4.02 Lesson Check Arithmetic Sequences (3)

The line passes through the two points (-2,-9) and (3,-16).

We can write an equation in point-slope form using the following formula for the line passing through the point (-2,-9) and having a slope of m = -7/5:

y - y1 = m(x - x1)

where m is the slope of the line and (x1,y1) is the line's point.

Inputting the values provided yields:

y - (-9) = (-7/5)(x - (-2))

Simplifying:

y + 9 = (-7/5)(x + 2)

To get rid of the fraction, multiply both sides by 5. This gives us:

5y + 45 = -7(x + 2)

As we enlarge and rearrange, we obtain:

7x + 5y + 59 = 0

This is the line's standard form equation.

We can locate two points on the line and plot them to create a graph of the line. We can utilise the y-intercept, which is obtained by solving for y while setting x=0 in the equation:

7(0) + 5y + 59 = 0

5y = -59

y = -59/5

The y-intercept is therefore (0,-59/5). Using the slope, we may also locate a different point on the line:

m = -7/5

Beginning at (-2,-9), we can travel 7 units left and 5 units right to reach a different point:

(-2+5, -9-7) = (3, -16) (3, -16)

Hence, the line traverses both of the following two points: (3,-16).

Here is the line's graph:

     |

     |

*    |

 \   |

  \  |

   \ |

    \|

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

     |

    3|

     |

    2|

     |

    1|

     |

     |

     |

     |

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The answer is:

[tex]\sf{y+9=-\dfrac{7}{5}(x+2)}[/tex]

Work/explanation:

We need to determine the line's equation.  I will begin by writing the equation in point slope form:

[tex]\mapsto\phantom{333}\sf{y-y_1=m(x-x_1)}[/tex]

where m = slope and (x₁,y₁) is the point

Plug in the data:

[tex]\sf{y-(-9)=-\dfrac{7}{5}(x-(-2)}[/tex]

Simplify

[tex]\sf{y+9=-\dfrac{7}{5}(x+2)}[/tex]

Answer:

x =~ 31.2

y =~ 18

z =~ 9

Step-by-step explanation:

cos 30º = 27 / x

0.8660 = 27 / x

x = 27 / 0.8660

x =~ 31.18

w = height (smaller trinagle w, y, z)

tg 30º = w / 27

0.5774 = w / 27

w = 27 * 0.5774

w = 15.5898

tg 30º = z / 15.5898

0.5774 = z / 15.5898

z = 0.5774 * 15.5898

z = 9.00155052

cos 30º = 15.5898 / y

0.8660 = 15.5898 / y

y = 15.5898 / 0.8660

y =~ 18

On average, Miguel loses half a dollar every time he plays.

To check all the events (6), we label the chips.

Suppose one chip with 1 is labeled R1 and the other B1 (as if they were red and blue). Now, lets take all combinations; for the first chip, we have 4 choices and for the 2nd chip we have 3 remaining choices.

Thus there are 12 combinations. Since we dont care about the order, there are only 6 combinations since for example R1, 3 is the same as 3, R1 for us.

The combinations are: (R1, B1), (R1, 3), (R1, 5), (B1, 3), (B1, 5), (3,5)

We have that in 1 out of the 6 events, Miguel wins 2$ and in five out of the 6 events, he loses one.

The expected value of this bet is: 1/6*2+5/6*(-1)=-3/6=-0.5$.

In general, the expected value of the bet is the sum of taking the probabilities of the outcome multiplied by the outcome; here, there is a 1/6 probability of getting the same 2 chips and so on.

On average, Miguel loses half a dollar every time he plays.

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Therefore , the solution of the given problem of percentage comes out to be F. 0.159.

A% is a figure or integer that in statistics is shown as a percentage of 100. Additionally, the words "pct.," "pct.," and "pc" are rarely used. But even so, the "%" indicator is commonly used to denote it. There are no dimensions; the percentage total is flat. Percentages are really just numbers because the numerator is always 100. To indicate that a number is a percentage, either the percent mark (%) or the phrase "percent" must appear before the number.

Here,

Divide by 100 to transform a percentage to a decimal. Therefore, we must locate a decimal between 0.11 and 0.17 in order to determine the figure that corresponds to a percentage between 11% and 17%.

The choices are as follows:

=> F. 0.159

=> G. 0.013

=> H. 11/1,000

=> J. 6/100

The decimal 0.159 falls within this range (0.11 to 0.17), so the solution is F. 0.159.

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The volume of the larger cylinder is (25/8)πR³.

Volume is a measure of the amount of space occupied by a three-dimensional object. It is typically measured in units of cubic meters, cubic centimeters, cubic feet.

Let's denote the radius and height of the smaller cylinder as r and h, respectively, and the radius and height of the larger cylinder as R and H, respectively.

We know that the lateral area of a cylinder is given by the formula:

Lateral area = 2πrh

Using this formula for the smaller and larger cylinders, we get:

25π = 2πrh

64π = 2πRH

Dividing both sides of the first equation by 2π, we get:

rh = 25/2π

We also know that the following formula determines a cylinder's volume:

Volume = πr²h

Using this formula for the smaller cylinder, we get:

150π = πr²h

Substituting rh = 25/2π from above, we get:

150π = πr²(25/2π)

r² = 12

Taking the square root of both sides, we get:

r = √12 = 2√3

Substituting this value into rh = 25/2π, we get:

h = 25/(2πr) = 25/(4π√3)

Now, we can use the ratio of the lateral areas to find the ratio of the heights:

25π/64π = (2πrh)/(2πRH)

25/64 = rh/RH

25/64 = (2√3)(25)/(4πR)(H)

H = 2hR/r

Finally, we can use the formula for the volume of the larger cylinder to find its volume:

Volume = πR²H

Volume = πR²(2hR/r)

Volume = 2πR³h/r

Substituting the values we found for r, h, and H, we get:

Volume = 2π(R³)(25/(4π√3))/(2√3)

Volume = (25/8)πR³

Therefore, the volume of the larger cylinder is (25/8)πR³.

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x² - x - 6 = 0 is the standard form quadratic equation with solutions of -2 and 3.

There are two roots—or solutions—to a quadratic equation with real or complex coefficients. These two possibilities may or may not be distinct, and they may or may not be true. The standard quadratic equation, Y=x2x6, can be solved using the set of solutions 2, 3, which are given.

The elements of a quadratic equation whose solutions are -2 and 3 are (x + 2) and (3, respectively) (x - 3).

The total value of the quadratic equation is 0.

We can add the two elements together to obtain the quadratic equation in standard form:

(x + 2) (x - 3) = x²-3x+2x-6

Simplifying this expression, we get:

x²- x - 6 = 0

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As a result of angle-angle-side (AAS) alignment, we have ABO, DBO, and AEO, CEO.

To prove △BEC≅△DEA:

First, we can see that segment AC and segment BD intersect at point O, which is the point of intersection of the angle bisectors of △BED.

Since AC and BD are angle bisectors, they are perpendicular bisectors of each other, so they divide the angles of △BED into congruent angles.

This means that angle ABO is congruent to angle DBO, and angle AEO is congruent to angle CEO.

Since angle ABO and angle AEO are vertical angles, they are congruent.

Similarly, angle DBO and angle CEO are vertical angles, so they are congruent.

Therefore, we have △ABO≅△DBO and △AEO≅△CEO by angle-angle-side (AAS) congruence.

Since BE is common to both triangles, we can conclude that △BEC≅△DEA by side-angle-side (SAS) congruence.

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the relationship between x and y given by the values (-3, -4), (2, 6), and (1, 4) is not a direct variation.

To determine whether the relationship between x and y is a direct variation, we need to check if y is directly proportional to x. In other words, we need to check if there is a constant of proportionality that relates the two variables.

To do this, we can calculate the ratio of y to x for each pair of corresponding values:

y/x = (-4)/(-3) = 4/2 = 2

y/x = 6/2 = 3

y/x = 4/1 = 4

If y is directly proportional to x, then the ratio of y to x should be the same for all pairs of corresponding values. However, in this case, the ratios are not equal. Therefore, the relationship between x and y is not a direct variation.

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Answer:

The conclusion is false.

Step-by-step explanation:

[tex]\frac{57}{3} = 19\\\\\frac{9}{3} = 3[/tex]

Although 3 and 9 do share some multiples, they must be divisible by 3. For example. 36/3 = 12, and 12 is divisible by 3. However, 57/3 = 19, which is not divisible by 3.

The height of the lighthouse from the top of the cliff is given as 25.3 cm

tan 37.2 = h / 825

h = 825 * 0.7590

h = 626.209

tan 38.3 = 626.209 + a / 825

825 tan 38.3 = 626.209 + a

651.55 = 626.209 + a

a = 651.55 - 626.209

a = 25 . 33 meters

Hence we can say that the height of the  lighthouse from the top of the cliff is given as 25 . 33 meters

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7 number of  students chose all three activities.

To find out how many students chose all three activities, we need to subtract the number of students who chose two activities from the total number of students. There are 76 students who chose sport, 64 students who chose music, and 40 students who chose drama. 12 students chose both music and sport, 7 chose both sport and drama, and 21 chose both music and drama. We can subtract these two-activity choosers from the total number of students to find out how many chose all three activities: 145 - (12 + 7 + 21) = 105. Therefore, 7 students chose all three activities.

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Velocity is given by the change in the distance divided by the change in the time, hence the following equation is built to model the relationship between these three variables:

v = d/t.

The parameters for the trip are given as follows:

Hence the velocity is obtained as follows:

230/7 = 32.86 miles per hour.

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the solution (x = 3/2, y = 5) satisfies both equations, and therefore it is the correct solution.

We can solve this system of equations using either substitution or elimination method.

Using substitution method:

From the second equation, we can solve for x in terms of y:

4x + 2y = 16

4x = 16 - 2y

x = 4 - (1/2)y

Substitute this expression for x into the first equation:

4x + 3y = 21

4(4 - (1/2)y) + 3y = 21

16 - 2y + 3y = 21

y = 5

Now, substitute y = 5 into either of the original equations to solve for x:

4x + 2y = 16

4x + 2(5) = 16

4x + 10 = 16

4x = 6

x = 3/2

Therefore, the solution to is x = 3/2 and y = 5.

We can check this solution by substituting x = 3/2 and y = 5 into both equations:

4x + 3y = 21

4(3/2) + 3(5) = 6 + 15 = 21

and 4x + 2y = 16

4(3/2) + 2(5) = 6 + 10 = 16

So the solution (x = 3/2, y = 5) satisfies both equations, and therefore it is the correct solution.

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Therefore , the solution of the given problem of ratio comes out to be  8:17 .

The simple formula "a / b," where "b" may be greater than zero, can be used to determine a group of integers "a" and "b" that appear to be equal. Two ratios are combined to form a ratio. If there were only one man and three ladies, the fraction would be 1:1. In the group, there are 3/4 women and 1/4 males. Parts can be a division between things, a figure, or a particular percentage of a component's volume.

Here,

Let x be the price of plastic per square centimetre used to create visage A.

Consequently, the price of the components for aspect A is:

=> 4πx

The metal sector's size on face B is:

=>  (20/360)π(2cm)² = (1/9)π cm²

The plastic sector's size on face B is:

=>  (340/360)π(2cm)² = (17/9)π cm²

Face B's entire surface size is:

=>  (1/9)π + (17/9)π = 2π cm²

Let y be the plastic's expense per square centimetre used to create face B.

Consequently, the price per cm2 of the metal used to create face B is three and a half times the price per cm2 of the plastic, leading to:

=> (3/2)y = (3/2)(2x) = 3x

The material cost for the metal section on face B is inversely correlated with its area, so:

=>  (1/9)π(3x)

The price of the plastic used in the section on face B is inversely proportional to its area, so:

=>  (17/9)πy

=> 8πx : 17πy = 8 : 17

The response is 8:17,

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In addition of fractions together student added the denominators together, which is the error in solution. You must have like denominators in order to add fractions together is the strategy for solution. Haley bought 7/8 pound of cheese.

We have a incorrect solution of problem consists addition and subtraction fractions. We have to identify error and solve problem correctly and also share a strategy for solution. Now the problem is defined as : Quantity of mozzarella cheese bought by Haley = [tex] \frac{3}{8} [/tex] pound

Quantity of parmesan cheese bought by Haley = [tex] \frac{1}{2} [/tex] pound

Total pound of cheese that she did bought = sum of parmesan cheese and mozzarella cheese

= [tex] \frac{3}{8} + \frac{1}{2} [/tex]

Incorrect work:

[tex] \frac{3}{8} + \frac{1}{2} = \frac{4}{10}[/tex] [tex]= \frac{2}{5} [/tex]

Identification and Explain the Error :

In the problem student correctly identified that this was addition of fractions however he was incorrect in solving. When adding fractions together you do not add denominators together.

He also forgot that you must have "like" denominators.

Correct Solution:

[tex] \frac{3}{8} + \frac{1}{2} [/tex]

For addition of fractions, make the same denominator in both of fractions. So, multiply the denominator and numentor in fraction 1/2 by 4.

[tex] = \frac{3}{8} + \frac{1 \times 4}{2\times 4} = \frac{3}{8} + \frac{4}{8} [/tex][tex] =\frac{7}{8} [/tex], which is the required solution.

Strategy: We do not add denominator together ever. You must have like denominators in order to add them together. Hence, Haley bought 7/8 pound of cheese.

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Complete question:

Error Analysis Adding and subtracting fractions :

Read the word problem. Look at the students work and solution. Identify the error and describe it. Solve the problem correctly. Then share a strategy this student could use to prevent the same error in the future. Haley bought 3/8 pound of mozzarella cheese and 1/2 pound of parmesan cheese. How many pound of cheese did she buy?

Incorrect work or solution : 3/8 + 1/2

= 4/10 = 2/5

Haley bought 2/5 pound of cheese.

The angle of elevation is approximately 36.87 degrees and the horizontal distance is 264.6 feet

a.

The diagram is attached below;

b. We can use the tangent function to find the angle of elevation:

tan(θ) = opposite / adjacent

tan(θ) = 300 / 400

tan(θ) = 0.75

θ = tan^(-1)(0.75)

θ ≈ 36.87 degrees

So the angle of elevation from the boat to the parasail rider is approximately 36.87 degrees.

c. We can use the Pythagorean theorem to find the horizontal distance from the parasail rider to the boat:

d = √(400^2 - 300^2)

d ≈ 264.6 feet

So the horizontal distance from the parasail rider to the boat is approximately 265 feet.

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The line is 13 units to the right (or shifted by -13 units).

A mathematical statement known as an equation consists of two algebraic expressions separated by equal signs (=) on either side.

It demonstrates the equality of the relationship between the printed statements on the left and right.

Left side equals right side in all formulas.

To find the values of unknowable variables, which stand in for unknowable quantities, you can solve equations.

A statement is not an equation if it lacks the equals sign.

When two expressions have the same value, a mathematical statement known as an equation will include the symbol "equal to" between them.

According to our question-

The ratio of y to x, which indicates how many units change in y when a specific number of units change in x, is used to express a line's slope.

A has a slope of 2 (x factor) = 2/1.

and as a result, y moves 2 units while x moves 1 unit.

B has a slope of 1 = 1/1.

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Answer:

[tex]30x +68[/tex]

Step-by-step explanation:

[tex]6(9+5x)+14[/tex]

[tex]\text{Distribute the 6}[/tex]

[tex]30x +68[/tex]

a. The number of students that failed in all subject is 4

b. The percentage of students who failed one of economic and mathematics is 5.2%

c. The probability that a student at random failed is 1/10

A set is defined as the collect of item with similarities.

n( of student that passed math only) = 6

n( of students that passed account only) = 9

n( of students that passed economics only) = 16-(4+3+2)

= 16-9 = 7

n ( of student that passed math and account only) = 5

n( of student that passed account and economics only) = 2

n ( of student that passed math and economics only) = 18-(5+3+y) = 6

= 18-8-y = 6

= 10-y = 6

y = 4

n( of students that passed all the three subjects) =19-( 5+2+x) = 9

= 19-7-x = 9

= 12 -x = 9

-x = -3

x = 3

a. Therefore no of student that failed all subject =

6+9+7+5+2+4+3+z = 40

36+z = 40

z = 40-36 = 4

b. Student that failed at least math and economics = 4+9 = 13

= 13/40 × 100 = 5.2%

c. probability that a student at random failed = 4/40 = 1/10

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The length of the slide itself in feet would be = 75 ft.

To calculate the length of the water slide the Pythagorean formula should be used which is written as follows;

C² = a² + b²

a = length of the tower = 21 ft

b = distance from the base of the tower = 72ft

C = length of the water slide = ?

c² = 21²+72²

C² = 441 + 5184

c² = 5625

C = √5625

C = 75 ft

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Answer: I belive the answer is 0lxoly+39.56

Step-by-step explanation:

SIKE

The length of the arc of the sector is approximately 34.6 millimeters when rounded to one decimal place.

What is angle ?

An angle is a geometric figure formed by two rays that share a common endpoint, called the vertex of the angle. The rays are often referred to as the sides of the angle. Angles are typically measured in degrees or radians and are used to describe the amount of turn between two lines or surfaces.

In a circle, the measure of the arc of a sector is proportional to the measure of the central angle that it subtends. Therefore, to find the length of the arc of the sector in this problem, we need to first find the measure of the central angle that corresponds to the inscribed angle of 312 degrees.

The measure of the central angle is equal to twice the measure of the inscribed angle, which is 312/2 = 156 degrees. This is because the inscribed angle subtends an arc that is half as long as the arc subtended by the central angle.

The length of the arc of the sector can be calculated using the formula:

arc length = (central angle / 360 degrees) x (2 x pi x radius)

Plugging in the values we have:

arc length = (156/360) x (2 x pi x 4mm)

arc length = (13/30) x (8pi)

arc length = (104/30) x pi

arc length ≈ 34.557mm

Therefore, the length of the arc of the sector is approximately 34.6 millimeters when rounded to one decimal place.

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Answer: 1/6

Step-by-step explanation:

1/1 - 5/6

2    6-5/6

2    1/6

The length of the dock on the scale drawing would be 13.89 inches (250/18 = 13.89).

On a scale drawing of a dock for a marina, 1 inch equals 18 feet. This means that for every inch on the scale drawing, it represents 18 feet in reality. The dock is 250 feet long. To calculate the length of the dock on the scale drawing, divide 250 by 18. This gives the result of 13.89. This means that the length of the dock on the scale drawing is 13.89 inches. This is because for every inch on the scale drawing, it represents 18 feet in reality. Therefore, the length of the dock on the scale drawing is 13.89 inches (250/18 = 13.89). To ensure accuracy, the calculation should be double checked. This can be done by multiplying 13.89 by 18 which should give the result of 250. This confirms that the length of the dock on the scale drawing is 13.89 inches.

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Answer:To solve this problem, we need to standardize the values of 55 and 63 to z-scores and then use the standard normal distribution table to find the area between those two z-scores.

The z-score for a value of 55 is:

z = (55 - 62) / 4 = -1.75

The z-score for a value of 63 is:

z = (63 - 62) / 4 = 0.25

Using a standard normal distribution table, we can find that the area to the left of z = -1.75 is 0.0401 and the area to the left of z = 0.25 is 0.5987. Therefore, the area between z = -1.75 and z = 0.25 is:

0.5987 - 0.0401 = 0.5586

To convert this area back to a percentage, we multiply by 100:

0.5586 * 100 ≈ 55.9%

Therefore, the answer is A. 55.9%.

Step-by-step explanation:

Answer:

Step-by-step explanation:

483278.4

The proportion of green apples can be estimated in the same way: expressions  Green apple proportion = (number of green apples) / (Total number of fruits)

In mathematics, you can multiply, divide, add, or subtract. An expression is formed as follows: Expression, number, and math operator Numbers, parameters, and functions make up a mathematical expression. It is possible to contrast phrases and expressions. Every mathematical statement that contains variables, numbers, and a mathematical action between them is referred to as an expression. For example, the phrase 4m + 5 is made up of the phrases 4m and 5, as well as the variable m from the provided equation, all separated by the mathematical sign +.

a) Because the basket contains four different types of fruits, the proportion of cherries can be computed as follows:

Cherry proportion = (number of cherries) / (Total number of fruits)

We can't compute the proportion of cherries without knowing how many fruits are in the basket.

b) The proportion of green apples can be estimated in the same way:

Green apple proportion = (number of green apples) / (Total number of fruits)

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[tex]g( \sqrt{55} ) = ( \sqrt{55} ) {}^{2} + 11 = 55 + 11 = 66[/tex]

[tex]f(g(55)) = f(66) = \sqrt{66 - 2} = \sqrt{64} = 8[/tex]

The value of the function f(g(√55)) is 3√7.

A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable).

We have the function:

f(x) = √ (x - 2)

and, g(x) = x² + 11.

As, g(√55)

= (√55)² + 11

= 55 + 11

= 66

So, f(g(√55))

= √ (65 - 2)

= √63

= 3√7

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