Triangle ABC is similar to triangle PQR, as shown below:

Two similar triangles ABC and PQR are shown. Triangle ABC has sides AB equals c, BC equals a, and AC equals b. Triangle PQR has

Which equation is correct?

I think that this is the x-intercept method:

First of all, we set y=x^3-6x^2+11x-6.

To find x, we need to let y=0. (y is also equals to f(x))

f(x)=x^3-6x^2+11x-6

In order to make f(x)=0, what does x have to be?

f(1)=0

So then we divide x^3-6x^2+11x-6 by x-1. Why? Because that will not give us a remainder.

(x^3-6x^2+11x-6)/(x-1) = x^2-5x+6

Now, we need to factorize it.

x^2-5x+6 = (x-2)(x-3)

So x=2,3

Answer:

[tex]sin \:C =\frac{8}{17}[/tex]

Step-by-step explanation:

In a right triangle, the sine of an angle is the length of the opposite side divided by the length of the hypotenuse.

[tex]sin \:x =\frac{O}{H}[/tex]

From the formula above we know that the sine of an angle is the opposite side divided by the hypotenuse. The opposite side is AB and has a length of 8. The hypotenuse is AC with a length of 17. So we can write

[tex]sin \:C =\frac{8}{17}[/tex]

Answer:

The heights of trees

Step-by-step explanation:

Because trees vary a lot, I would say trees.

Eye color does not vary that much and this is true for new born babies.but trees on the other hand are a different story. They range from  0 to hundreds of feet so it would be a good idea to bind.

Answer:

The heights of trees

Step-by-step explanation:

I did the lesson and got 100% also the guy above explains it perfectly.

The graph which represent a exponential function is:

                      Graph A.

Exponential function--

A exponential function is a function which is in the form of:

                   [tex]y=f(x)=ab^x[/tex]

where a and b are integers and a,b≠0.

From the given graphs, the graph which one that  represents a exponential function is:  Graph A.

An exponential graph represents a mathematical relationship where the dependent variable (y) is a constant raised to the power of the independent variable (x). This results in a curved graph that either increases or decreases rapid

A exponential function is a function which is in the form of:

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

where a and b are integers and a,b≠0.

On the other hand, the graph of a exponential function either increases continuously or decreases continuously and rapidly in it's domain.

The x-axis is always an asymptote for the graph.

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1,2,3,6,7,14,21,42

1.2,4,7,14,28

A, D and E are correct

given ( x - 4 ) is a factor then x =  4 is a root

the remainder on division by (x - 4 ) = 0 as indicated by the 0 on the right side of the quotient

(x - 4 ) is a factor of 3x² - 13x + 4 → A

the number 4is a root of f(x) = 3x² - 13x + 4 → D ( explained above )

thus 3x² - 13x + 4 ÷ (x - 4 ) = 3x - 1 → E

the quotient line 3 - 1 0

3 and - 1 are the coefficients of the linear quotient and 0 is the remainder

The correct options are [tex]\boxed{{\mathbf{Option A, D and E}}}[/tex].

Further explanation:

In any synthetic division, the dividend polynomial [tex]F\left( x \right) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + {a_{n - 2}}{x^{n - 2}} +  \cdots {a_0}[/tex] and the divisor polynomial [tex]g\left( x \right) = x - b[/tex] can be written as,

[tex]\begin{aligned}b\left){\vphantom{1{\underline {\begin{array}{*{20}{c}}3&{ - 13}&4\\{  }&{12}&{ - 4}\end{array}} }}}\right.\!\!\!\!\overline{\,\,\,\vphantom 1{{\underline {\begin{array}{*{20}{c}}&{a}_n&{ a_{n-1}}&_\cdot_\cdot_\cdot{a_0}\\{    }&{}&{ }\end{array}} }}} \hfill \\\begin{array}{*{20}{c}}{{\text{    }}{c}_n}&_\cdot_\cdot_\cdot{ c_0}&{{\text{    }}0}\end{array} \hfill\\\end{aligned}[/tex]

Here, the monic polynomial   is divided by the polynomial   that provides the polynomial   after division that is also a factor of the polynomial  .

Given:

The synthetic division is given below.

[tex]\begin{aligned}4\left){\vphantom{1{\underline {\begin{array}{*{20}{c}}3&{ - 13}&4\\{  }&{12}&{ - 4}\end{array}} }}}\right.\!\!\!\!\overline{\,\,\,\vphantom 1{{\underline {\begin{array}{*{20}{c}}&3&{ - 13}&4\\{    }&{12}&{ - 4}\end{array}} }}} \hfill \\\begin{array}{*{20}{c}}{{\text{    }}3}&{ - 1}&{{\text{    }}0}\end{array} \hfill\\\end{aligned}[/tex]  

Step by step explanation:

We have to determine the answer among all the options.

Option A: [tex]\left( {x - 4} \right)[/tex] is a factor of [tex]3{x^2} - 13x + 4[/tex].

It can be observed from the given synthetic division the polynomial is [tex]g\left( x \right) = \left( {x - 4} \right)[/tex] that is divisible by the polynomial [tex]F\left( x \right) = 3{x^2} - 13x + 4[/tex].

Therefore, the polynomial [tex]g\left( x \right) = \left( {x - 4} \right)[/tex] is a factor of the polynomial [tex]F\left( x \right) = 3{x^2} - 13x + 4[/tex].

Therefore, the option A is correct option.

Option B: [tex]\left( {x + 4} \right)[/tex] is a factor of [tex]3{x^2} - 13x + 4[/tex].

From the option A, it has been proved that [tex]g\left( x \right) = \left( {x - 4} \right)[/tex] is a factor of the polynomial  

[tex]F\left( x \right) = 3{x^2} - 13x + 4[/tex]

Therefore, [tex](x+4)[/tex] is a not factor of [tex]3{x^2} - 13x + 4[/tex].

Thus, the option B is not correct option.

Option C: The number [tex]-4[/tex] is a root of  [tex]F\left( x \right) = 3{x^2} - 13x + 4[/tex].

From the option A, it has been proved that [tex]g\left( x \right) = \left( {x - 4} \right)[/tex] is a factor of the polynomial  

[tex]F\left( x \right) = 3{x^2} - 13x + 4[/tex]

Now substitute 0 for [tex]g\left( x \right)[/tex] in the equation [tex]g\left( x \right) = \left( {x - 4} \right)[/tex] to find the root of the polynomial [tex]F\left( x \right) = 3{x^2} - 13x + 4[/tex] as,

[tex]\begin{aligned}0&= \left( {x - 4} \right) \hfill\\x&= 4 \hfill\\\end{aligned}[/tex]  

It can be seen that the value of [tex]x[/tex] is 4 it means [tex]-4[/tex] is not a root of the polynomial [tex]F\left( x \right) = 3{x^2} - 13x + 4[/tex].

Therefore, the option C is not correct option.

Option D: The number [tex]4[/tex] is a root of [tex]F\left( x \right) = 3{x^2} - 13x + 4[/tex].

It can be seen that the value of [tex]x[/tex] is 4 in option C it means [tex]4[/tex] is a root of the polynomial [tex]F\left( x \right) = 3{x^2} - 13x + 4[/tex].

Therefore, the option D is correct option.

Option E: [tex]\left( {3{x^2} - 13x + 4} \right) \div \left( {x - 4} \right) = \left( {3x - 1} \right)[/tex]  

The option E is also correct as [tex](x-4)[/tex] is the factor of the polynomial  [tex]F\left( x \right) = 3{x^2} - 13x + 4[/tex].

Option F: [tex]\left( {3{x^2} - 13x + 4} \right) \div \left( {x + 4} \right) = \left( {3x - 1} \right)[/tex]  

The option F is not correct as [tex]\left( {x + 4} \right)[/tex] is not the factor of the polynomial [tex]F\left( x \right) = 3{x^2} - 13x + 4[/tex].

Result:  

Therefore, the correct options are [tex]\boxed{{\mathbf{Option A, D and E}}}[/tex].

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Answer details:

Grade: Medium school

Subject: Mathematics

Chapter: Synthetic division

Keywords: Synthetic division, polynomial, monic polynomial, function, factor, real number, root, divisible, addition, remainder, quotient, divisor, dividend.

Answer:

1.827 x 10^-23

Step-by-step explanation:

(1.994 x 10^-23) - (1.67 x 10^-24) = 1.827 x 10^-23

you can use the formula for sample means:

[tex][tex]z(70) = \frac{70-75}{5 / \sqrt{64}} = -8[/tex][/tex]

where m stands for a value of the sample mean.

We are looking for the value, specifically for its two borderline values: 70 and 80

and their cumulative probabilities, p(z(80)) and p(z(70)). The difference p(z(80))-p(z(70))

will give the probability that m falls in the interval [70,80]

So let's get cracking at it:

[tex]z(70) = \frac{70-75}{5 / \sqrt{64}} = -8[/tex]

[tex]z(80) = \frac{80-75}{5 / \sqrt{64}} = 8[/tex]

These are very large values of z. You may notice that any z-table online won't even bother covering range that high - the probabilities for these values are virtually 0 (in the neg case) 1 (in the pos case).

This means that numerically the probability of the sample mean of 64 samples falling within the range of 1 standard deviation is very close to 1

So the answer choice  should be 1.0

Answer:

(a) Sandra earns $15 per hour

(b) Sandra earns more per hour

Step-by-step explanation:

(a) y=15x tells Sandra's earnings for x hours. To find Sandra's earnings for 1 hour, let x=1.

... y = 15·1 = 15 . . . . Sandra earns $15 in 1 hour.

(b) To find Susan's earnings you can choose any number of hours and divide that into the corrsponding number of dollars. The math is really easy if you choose the last table entry:

... $80/(10 hours) = $8/hour

Sandra's earnings of $15 per hour are more than Susan's earnings of $8 per hour.

_____

Another way to compare the wages is to pick one of the hour numbers from the table and compute Sandra's equivalent earnings. Again, if we choose 10 hours, the math is easy

... y = 15·10 = 150

Sandra earns $150 for 10 hours' work, while Susan only earns $80 for the same time.

So based on the table, Susan's equation describing her earnings is:

y=8x   (x being the hours, y amount earned)

(a) Sandra earns $8 per hour (the factor in front of x in the equation)

(b) Sandra's equation is y=15x which means for every unit of time she adds $15, so her hourly pay is $15. Therefore Sandra earns ($7) more per hour than Susan.

If rain is falling at a rate of 2.56 inches per hour, one would expect 9.6 inches of rainfall after 3.75 hours.

The question pertains to the calculation of total rainfall when the rate of rainfall and time of rainfall are known.

        So, 2.56 inches/hour * 3.75 hours = 9.6 inches total rainfall.

Hence, if rain is falling at a rate of 2.56 inches per hour, you would expect 9.6 inches after 3.75 hours

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Answer-

The average U.S. public debt per American is approximately $29,400

Solution-

Given in the question,

U.S public debt as of October 2010 = $9.06×10¹²

Population of U.S in 2010 = 3.08×10⁸

We have to calculate the average U.S. public debt per American,

[tex]\text{Average debt per American}=\frac{\text{Total debt}}{\text{Population}}\\\\=\frac{9.06\times 10^{12}}{3.08\times 10^8}\\\\=2.9415\times 10^{(12-8)}\\\\=2.9415\times 10^4 \approx 2.94\times 10^4=29,400[/tex]

Therefore, the average U.S. public debt per American is $29,400 (after rounding off)

A linear function is the equation of a line. Here, you are given two points on the line, one of which is the y-intercept, and asked to write the equation. It can work reasonably well to use the 2-point form of the equation of a line.

For points (x1, y1) and (x2, y2), the equation of the line through them can be written as

... y = (y2 -y1)/(x2 -x1)·(x -x1) + y1

When it is possible to use x1 = 0, this is simplified somewhat, so we choose

... (x1, y1) = (0, -5) . . . . . . from f(0) = -5

... (x2, y2) = (5, -1) . . . . . .from f(5) = -1

Putting these values in the above formula, we get ...

... y = (-1 -(-5))/(5 - 0)·(x -0) -5

... y = 4/5x -5 . . . . . simplified

This can be put in the desired functional form using f(x) instead of y:

... f(x) = 4/5x -5

Here we're given two points on a line:  (0,-5) and (5, -1).  Going from the first point to the second, x increases by 5 and y increases by 4.  Thus, the slope of the line is

m = 4/5 = 1.         Use the point-slope formula to find the equation of this line:

                                   y-(-5) = (4/5)(x).  Thus, y = (4/5)x - 5 or f(x) = (4/5)x - 5

Answer:

3.5 m

Step-by-step explanation:

The perimeter is the sum of the lengths of the three sides of the triangle, so is the sum of two legs and the base.

... 7.5 m = (2 m) + (2 m) + base

... 3.5 m = base . . . . . . subtract 4 m

base = 3.5m

Since the triangle is isosceles both legs are equal to 2m

base = 7.5 - 2 - 2 = 3.5m

If you multiply the original equation by 8, you get

... 8(1/4x -1/2y) = 8(8)

... 8/4x -8/2y = 64

... 2x -4y = 64 . . . . . matches choice b)

area = 20 units²

construct a square of side 6 units around the given figure, then the area of the figure is given by area of square minus 4 congruent triangles

A = 6² - 4 × [tex]\frac{1}{2}[/tex] × 4 × 2

  = 36 - 16 = 20 units²

Given expression is

-33/-10- (-45.3+ 35.2)

Problem says that it is simplified.

After simplification it gives:

-33/-10+?

Now we need to find the unknown value that goes in place of ?.

Notice that it is expression having only constants so we will get a unique answer not the smallest or largest value.

Now let's find the value of unknown that goes in place of ?

-33/-10- (-45.3+ 35.2)= -33/-10+?

we see that -33/-10 is on both sides so we can remove that

- (-45.3+ 35.2)= ?

Now let's simplify the left side by distributing the negative sign

+45.3 - 35.2= ?

10.1 = ?

Hence the unknown value is 10.1 that goes in place of ?.

Answer:

answer is 3 on edge

Step-by-step explanation:

wwww

It should be no surprise that the volume of a 1-inch cube is 1 cubic inch. Assuming those 36 cubes completely fill the interior volume of the figure, that figure's volume is

... 36 cubic inches

This is known as Einstein's proof, not because he was the first to come up with it, but because he came up with it as a 15 year old boy.

Here the problem is justification step 2.  The written equation

BC ÷ DC = BC ÷ AC

is incorrect, and wouldn't get us our statement 2, which is correct.

For similar triangles we have to carefully pair the corresponding parts to get our ratios right:

ABC ~ BDC means AB:BD = BC:DC = AC:BC so BC/DC=AC/BC.

Justification 2 has the final division upside down.

11. {OD, OH} . . . or . . . {OB, OJ}

12. b. Open a compass to distance AB

13. 17 square units

11. Opposite rays are rays that together make a line. The only point they have in common is their mutual end point.

Given the letter names of points on lines, one can name opposite rays by choosing 3 collinear points, and selecting the middle one of those for the mutual end point. That point's name will be the first letter of each of the ray names. One ray will be named by one of the remaining points, and the other will be named by the other remaining point.

Here, we can choose points H, O, D on the same line. Then rays in opposite directions will be OD and OH, each having point O as its end point.

Similarly, we can choose points B, O, J on the other line and name opposite rays OB and OJ.

___

12. The general sequence of steps for copying a line segment is to draw a line segment longer than the one you want to copy, mark a point on it, set the compass to the length of the given segment, then use the marked point as one end and mark the distance from that point using the compass.

Here, the target line and reference point (C) have been made already, so the next step is open the compass to distance AB.

___

13. The unadulterated triangle has legs of 6 and 8 units, so an area of ...

... area = (1/2)bh = (1/2)(6)(8) = 24 . . . . square units.

One can count the squares within that area that have been removed from the triangle. There are 7 of them, so the area is ...

... 24 -7 = 17 . . . . square units

There are other ways to find the area, including dividing the triangle into smaller pieces. Doing the latter, one needs to be careful, because the slanted side only goes through a grid intersection at points (1, 1), (4, 5), and (7, 9).

If one uses two triangles with legs 3 and 4, and a central rectangle of 2×3, then the area adds up to 6+6+5 = 17 square units, because there is one square taken out of the lower right corner of the upper triangle.

Let the width of rectangle be X inches

Then according to question

Length of rectangle will be 3X

Also, we know that

Area of rectangle is = L×B

108 = X . 3X

3X^2 = 108

Divide both sides by 3

X^2 = 36

Taking squareroot of both sides

X=6

Therefore, width of rectangle is 6 inches

Answer:

1. 6 in.

Step by step explanation:

Let the width of our rectangle be x inches.

We have been given that the length of our rectangle is three times the width, therefore, length of our triangle will be 3x inches.

We are told that area of the rectangle is 108 square inches.

Since we know that [tex]\text{Area of a rectangle}=\text{Width}\cdot\text{Length}[/tex]

Let us substitute our given information in area formula.

[tex]108=x\cdot 3x[/tex]

Upon multiplying 3x by x we will get,

[tex]108=3x^{2}[/tex]

Let us divide both side of our equation by 3.

[tex]\frac{108}{3} =\frac{3x^{2}} {3}[/tex]

[tex]36=x^{2}[/tex]

Upon taking square root of both sides of our equation we will get,

[tex]6\pm=x[/tex]

Since width cannot be negative, therefore, width of our rectangle will be 6 inches and 1st option is the correct choice.

16%

the percent tips is calculated as

[tex]\frac{tips}{total bill}[/tex] × 100%

total bills = 23.59 + 40.65 + 30.50 + 15.68 = 110.42

percent tips = [tex]\frac{17.67}{110.42}[/tex] × 100% = 16%

the y-axis, that is x = 0

the equation of a parabola in the form f(x) = ax²

has it's vertex at the origin and since the axis of symmetry passes through the vertex then

the axis of symmetry is the y- axis

Answer:

The investment has doubled in worth by the start of 15th year.

Step-by-step explanation:

We are Provided a table which tells us that principal amount of $100 is invested on a interest rate of 5% for 15 years compounded annually and we are asked to find out by the start of which year our investment will be doubled.

We can see that after 14 years our amount is doubled but this can't be our answer as amount is doubled ($207.90) after 14 years not in the start of 14th year. We can see that by the start of 15 year our amount is doubled that is $207.90.

Therefore, by the start of 15th year our investment will be doubled.

Using a somewhat liberal interpretation of "the angles shown in the diagrams", we can conclude that the appropriate choice is ...

... A. The value of x should be 75, because the angles shown in the diagrams are congruent.

_____

There are 8 angles shown in each of 4 diagrams. All 32 angles are not congruent. Taken literally, "the angles shown in the diagrams are congruent" is a false statement. Rather, the marked pair of angles in each diagram are congruent angles.

Answer:

Card 7

The hourly rate of change/slope is $75 per hour.

Card 8

Frastastic is charging $0.48 per ounce.

Step-by-step explanation:

Card 7

Lets take the set up fee as "$[tex]s[/tex]" and,

Hourly rate as $[tex]x[/tex]per hour.

Now we can write equations for 3 hour event and 5 hour event.

[tex]s+3x=275[/tex] <------- (1)

[tex]s+5x=425[/tex] <------- (2)

Now we need to solve the simultaneous equations to find [tex]s[/tex] and [tex]x[/tex].

Subtract (1) from (2)

⇒[tex]s+5x-(s+3x)=425-275[/tex]

[tex]s+5x-s-3x=150[/tex]

[tex]2x=150[/tex]

[tex]x=75[/tex]

So the hourly rate of change/slope is $75 per hour.

Therefore, Raymond is right.

Card 8

You can take Amount (oz) as x values and Cost as y values and plot a graph.

Then the Slope of the graph will give you the rate Frastastic is charging per yogurt.

Slope of a linear graph is given by the following equation;

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

Slope(m)=[tex]\frac{6.72-2.4}{14-5}[/tex]

Slope(m)=[tex]\frac{4.32}{9}[/tex]

Slope(m)=[tex]0.48[/tex]

Therefore, Frastastic is charging $0.48 per ounce.

This is less than $0.50 per ounce charged by Fromazing. Therefore, Sterling is right.

Solution

When we add 1/3 which is rational to an irrational number, we get an irrational number.

pi - is a rational number. √2, √3 are also rational numbers

So the possibilities  are pi + 1/3,

2pi + 1/3,

3pi+1/3,

√3 + 1/3,

√2 + 1/3...

I hope this will help you.

Thank you.

Answer:

$48

Step-by-step explanation:

1. 192/12 = 16

2. 16 * 3 = 48

3. Your answer is 48

192/12 = $16 * 3 = $48. $48 dollars was how much he was paid for 3 weeks.

The problem asks for the equations of the lines, so you use the information given in the graph to help you write the equations.

The given line has a "rise" of -4 for a "run" of 1, hence a slope of -4/1 = -4. This is the information you need from that line to write the equations of lines through the given point, (4, 3).

(a) The parallel line will have the same slope. You are asked for "an equation", so the simplest is to provide the necessary equation in point-slope form.

The equation of a line with slope m through point (h, k) is ...

... y - k = m(x - h)

For m = -4, (h, k) = (4, 3) your parallel line is ...

... y - 3 = -4(x -4)

(b) The perpendicular line will have a slope that is the negative reciprocal of that of the given line: -1/-4 = 1/4. Using the same point-slope form, your perpendicular line has equation ...

... y - 3 = (1/4)(x - 4)

Inputs are the first number output is the second. So youre outputs are 2, -1, -1, and 3. so i would say b

The set of outputs for the function represented by the scatterplot is {-1, 2, 3}, which matches Option B from the given choices.

You are looking to identify the set of outputs (also known as the range) of a given function based on a scatterplot. The output of a function is comprised of the second element in each ordered pair, which represents the y-value on the Cartesian plane. In this case, the scatterplot includes the points (-3, 2), (-2, -1), (2, -1), and (3, 3).

To find the correct set of outputs, we need to list the y-values from each point: 2, -1, -1, and 3. When listing a set, we do not repeat values, so the set of outputs for this function would be {2, -1, 3}. Therefore, the correct answer is Option B) {-1, 2, 3}.