means when you add 7 to y you get x.

So the equation for this is 7+y=x.

We want x as a function of y. So we want to replace x with f(y).

This means we have the function 7+y=f(y) or f(y)=7+y by symmetric property of equality.

Your answer is B.

Answer:

the answer is  x = f(y) = 7 + y

Step-by-step explanation:

Answer:

3(x + 5)(x+ 5)  

Step-by-step explanation:

3x² + 30x + 75

Here's one way to do it.

Step 1. Factor out the highest common factor of all three terms.

You can factor out a 3.

3(x² +10x +25) = 0

Step 2. Factor the remaining polynomial

We must find two numbers whose product is 25 and whose sum is 10.

A little trial-and-error gives us 5 and 5.  

5 × 5 = 25; 5 + 5 = 10

We can factor the expression as

3(x + 5)(x + 5)

To factorize a polynomial implies that, we want to get several algebraic expressions from the polynomial.

When [tex]3x^2 +30x + 75[/tex] is factored, the result is: [tex]3(x + 5)(x + 5)[/tex]

Given

[tex]3x^2 +30x + 75[/tex]

Factor out 3

[tex]3x^2 +30x + 75 = 3 \times (x^2 + 10x + 25)[/tex]

Expand the bracket

[tex]3x^2 +30x + 75 = 3 \times (x^2 + 5x + 5x + 25)[/tex]

Factorize

[tex]3x^2 +30x + 75 = 3 \times (x(x + 5) + 5(x + 5))[/tex]

Factor out x + 5

[tex]3x^2 +30x + 75 = 3 \times ((x + 5)(x + 5))[/tex]

So, we have:

[tex]3x^2 +30x + 75 = 3(x + 5)(x + 5)[/tex]

Hence, the factors of [tex]3x^2 +30x + 75[/tex] are 3 and (x + 5)

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Answer:

The quotient to the problem 806 ÷ 6 is 134.3333 and so forth.

The quotient of the expression 806 ÷ 6 will be 134.33.

What is Division method?

Division method is used to distributing a group of things into equal parts.

Given that;

The expression is,

806 ÷ 6

Now, Solve the expression by using long division as;

6 ) 806 ( 134.333.....

  - 6

---------

    20

  - 18

 -----------

      26

   -  24

    -------

         20

       - 18

       -------

           2

Thus, The quotient of the expression 806 ÷ 6 will be 134.33.

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Answer:

1 kg/m³

Step-by-step explanation:

Density is mass divided by volume.

D = M / V

D = 10 kg / 10 m³

D = 1 kg/m³

Answer:

1 kg/m³

Step-by-step explanation:

Answer:

-21 =x

Step-by-step explanation:

3x – 10 = 5x + 32

Subtract 3x from each side

3x-3x – 10 = 5x-3x + 32

-10 =2x +32

Subtract 32 from each side

-10 -32 = 2x+32-32

-42 = 2x

Divide each side by 2

-42/2 = 2x/2

-21 =x

Answer: [tex]x=-21[/tex]

Step-by-step explanation:

To solve the equation you need to find the value of "x".

First, you need to subtract 32 from both sides of the equation:

[tex]3x - 10 -32= 5x + 32-32\\\\3x - 42= 5x[/tex]

 Now you must subtract [tex]3x[/tex] from both sides of the equation:

[tex]3x - 42-3x= 5x-3x\\\\- 42= 2x[/tex]

And finally divide both sides of the equation by 2. Then:

[tex]\frac{-42}{2}=\frac{2x}{2}\\\\x=-21[/tex]

Answer:

a. 3/4

Step-by-step explanation:

The expected value is the sum of each outcome times its probability.

If n is the total number of marbles, then the expected value is:

E = (3/8) (n) + (3/8) (n)

E = 3/4 n

sorry wrong question-

Set up a ratio with the height over the shadow.

Let the height of the tower be X

The ratio of the tower is x/120

The ratio if the sign is 10/8

Set them to equal and solve for x:

x/120 = 10/8

Cross multiply:

8x = 1200

Divide both sides by 8:

x = 1200/8

x =150

The answer is B. 150 ft.

Answer:

B.  150 feet.

Step-by-step explanation:

The tower and the street sign and their shadows form 2 similar triangles.

So the corresponding lengths are in the same ratio, then:

120 / 8 = h / 10  where h is the height of the tower.

8h = 120 * 10

h = 120 * 10 / 8

= 150 feet.

Answer:

[tex]f(x) = - 5 {(x - 1)}^{2} [/tex]

Step-by-step explanation:

The given parent function is:

[tex]f(x) = {x}^{2} [/tex]

If we shift to the right by 1 unit, the function becomes:

[tex]f(x) = {(x - 1)}^{2} [/tex]

If we stretch by a factor of 5, the function becomes,

[tex]f(x) =5 {(x - 1)}^{2} [/tex]

Finally reflecting over the x-axis gives:

[tex]f(x) = - 5 {(x - 1)}^{2} [/tex]

Answer:

f'''(x)=-5(x-1)^2

Step-by-step explanation:

Given:

f(x)= x^2

Shifting 1 unit to right means subtracting 1 from the function i.e

f(x-1)=(x-1)^2

f'(x)=(x-1)^2

now vertically stretching above f'(x)  by a factor of 5:

5f'(x)=5(x-1)^2

f''(x)=5(x-1)^2

finally reflecting above f'''(x) over the x-axis:

-f''(x)=-5(x-1)^2

f'''(x)=-5(x-1)^2 !

Answer:

Step-by-step explanation:

[tex]2x-3(x+4)=-5\qquad\text{use the distributive property}\ a(b+c)=ab+ac\\\\2x+(-3)(x)+(-3)(4)=-5\\\\2x-3x-12=-5\qquad\text{add 12 to both sides}\\\\-x=7\qquad\text{change the signs}\\\\x=-7[/tex]

Step-by-step explanation:

the triangle is obtuse...has an angle greater than 90deg

also it is scalene...all 3 sides are different length

The triangle is a scalene and obtuse triangle

A scalene triangle is a type of triangle with three different length sides and three interior angles that add up to 180 degrees. Scalene triangles have no equal of parallel sides , hence there is no line of symmetry. The interior angles of a scalene triangle can be acute , right or obtuse angles.

Given data ,

Let the triangle be represented as ΔABC

Now , the measure of sides of the triangle are

The measure of side AB = 11.9

The measure of side BC = 7

The measure of side AC = 6

And , the measure of ∠ABC = 132°

So , the sides of the triangle are having different lengths so it can be classified as a scalene triangle

And , the angle is obtuse , so it is also an obtuse triangle

Hence , the triangle is scalene

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Answer:

There are infinite numbers that added or multiplied will give the same answer. But, if the require this two numbers to be the same, then we need to solve the following equation to see what we found out:

x + y = x * y

Given that we want the numbers to be equal, then y=x.

x + x = x * x

2x = x^2

x^2 - 2x = 0

x(x-2) = 0

We find that NOT only x=2 meets this requirement but also x=0.

Therefore, there is another number that when added, multiplied or squared gives the same result. The other number is zero!

We can prove it:

0 × 0 = 0

0 + 0 = 0

Using the slope-intercept form, the equation of the line with slope 6 passing through the point (7,49) is obtained as y = 6x + 7.

To find the equation of a line given the slope and a point it passes through (7,49), we'll use the slope-intercept form of a line, which is y = mx + b, where m is the slope and b is the y-intercept. Since we are given the slope to be 6, we can substitute this into the equation, getting y = 6x + b. To find b, the y-intercept, we substitute the coordinates of the given point into the equation: 49 = 6(7) + b. After solving, we get b = 49 - 42, which simplifies to b = 7. Therefore, the complete equation of the line is y = 6x + 7.

Answer:

12.9

Step-by-step explanation:

Intersecting chord segments are proportional, so:

35×a = 15×30

a = 12.9

The value of a in the intersecting chords is 12.9

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

The intersecting chords

Using the theorem of intersecting chords, we habe the following equation

a * 35 = 15 * 30

This gives

a = 15 * 30/35

Evaluate

a = 12.9

Hence, the value of a is 12.9

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Answer:

93

Step-by-step explanation:

$11.55 per hour.

$1074.15 in two weeks.

1074.25 ÷ 11.55 = 93.

Answer:

93 hours.

Step-by-step explanation:

The question states that $1074.15 has been earned in a two weeks time (in 14 days total). The hourly wage rate is $11.55. To calculate the total number of hours worked in the two week time, following formula will be used:

Number of hours worked = Total income / Hourly wage rate.

Number of hours worked = 1074.15/11.55 = 93 hours.

Therefore, 93 hours have been worked in 2 weeks to earn the total money of $1074.15!!!