A sum of money is shared such that Mark gets twice as much as Sofia and Tina gets ½ of Sofia's share. Express these amounts in the form of a proportion. If Tina receives $1 200, how much do Mark and Sofia each receive? ​

[tex]\begin{cases} x = 6 &\implies x -6=0\\ x = -5 &\implies x +5=0\\ \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{original~polynomial}{a ( x -6 )( x +5 ) = \stackrel{0}{y}}\hspace{5em}\stackrel{\textit{since the leading coefficient is 4}}{a=4} \\\\\\ 4(x-6)(x+5)=y\implies 4(x^2-x-30)=y\implies \boxed{4x^2-4x-120=y}[/tex]

Answer:

Step-by-step explanation:

To find two points on the line represented by the function r(x) = 4 - 32x, we can arbitrarily choose two values of x and then find the corresponding values of r(x) using the equation.

For example, if we choose x = 0, then r(0) = 4 - 32(0) = 4. So the point (0,4) is on the line.

Similarly, if we choose x = 1, then r(1) = 4 - 32(1) = -28. So the point (1,-28) is on the line.

Therefore, we have two points (0,4) and (1,-28) on the line represented by the function r(x) = 4 - 32x. These two points can be used to graph the function.

Answer: 1407.4

Step-by-step explanation:

Answer:

If the roots of a quadratic equation are given as x = -1 ± 2i, then the equation can be written in the form:

(x - (-1 + 2i))(x - (-1 - 2i)) = 0

Simplifying this expression, we get:

(x + 1 - 2i)(x + 1 + 2i) = 0

Expanding the left side of this equation, we get:

x^2 + x(2 - 2i + 2 + 2i) + (1 - 2i)(1 + 2i) = 0

Simplifying this expression, we get:

x^2 + 2x + 5 = 0

Therefore, the quadratic equation with roots of x = -1 ± 2i is:

x^2 + 2x + 5 = 0

The resultant vector for her trip is  <-9, 14>, where the units are miles.

Let's define north as the positive y-axis and east as the positive x-axis.

First we know that she goes 9 mileswestand 4 miles north of her house, then that can be represented by the vector:

<-9, 4>

Then she travels at a speed of 30mi/h for 20 minutes, we know that:

1hour = 60min

then:

20 min = (20/60) hours = (1/3) hours.

The distance that she travels in that time is:

d = (30 mi/h)*(1/3)h = 10 mi

So she travles 10 miles north, this time the vector is <0, 10>

To get the resultant vector we just need to add the two ones we have:

<-9, 4> + <0, 10>  = <-9 + 0, 4 + 10> = <-9, 14>

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Check the 1st picture below.

from the template below, hmm we can say that a horizontal shift of +5 simply means B = 1 and C = 5, so C/B = 5

[tex]y=x^3\hspace{4em}y=1(1x+0)^3+0\implies y=1(1x+5)^3+0\implies y=(x+5)^3[/tex]

Check the 2nd picture below.

Answer: The given equation of the line is y + 2 = -1/2 x, or equivalently, y = -1/2 x - 2. This equation is in slope-intercept form, y = mx + b, where the slope m is -1/2.

A line parallel to this line will have the same slope, so its equation will also be in the form y = -1/2 x + b, where b is the y-intercept that we need to find.

To find b, we can use the fact that the line passes through the point (2, -7). Substituting x = 2 and y = -7 into the equation, we get:

-7 = -1/2 (2) + b

Simplifying and solving for b, we get:

b = -7 + 1 = -6

Therefore, the equation of the line parallel to y + 2 = -1/2 x that passes through the point (2, -7) is:

y = -1/2 x - 6

Step-by-step explanation:

The algebraic expression that represents the verbal expression is given as follows:

(j + m)/7.

The algebraic expression representing the situation is obtained applying the proportions in the context of the problem.

The verbal expression is given as follows:

"Jaylen brought j crackers and combined them with Marvin’s m crackers. They then split the crackers equally among 7 friends.".

The total number of crackers is then given as follows:

j + m.

The crackers were divided evenly among the seven people, hence the expression representing the number of crackers by each person is obtained applying the proportion as follows:

(j + m)/7 -> total divided by number of people.

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Answer:

7p(p-3) (if you're trying to find P then p is 3)

Step-by-step explanation:

First multiply the bottom so you get 9p^2 - 72p and then you can remove 1 variable of p from the bottom and top to get 63(p^3-11p^2+24p) and take one p out for 63p(p^2-11p+24) and factor for 63p(p-8)(p-3)/9(p-8) and take p-8 out of both. Then, you're left with 63p(p-3)/9 to get 7p(p-3) and to solve for p if the equation is equal to 0 -> 7(p)(p-3) = 0

-> 7p^2-21p

->7p^2 = 21p

7p = 21

p = 3

Answer: 19 units

Step-by-step explanation:

Just subtract the y values and find the absolute value since the x values are the same.

Answer:

a person randomly grabs candies out of a bag there are 480 candles in the bag the person looked and they pulled out 20% if the candies were yell0ow and 15% of them were green how many more were yellow than green

Step-by-step explanation:

it's B

Answer: 5 more yellow candies than 15 candies, or 5

Step-by-step explanation: Think, if there are 20 yellow candies and 15 of them green;

You need to take away 20 - 15 which will be 5, 5 more yellow candies than 15 candies

Keep up the good work=)

The rules that are functions on an appropriate domain are A and D, and other solutions are shown below

Given that

g(x) = 4x² + x + 7

For g(-2), we substitute -2 for x in g(x) = 4x² + x + 7

So, we have

g(-2) = 4(-2)² + (-2) + 7

Evaluate

g(-2) = 21

Here, we have

A(t) = 3- 2t² + 4^t.

To calculate A(2) - A(1), we calculate A(2) and A(1) and then subtract the values

So, we have

A(2) = 3 - 2(2)² + 4² = 11

A(1) = 3 - 2(1)² + 4¹ = 5

So, we have

A(2) - A(1) = 11 - 5

A(2) - A(1) = 6

Here, we have

g(x) = 3(x - 6)² + 1

The vertex above is

vertex = (6, 1)

Because the leading coefficient (a) is greater than 0

i.e. a = 3, then the vertex is a minimum

So, the range is y ≥ 1

Here, we have the following definition

a ♡ b = (a + b)/2.

This means that

(5 ♡ 7) ♡ 11 = (5 + 7)/2 ♡ 11

(5 ♡ 7) ♡ 11 = 6 ♡ 11

So, we have

(5 ♡ 7) ♡ 11 = (6 + 11)/2

(5 ♡ 7) ♡ 11 = 8.5

The rules that are functions on an appropriate domain are A and D.

Rules B, C, E, and F are not functions on an appropriate domain

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Complete question

Let g(x) = 4x^2 + x + 7. Find g(-2)

Let A(t) = 3- 2t^2 + 4^t. Find A(2) - A(1).

Let g(x) = 3(x - 6)^2 + 1. Find the range of g. Write your answer in interval notation.

Define the operation ♡ so that a ♡ b = (a + b)/2.

Find the value of (5 ♡ 7) ♡ 11

Which of these rules are functions on an appropriate domain?

Select all that apply.

A inputs a real number, subtracts 100, then outputs the square root of the result if it is real.

B inputs a date, then outputs the name of a person born on that date.

C inputs a real number x, then outputs a real number y that is less than 1/1000 larger than x

D inputs a real number x, then outputs the result of the calculation (x^2-1)/(x-1)(x+1)

E inputs a real number x, then a coin is flipped. If the coin flips heads, the output is x+2. If the coin flips tails, the output is x+3.

F inputs a person, then outputs their birth date.

the probability that the CDs are in alphabetical order is 1/12528

Probability is a branch of mathematics that deals with the study of random events or outcomes. It is the measure of the likelihood or chance of an event occurring, expressed as a number between 0 and 1. An event with a probability of 0 is impossible, while an event with a probability of 1 is certain to occur.

Given by the question.

Let's assume that the CDs are labeled as CD1, CD2, CD3, ..., CD9.

If CD1 is placed first, then there is only one way to place the remaining CDs in alphabetical order.

If CD2 is placed first, then there is only one way to place CD1 and CD3-CD9 in alphabetical order.

If CD3 is placed first, then there are two ways to place CD1-CD2 and CD4-CD9 in alphabetical order arrange (CD1-CD2-CD4-CD5-CD6-CD7-CD8-CD9 or CD2-CD1-CD4-CD5-CD6-CD7-CD8-CD9).

Similarly, for CD4-CD9, there are 3, 4, 5, 6, and 7 ways to place the remaining CDs in alphabetical order.

Therefore, the total number of ways to arrange the CDs in alphabetical order is:

[tex]1 + 1 + 2 + 3 + 4 + 5 + 6 + 7 = 29[/tex]

Hence, the probability of the CDs ending up in alphabetical order is:

[tex]1/12528[/tex]

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Answer:

For a polygon to be regular, it must satisfy the following two conditions:

All of its interior angles must be congruent.

All of its sides must be congruent.

Let n be the number of sides of the polygon. Then the measure of each interior angle of a regular n-gon is given by:

180(n-2)/n degrees

For a 25-gon, the measure of each interior angle is:

180(25-2)/25 = 156.8 degrees

Therefore, option D) "All interior angles and all sides are congruent" must be true for a 25-gon to be regular.

Answer:

Step-by-step explanation:

The length of one side of the box is approximately 8.5 inches whose circumference is 37.7 inches.

Circumference is the distance around the edge of a circle or any circular object. It can be found by multiplying the diameter of the circle by pi (π), which is approximately 3.14159. Alternatively, it can be found by multiplying twice the radius of the circle by pi.

The diameter of the circle inscribed in the square is equal to the side length of the square. Since the circumference of the pizza crust is 37.7 inches, the diameter of the circle is:

d = circumference / pi = 37.7 / pi ≈ 12

Therefore, the radius of the circle is r = d/2 ≈ 6. To find the side length of the square, we need to find the diagonal of the square, which is equal to twice the radius of the circle. As a result, the square's length of one side is:

s = diagonal / √(2) = 2r / √(2) = 6 √(2) ≈ 8.5

Therefore, the length of one side of the box is approximately 8.5 inches.

The circumference is an important measurement in geometry and is used to calculate the perimeter of circles, as well as to determine the length of curved lines in various other contexts.

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Answer:

5.91

Step-by-step explanation:

formula of a circle = (x-h)^2 +*y-k)^2 = r^2

thus, to get the radius, square root 35 which when rounded two deicmal places is 5.91

Answer:

22 degree

Step-by-step explanation:

We know

∠1 and ∠2 combined must be 90 degrees. We know angle 2 is 68 degrees.

What is the measure of angle 1?

90 - 68 = 22 degrees

So, angle 1 is 22 degrees.

Answer: 64x

Step-by-step explanation:

Subtract 6 from 20 to get 14.

Multiply 14 and  1/2 to get 14/2

Divide 14 by 2 to get 7.

Add 1 and 7 to get 8.

Multiply 8 and 8 to get 64.

​

.

The distance between zero and point P is 5 units.

To find the distance between two points on the number line, we must take the difference between the values ​​corresponding to those points.

The value of 0 is trivially 0.

The value of P is between -4 and -6, so we conclude that:

P = -5

Taking the difference between these values ​​we obtain:

distance = 0 - (-5)

distance = 0 + 5 = 5

The distance between those values ​​is 5 units.

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Answer:

When we raise a power to another power, we multiply the exponents. So:

(6^7)^8 = 6^(7*8) = 6^56

Therefore, (6^7)^8 can be written in the form 6^56, where k=56.

Step-by-step explanation:

Answer:

1385 square feet

Step-by-step explanation:

42ft÷2=21ft to get the radius

21^2=441ft

441×π=441π

441π=1385.44236

=1385 to the nearest whole number

"Given that they passed the test" says we're only considering the 21 students who passed the test.

There were 9 students who didn't pass the test and 7 of those didn't complete the HW.  This means that 2 students who did the HW didn't pass the test.  

Since 20 students completed the HW and 2 didn't pass the test, 18 students completed the HW and passed the test.

18/21 = 6/7 or 85.71%

Your probability is 85.71%.

Answer: Since we know that the two variables are independent, we can use the addition rule of probability to find the probability of selecting a student who is either female or a psychology major:

P(female or psychology major) = P(female) + P(psychology major) - P(female and psychology major)

Since the variables are independent, we know that P(female and psychology major) = P(female) x P(psychology major) = 0 x 0.1 = 0.

Therefore:

P(female or psychology major) = 0 + 0.1 - 0 = 0.1

So the probability of randomly selecting a student who is either female or a psychology major is 0.1.

Step-by-step explanation:

The dimensions of the original piece of cardboard are 15 inches by 23 inches.

Let's start by defining the variables we'll need for this problem:

w = width of the original piece of cardboard (in inches)

l = length of the original piece of cardboard (in inches)

h = height of the box (in inches)

We know that the length of the original piece of cardboard is 8 inches more than the width, so we can write:

l = w + 8

When Lorene cuts out 4-inch squares from each corner of the rectangular sheet, the resulting height of the box will be 4 inches. Therefore, the height of the box is:

h = 4

The base of the box will have dimensions (w - 8) inches by (l - 8) inches, since 4 inches are removed from each side of the width and length. The volume of the box is given as 420 cubic inches, so we can write:

V = (w - 8)(l - 8)(4)

Substituting the expression for l in terms of w, we get:

420 = (w - 8)(w + 8 - 8)(4)

420 = (w - 8)(w)(4)

105 = w(w - 8)

Solving for w using the quadratic formula, we get:

w = 15 or w = -7

Since w must be a positive value, we can discard the solution w = -7. Therefore, the width of the original piece of cardboard is:

w = 15 inches

Using the expression for l in terms of w, we can find the length of the original piece of cardboard:

l = w + 8 = 23 inches

Therefore, the dimensions of the original piece of cardboard are 15 inches by 23 inches.

Answer: I'd be happy to help you with these problems!

1a.

i. Cardinal: There were 10 apples in the basket.

ii. Ordinal: He came in first place in the race.

iii. Nominal: My favorite color is blue.

i. 1997 in Roman numerals is MCMXCVII.

ii. 15-03-2023 in Roman numerals is XV-III-MMXXIII.

2a.

2√128 - √243 + 5√125 + √27 - √80 - 162 + √98

First, simplify the square roots:

2√64·2 - 3√81·3 + 5√25·5 + √9·3 - √16·5 - 162 + √49·2

Next, simplify further:

2·8√2 - 3·3√3 + 5·5√5 + 3√3 - 4√5 - 162 + 7√2

Combine like terms:

15√2 - √3 + 8√5 - 162 - 4√5 + 7√2

Simplify further:

22√2 + 4√5 - √3 - 162

b.

1.3555555555.... can be written as 1.36 in 7º.

c.

Village A has a funeral every 10 days, so it will have a funeral on March 4th, March 14th, March 24th, etc.

Village B has a funeral every 15 days, so it will have a funeral on March 4th, March 19th, April 3rd, etc.

Village C has a funeral every 20 days, so it will have a funeral on March 4th, March 24th, April 13th, etc.

To determine the next time all three villages will have a funeral on the same day, find the least common multiple of 10, 15, and 20.

10 = 2 x 5

15 = 3 x 5

20 = 2 x 2 x 5

The least common multiple is 2 x 2 x 3 x 5 = 60.

So, the next time all three villages will have a funeral on the same day is 60 days after March 4th, which is May 3rd, 2023.

Comment if you have any more questions.

Answer:

Step-by-step explanation:

To solve for x in the equation x^2 - 16 = 0, we can use the difference of squares formula:x^2 - 16 = (x + 4)(x - 4) = 0

Now we can set each factor equal to zero and solve for x:x + 4 = 0 or x - 4 = 0x = -4 or x = 4
Therefore, the solutions to the equation x^2 - 16 = 0 are x = -4 and x = 4.

Tο find the cοst οf 10 such pencils, we need tο multiply the price οf οne pencil by 10. We can write 1½ as a mixed number οr as an imprοper fractiοn.

A prοpοrtiοn is an equatiοn that sets twο ratiοs at the same value. Fοr instance, yοu cοuld express the ratiο as fοllοws: 1: 3 if there is 1 bοy and 3 girls (fοr every οne bοy there are 3 girls) There are 1 in 4 bοys and 3 in 4 girls. 0.25 are male (by dividing 1 by 4)

A mixed number is a whοle number and a fractiοn tοgether, such as 1½. An imprοper fractiοn is a fractiοn where the numeratοr (tοp number) is larger than the denοminatοr (bοttοm number), such as 3/2. Tο multiply a mixed number by a whοle number, we can either cοnvert the mixed number tο an imprοper fractiοn first, οr multiply the whοle number by the whοle part and the fractiοnal part separately and then add them tοgether. Fοr example:

1½ × 10 = (1 + ½) × 10 = (1 × 10) + (½ × 10) = 10 + 5 = 15

Or

1½ × 10 = (3/2) × 10 = (3 × 10) / 2 = 30 / 2 = 15

Either way, we get the same answer: The cοst οf 10 such pencils is 15.

Tο find the cοst οf 20 such pencils, we can use the same methοd but multiply by 20 instead οf 10. Fοr example:

1½ × 20 = (1 + ½) × 20 = (1 × 20) + (½ × 20) = 20 + 10 = 30

Or

1½ × 20 = (3/2) × 20 = (3 × 20) / 2 = 60 / 2 = 30

Either way, we get the same answer: The cοst οf 20 such pencils is 30.

Tο check if the price is directly prοpοrtiοnal tο the number οf pencils, we need tο see if the ratiο οf the price tο the number οf pencils is cοnstant. A ratiο is a cοmparisοn οf twο quantities using divisiοn. A cοnstant ratiο means that nο matter hοw many pencils we buy, the price per pencil will always be the same. Fοr example:

Price / Number οf Pencils = Cοnstant Ratiο 15 / 10 = Cοnstant Ratiο 30 / 20 = Cοnstant Ratiο

We can simplify these fractiοns by dividing bοth numbers by their greatest cοmmοn factοr (GCF). The GCF οf 15 and 10 is 5, sο we divide bοth numbers by 5:

(15 / 5) / (10 / 5) = Cοnstant Ratiο 3 / 2 = Cοnstant Ratiο

The GCF οf 30 and 20 is alsο 5, sο we divide bοth numbers by 5:

(30 / 5) / (20 / 5) = Cοnstant Ratiο 6 / 4 = Cοnstant Ratiο

Nοw we can cοmpare these simplified fractiοns and see if they are equal. If they are equal, then it means that the price per pencil is always the same regardless οf hοw many pencils we buy. If they are nοt equal, then it means that the price per pencil changes depending οn hοw many pencils we buy.

3 /2= ?6 /4 12 /8= ?12 /8 12 /8=12 /8

Since these fractiοns are equal, it means that:

Price / Number οf Pencils = Cοnstant Ratiο Price ∝ Number οf Pencils

The symbοl ∝ means “is prοpοrtiοnal tο”. Sο this equatiοn tells us that yes, the price is directly prοpοrtiοnal tο the number οf pencils because fοr every increase in οne unit in οne quantity (number οf pencils), there is a cοrrespοnding increase in a fixed amοunt in anοther quantity (price).

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