The work of a student to solve a set of equations is shown: Equation A: y = 4 − 2z Equation B: 4y = 2 − 4z Step 1: −4(y) = −4(4 − 2z) [Equation A is multiplied by −4.] 4y = 2 − 4z [Equation B] Step 2: −4y = 4 − 2z [Equation A in Step 1 is simplified.] 4y = 2 − 4z [Equation B] Step 3: 0 = 6 − 6z [Equations in Step 2 are added.] Step 4: 6z = 6 Step 5: z = 1 In which step did the student first make an error? (5 points)

Answer:

The measures of the angles of the triangle are 51° , 64.5° , 64.5°

OR

The measures of the angles of the triangle are 129° , 25.5° , 25.5°

Step-by-step explanation:

* Lets explain the meaning of the inscribed triangle in a circle

- If a triangle inscribed in a circle, then the vertices of the triangle lie

 on the circumference of the circle and each vertex is an inscribed

 angle in the circle subtended by the opposite arc

- Fact in the circle the measure of the inscribed angle is 1/2 the

 measure of its subtended arc

* Now lets solve the problem

- Δ ABC is an isosceles with the base BC

∴ AB = AC

∴ m∠B = m∠C

- Δ ABC is inscribed in a circle

∴ ∠A is inscribed angle subtended by arc BC (minor or major)

# The measure of the minor arc is less than 180° and the measure of

  the major arc is greater then 180° and the sum of the two arcs

  equals the measure of the circle which is 360°

∴ ∠B subtended by arc AC

∴ ∠C subtended by arc AB

∵ The measure of the arc BC = 102°

- There is two cases in this question

(1) If the angle A subtended by the minor arc BC

(2) If the angle A subtended by the major arc BC

- Lets solve case (1)

∵ ∠A is an inscribed angle subtended by the minor arc BC

∴ m∠A = 1/2 the measure of the arc BC

∵ The measure of the arc BC is 102°

∴ m∠A = 1/2 × 102 = 51°

∵ The sum of the measures of the interior angles of a triangle is 180°

∴ m∠A + m∠B + m∠C = 180°

∴ 51 + m∠B + m∠C = 180° ⇒ subtract 51 from both sides

∴ m∠B + m∠C = 129°

∵ m∠B = m∠C ⇒ isosceles Δ

∴ m∠B = m∠C = 129/2 = 64.5°

* The measures of the angles of the triangle are 51° , 64.5° , 64.5°

- Lets solve case (2)

∵ ∠A is an inscribed angle subtended by the major arc BC

∴ m∠A = 1/2 the measure of the arc BC

∵ The measure of the minor arc BC is 102°

∵ The measure of the circle is 360°

∴ The measure of the major arc = 360 - 102 = 258°

∴ m∠A = 1/2 × 258 = 129°

∵ The sum of the measures of the interior angles of a triangle is 180°

∴ m∠A + m∠B + m∠C = 180°

∴ 129 + m∠B + m∠C = 180° ⇒ subtract 129 from both sides

∴ m∠B + m∠C = 51°

∵ m∠B = m∠C ⇒ isosceles Δ

∴ m∠B = m∠C = 51/2 = 25.5°

* The measures of the angles of the triangle are 129° , 25.5° , 25.5°

Answer:

13.7 cm²

Step-by-step explanation:

area of yellow region = area of square - area of 4 quarter circles

area of square = 8² = 64

area of 4 congruent quarter circles = area of circle

area of circle with radius = 4

A = π × 4² = 16π

yellow region = 64 - 16π ≈ 13.7 cm²

Answer:

She needs to buy 8 packages.

Step-by-step explanation:

Divide 44 (cookies) by 6 (cookies in package) to get 7.33 (packages). Since you can't buy 7.33 packages, you will have to buy 8. Hope this helps!

Answer:

The logarithmic function modeled by the table would be

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

So it would seem to be answer B.

The fundamental identity used in the second step of the proof is sin^2(θ) + cos^2(θ) = 1

An identity is true for general case, and not only for special cases. The proof for given statement is derived by using Pythagorean identity.

[tex]sin^2(\theta) + cos^2(\theta) = 1\\\\1 + tan^2(\theta) = sec^2(\theta)\\\\1 + cot^2(\theta) = csc^2(\theta)[/tex]

Given statement is [tex]\sin^2\theta - \cos^2\theta\sin^2\theta =\sin^4\theta\\\\[/tex]

Taking its left side:

[tex]\begin{aligned}\sin^2\theta(1 - cos^2\theta) &= \sin^2\theta \times sin^2\theta \\&= sin^4\theta \end{aligned}[/tex]

(from first Pythagorean identity).

Thus, the given statement is proved using Pythagorean identity (first) that [tex]sin^2\theta + cos^2\theta = 1\\[/tex]

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[tex]\bf \textit{surface area of a cone}\\\\ SA=\pi rs+\pi r^2~~ \begin{cases} r=&radius\\ s=&slant\\ &height\\ \cline{1-2} SA=&16.8\pi \\ r=&3 \end{cases}\implies 16.8\pi =\pi (3)s+\pi (3)^2 \\\\\\ 16.8\pi =3\pi s+9\pi\implies 16.8\pi -9\pi =3\pi s\implies 7.8\pi =3\pi s \\\\\\ \cfrac{7.8\pi }{3\pi }=s\implies 2.6=s[/tex]

Answer:

D

Step-by-step explanation:

The roots are where the curve crosses the x-axis.  So that's x=-3 and x=4.

Answer D.

Answer:

D

Step-by-step explanation:

The roots are the points on the x- axis where the function has a value of zero. That is where the graph crosses the x- axis

The graph crosses the x- axis at x = - 3 and x = 4, hence

Roots of the quadratic function are x = -3, 4 → D

Answer:

-6

Step-by-step explanation:

You are finding the dot product of vectors <2, 4> and <1, -2>.  Recall that the dot product is a SCALAR.  In this case  <2, 4> · <1, -2> = 2(1) + 4(-2)>, or

2 - 8, or -6.  

Answer:

c

Step-by-step explanation:

Given the roots are x = - 1 + 4i and x = - 1 - 4i then the factors are

(x - (- 1 + 4i))(x - (- 1 - 4i))

= (x + 1- 4i )(x + 1 + 4i )

= (x + 1)² - 16i² → [ i² = - 1 ]

Expand and simplify

= x² + 2x + 1 + 16

Hence

x² + 2x + 17 = 0 → c

Answer:

Step-by-step explanation:

Recall that we use the first derivative to discover where a function is increasing or decreasing; it's increasing where the first derivative is + and decreasing where the first derivative is -.

The derivative of y = x^4 - 8x^2 + 16 is  dy/dx = 4x^3 - 16x, or 4x(x^2 - 4).

This can be factored further:  4x(x - 2)(x + 2).

Set this equal to zero and solve for the three critical values:

{-2, 0, 2}.

Set up a total of four intervals:  (-∞, -2), (-2, 0), (0, 2), (2, ∞).

Now choose a test point within each interval:  {-3, -1, 1, 3}.

Evaluate the first derivative, dy/dx, at each of these four test points.  Rule:  if the first derivative is +, we know the function is increasing on that interval; if -, the function is decreasing.

At x = -3, dy/dx = 4x(x - 2)(x + 2) becomes (-)(-)(-), which is -, so we know that the function is decreasing on interval (∞, -2).

At x = -1, dy/dx is (-)(-)(+), which is +, so we know that the function is increasing on (-2, 0).

At x = 1, dy/dx is (+)(-)(+), which is -, so we know that the function is decreasing on (0, 2).

Finally:  at x = 3, dy/dx is (+)(+)(+), so we know that the function is increasing on (2, ∞ ).

Answer:

87 million tickets are sold every year.

Step-by-step explanation:

Number of Americans attending classical music concerts:

30 million

Average number of times a spectator attends a concert in a year

2.9 times

Then the amount of tickets sold "x" for concerts of classical music in a year, will be equal to the number of people attending for the number of times they attend. So:

[tex]x= 30 * 2.9[/tex]

Where "x" is given in units of millions.

[tex]x = 87[/tex]

Hello There!

X=9

If we substitute X in this equation, it would say 3*9 which equals 27 subtract 6 and you get 21 which is true.

the value of x that satisfies the equation 3x - 6 = 21 is x = 9.

To find the value of x in the equation 3x - 6 = 21, we need to isolate the variable x on one side of the equation. We can do this by performing inverse operations.

First, we add 6 to both sides of the equation to cancel out the -6 term: 3x - 6 + 6 = 21 + 6, which simplifies to 3x = 27.

Next, we divide both sides of the equation by 3 to isolate x: (3x)/3 = 27/3, resulting in x = 9.

Therefore, the value of x that satisfies the equation 3x - 6 = 21 is x = 9. When we substitute x = 9 back into the equation, we get 3(9) - 6 = 21, which simplifies to 27 - 6 = 21, confirming that x = 9 is indeed the correct solution.

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To find the best fit model for the given data, draw a scatterplot and fit the models given. For each model, calculate the regression equation and R-squared value. Compare the R-squared values, the model with the highest value is the best fit.

To construct a scatterplot and identify the mathematical model that fits the data, first plot the points of weight and gallons of juice on a graph, with weight as the x-coordinate (independent variable) and juice as the y-coordinate (dependent variable). Next, try to draw each of the models given (linear, exponential, power and logarithmic) through the data points.

For each model, you'll need to calculate the regression equation and the R-squared value. The R-squared value is a statistical measure that shows the proportion of variance for a dependent variable that's explained by an independent variable. It ranges from 0 to 1, with 1 indicating a perfect fit.

After this, you should compare the R-squared values of different models. The model giving the highest R-squared value would be the best fit for your data. This model's regression equation will be the most accurate for the prediction of the dependent variable over the scope of your data.

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The best mathematical model that fits the given data is a linear model. The regression equation is y = -13.07 + 0.079x

To find the regression equation of the best model, we need to compare the values of R2 for different models. Let's start by constructing a scatterplot using the given data:

To solve this, you need to input the data points of the x (pounds of oranges) and the y (gallons of orange juice) into a calculator or statistics software to calculate the best regression model fit. The data points are 4321, 5012, 5239, 5366, 8978, 25413 for x and 341.3, 391.5, 399.6, 417.2, 656.1, 1927.3 for y.

From the scatterplot, we can see that the data points form a linear pattern. Therefore, the best mathematical model that fits the data is a linear model. Using a calculator or computer, we can find the regression equation:

y = -13.07 + 0.079x

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ANSWER

(-12,8)

EXPLANATION

The equation of the parabola is given as:

[tex]y = - \frac{1}{12} {x}^{2} - 2x - 1[/tex]

We can rewrite this in the vertex form as:

[tex] {(x + 12)}^{2} = - 12(y - 11)[/tex]

This implies that

[tex]4p = 12[/tex]

p=3

The vertex of this parabola is (-12,11)

The focus is

(-12,11-3)

=(-12,8)

Answer: -12, 8

Step-by-step explanation:

ANSWER

20 units.

EXPLANATION

We want to find the distance between (-8,-8) and (4,8).

We use the distance formula:

[tex]d = \sqrt{ {(x_2-x_1)}^{2} + {(y_2-y_1)}^{2} } [/tex]

We substitute the points into the formula to get:

[tex]d = \sqrt{ {(4 - - 8)}^{2} + {(8 - - 8)}^{2} } [/tex]

We simplify to get;

[tex]d = \sqrt{ {(12)}^{2} + {(16)}^{2} } [/tex]

[tex]d = \sqrt{144+ 256} [/tex]

[tex]d = \sqrt{400} [/tex]

[tex]d = 20[/tex]

The distance between the two points is 20 units.

Answer:

Distance = 20 units

Step-by-step explanation:

Points to remember

Distance formula

Length of a line segment with end points (x1, y1) and (x2, y2) is given by,

Distance = √[(x2 - x1)² + (y2 - y1)²]

To find the distance between give 2 points

Here (x1, y1) = (-8, -8)  and (x2, y2) = (4, 8)

Distance = √[(x2 - x1)² + (y2 - y1)²]

 = √[(4 - -8)² + (8 - -8)²]

 = √[(4 + 8)² + (8 + 8)²]

 = √[12² + 16²] =  √[144 + 256)

  =  √400 = 20

Therefore distance = 20 units

Answer: the height would be 15

Step-by-step explanation:

v= pi*radius^2*height

1215pi mm= pi*9^2*height

1215pi mm= pi*81*height

(divide by pi on both sides, which isolates pi on both sides)

1215 mm= 81*height

(divide by 81 on both sides, which would isolate 81 on the right side of the equation)

1215/81= 15= height

The height of the cylinder is 15 mm.

To find the height of a cylinder, we can use the formula for the volume of a cylinder, which is V = πr²h, where V is the volume, r is the radius, and h is the height.

Given that the volume is 1215π mm and the radius is 9 mm, we can plug these values into the formula and solve for h.

1215π = π(9)²h

Simplifying the equation, we have:

1215 = 81h

Dividing both sides by 81, we find:

h = 15 mm

Therefore, the height of the cylinder is 15 mm.

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It should be 40 degrees

Answer:

60 deg

Step-by-step explanation:

The measure of angle LPM is the sum of the measures of arcs LM and KN divided by 2.

m<LPM = (40 + 80)/2 = 60

Answer: m<LPM = 60 deg

It is 5/3 because it going up by 5 over by 3 hope this helps.

Answer:

D' ..................> (-2,2)

C'' .................> (3,-1)

A''' ................> (4,-2)

B'' .................> (4,2)

Step-by-step explanation:

Before we begin, we should remember the rotation rules:

1- Rotation 90° clockwise:

Coordinates (x,y) become (y,-x)

2- Rotation 90° counter clockwise:

Coordinates (x,y) become (-y,x)

3- Rotation 180° clockwise:

Coordinates (x,y) become (-x,-y)

4- Rotation 270° counter clockwise:

Coordinates (x,y) become (y,-x) ...........> same as rotation 90° clockwise

Now, let's consider the given problem:

1- Coordinates of D' if polygon ABCD rotates 90° counter clockwise to create A'B'C'D'

Original coordinates of point D are (2,2)

Rotation 90° counter clockwise means that the new coordinates will be (-y,x) which is (-2,2)

2- Coordinates of C'' if polygon ABCD rotates 90° clockwise to create A''B''C''D''

Original coordinates of point C are (1,3)

Rotation 90° clockwise means that the new coordinates will be (y,x) which is (3,-1)

3- Coordinates of A''' if polygon ABCD rotates 180° clockwise to create A'''B'''C'''D'''

Original coordinates of point A are (-4,2)

Rotation 180° clockwise means that the new coordinates will be (-x,-y) which is (4,-2)

4- Coordinates of B'' if polygon ABCD rotates 270° counter clockwise to create A''B''C''D''

Original coordinates of point B are (-2,4)

Rotation 270° counter clockwise means that the new coordinates will be (y,-x) which is (4,2)

Hope this helps :)

A: 5.50/p

B: 4.20/p

total 602.10

ratio 1:4 =5

602.10 ÷ 5 =120.42

A= 120.42

(B= 120.42×4= 481.86)

120.42÷ 5.50= 21.8945.....

Answer: 27 pounds

Step-by-step explanation:

We know that Coffee A costs $5.50 per pound, Coffee B costs $4.20 per pound and that this month the total cost was $602.10, then:

[tex]5.50A+4.20B=602.10[/tex]  (2)

We know that this month were used used four times as many pounds of type B coffee as type A, then:

[tex]B=4A[/tex]  (1)

So, you need to substitute (1) into (2) and solve for A:

[tex]5.50A+4.20(4A)=602.10\\22.3A=602.10\\A=27[/tex]

[tex]\bf \begin{array}{|c|ll} \cline{1-1} \textit{a\% of b}\\ \cline{1-1} \\ \left( \cfrac{a}{100} \right)\cdot b \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{5\% of 50000}}{\left( \cfrac{5}{100} \right)50000}\implies 500[/tex]

The realtor receives a 5% commission from the sale of a property. For a property sold at $50,000, the realtor's commission would be $2,500, calculated by multiplying the sale price by the commission rate.

To calculate the amount the realtor receives from the sale of a property when they earn a commission of 5%, you can use the following formula:

Commission = Sale Price * Commission Rate

In this case, the sale price of the property is $50,000 and the commission rate is 5%, which is 0.05 when expressed as a decimal.

Thus, the commission the realtor earns is calculated as follows:

Commission = $50,000 * 0.05

Commission = $2,500

Therefore, the realtor would receive $2,500 from the sale of a property that sells for $50,000.

The answer is:

The father's step is 80 cm

The son's step is 60 cm.

To solve the problem, we need to write equations in order to create a relation between the father's and son's steps, and the distance covered by them.

We know thay they covered 240 meters, and the father's step is 20 cm  (0.2m)greater than his son's step, so, we can write the following relation:

[tex]NumberOfSteps_{Son}=\frac{240m}{x}[/tex]

and,

[tex]NumberOfSteps_{Father}=\frac{240m}{x+0.2m}[/tex]

Now, if the know that the father made 100 fewer steps than his son, we have:

[tex]NumberOfSteps_{Son}-100=\frac{240m}{x+0.2m}[/tex]

Then, substituting the first equation into the second equation, we have:

[tex]\frac{240m}{x}-\frac{240m}{x+0.2m}=100\\\\\frac{240m(x+0.2m)-240mx}{x(x+0.2m)}=100\\\\240mx+0.2m*240m-240mx=(x)(x+0.2m)*100\\\\48m^{2}=100*(x^{2}+0.2mx)=100x^{2} +20mx\\\\48m^{2}=100x^{2}+20mx\\\\100x^{2}+20mx-48m^{2}[/tex]

We have a quadratic equation:

[tex]100x^{2}+20mx-48m^{2}=0[/tex]

Where,

[tex]a=100\\b=20\\c=-48[/tex]

So, solving it applying the quadratic formula, we have:

[tex]\frac{-b+-\sqrt{b^{2} -4ac} }{2a}[/tex]

[tex]\frac{-20+-\sqrt{20^{2} -4*100*-48} }{2*100}=\frac{-20+-\sqrt{400+19200} }{200}\\\\\frac{-20+-\sqrt{400+19200} }{200}=\frac{-20+-\sqrt{19600} }{200}\\\\\frac{-20+-\sqrt{19600} }{200}=\frac{-20+-(140)}{200}\\\\x_{1}=\frac{-20+140}{200}=0.6\\\\x_{2}=\frac{-20-140}{200}=-0.8[/tex]

Then, since distance cannot be negative, we need to take the positive value, so, the son's step is 0.6m or 60 cm.

Now, calculating the father's step, we have:

[tex]FatherStep=SonStep+20cm\\\\FatherStep=60cm+20cm=80cm[/tex]

Hence, we have that the father's step is 80 cm.

Have a nice day!

The distance of the father and son will be 80cm and 60cm respectively.

The number of son's step will be:

= 240/x

The number or father's step will be:

= 240/x + 0.2

Solving further, we'll have 100x² - 20mx - 48m² and using quadratic formula, this will be x1 = 0.6 and x2 = -0.8.

The son's step will be:

= 0.6 × 100 = 60cm

The father's step will be:

= 60cm + 20cm = 80cm

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The amount of money that can be saved without having a negative actual net income is: $170 can be saved resulting in an actual net income of $0.

Predict how much money can be saved without having a negative actual net income.

Monthly Budget  (is an itemized list of expected income and expenses that helps you to plan how the money will be spent or saved and track of spending habits.)

Budgeted Amount  (is an itemized allotment of funds, time for a given period)

Actual Amount  (is the particular year in which the amount is spent)

Income  (business receives in exchange to provide a good /service /through investing capital )

Wages  (is monetary compensation paid by  employer to employee in exchange for work done)

Savings Interest  (is money the you earn in return for holding your savings in an account.)

$1150

$25

$900

$25

Expenses

Rent

Utilities

Food

Cell Phone

Savings

$400

$100

$250

$75

$200

$400

$80

$200

$75

$____

Net Income

$150

$____

How much money can be saved without having a negative actual net income?

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Answer:

$170 can be saved resulting in an actual net income of $0.

Step-by-step explanation:

Separate the shape into two figures, a semi circle and a rectangle. Find the area of each and add.

See attached picture:

Answer:  The area of the figure is 29.13 sq. units.

Step-by-step explanation:  We are given to find the area of he figure shown in the graph.

The figure is made up of a triangle and a semi-circle.

Let us divide the figure into two parts, triangle ABC with base BC and altitude AD and the circle with diameter BC as shown in the attached image below.

From the graph, we note that

AD = 5 units  and  BC = 6 units.

Since BC is the diameter of the semicircle, so its radius will be

[tex]r=\dfrac{1}{2}\times BC=\drac{1}{2}\times6=3~\textup{units}.[/tex]

So, the area of the semicircle is given by

[tex]A_{sc}=\dfrac{\pi r^2}{2}=3.14\times \dfrac{3^2}{2}=3.14\times 4.5=14.13~\textup{sq. units}.[/tex]

Also, the area of the triangle ABC is given by

[tex]A_t=\dfrac{1}{2}\times AD\times BC=\dfrac{1}{2}\times5\times6=15~\textup{sq. units}.[/tex]

Therefore, the total area of the figure will be

[tex]A=A_{sc}+A_t=14.13+15=29.13~\textup{sq. units}.[/tex]

Thus, the area of the figure is 29.13 sq. units.

Answer:

$22.50

Step-by-step explanation:

The cost of admission for the 5 coaches was ...

5×$27.50 = $137.50

So the cost of admission for the 12 players was ...

$407.50 -137.50 = $270.00

The cost for one player was 1/12 that amount, or ...

$270/12 = $22.50

The admission cost for each player was $22.50.

The admission cost for each player is $22.50.

To find the admission cost for each player, we need to subtract the cost of admission for the coaches from the total cost of admission for all the coaches and players. The cost for five coaches is $((5  imes 27.50) = 137.50). So, the cost for the 12 players is $(407.50 - 137.50 = 270).

Therefore, the admission cost for each player is $22.50.

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Answer:

A

Step-by-step explanation:

The geometric mean of 2 values a and b is [tex]\sqrt{ab}[/tex]

The 7 th term is the geometric mean of the 6 th and 8 th terms

Hence

[tex]a_{7}[/tex] = [tex]\sqrt{6(216)}[/tex] = 36 → A

Hello There!

-What We Know-

Erin made 66 deviled eggs for a party.

After an hour 8 eggs were left.

X amount of eggs were eaten within the first hour.

——————————————————————————

We subtract 8 eggs how many were left after the hour

From how many eggs we had in total.

66-8=58.

58 eggs were eaten within the hour.

Answer:

58 eggs

Step-by-step explanation:

Answer:

x = 60°

Step-by-step explanation:

The measure of x is one- half the difference of the measures of the intercepted arcs.

The minor arc = 120°, hence

The major arc = 360° - 120° - 240°, so

x = 0.5(240 - 120) = 0.5 × 120° = 60°

Answer:

perpendicular lines

Step-by-step explanation:

The axis of symmetry for the parabola [tex]y = -5x^2 - 10x - 13[/tex] is the line x = -1.

To find the axis of symmetry for the parabola given by the equation [tex]y = -5x^2 - 10x - 13[/tex], we need to use the formula x = -b/(2a), where a and b are the coefficients of x² and x respectively from the standard form of a quadratic equation [tex]ax^2 + bx + c.[/tex]

In this case, a is -5 and b is -10. Plugging these values into the formula, we get:

x = -(-10)/(2*(-5))

x = 10/(-10)

x = -1

Therefore, the axis of symmetry for the parabola y = -5x² - 10x - 13 is the line x = -1.