HELP ASAP Leadership positions for a college social club are up for electionThree people are needed to fill the positions of president vice president, and secretaryOnly 10 members are willing to serve in these positions. Each candidate is eligible for cach position, and a candidate can only hold one leadership position. How many different ways can the three positions be filled using the 10 possible candidates?

Solve for the first variable in one of the equations, then substitute the result into the other equation.

Answer:

Point Form:

(−8,−17)

Equation Form: x=−8,y=−17

Answer:

x=-4, y=5

Step-by-step explanation:

Because y=-3x-7 and y=x+9, so y=y, and -3x-7=x+9

-3x-7=x+9,

so 4x=-16

      x=-4

so we put x=-4 to y=-3x-7

      y=(-3)* (-4)-7

      y=12-7

       y=5

the image of the point (x,y) after the rotation is (y,-x).

what is a graph?

A graph is a visual representation of data, relationships or information that is often used in mathematics, science, engineering, economics, and other fields.

A graph consists of a set of points, called vertices or nodes, that are connected by lines or curves, called edges or arcs. The vertices represent objects or events, while the edges represent the relationships or connections between them.

To apply the rotation of 90 degrees clockwise about the origin to the point (x,y), we use the following formula:

r(90,0)(x,y) = (y, -x)

To visualize this transformation, we can plot the preimage point (x,y) and the image point (y,-x) on a coordinate plane, and draw an arrow to represent the rotation.

Therefore, the image of the point (x,y) after the rotation is (y,-x).

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Answer:

First, let's find the total number of crayons produced each day:

500 boxes/day x 16 crayons/box = 8,000 crayons/day

Since each crayon uses 400 milliliters of melted wax, the total amount of wax used in one day can be found by multiplying the number of crayons by the amount of wax used per crayon and converting from milliliters to liters:

8,000 crayons/day x 400 mL/crayon x 1 L/1000 mL = 3,200 liters/day

Therefore, Bright Designs Crayon Factory uses 3,200 liters of melted wax each day.

Step-by-step explanation:

Answer:

Bright Designs Crayon Factory uses 3,200 liters of melted wax each day.

Step-by-step explanation:

The new function after the parent function f(x)=2^(x) is horizontally shrunk by a factor of two, translated right by 10 and translated 5 units down is f(x) = 2^((x-10)/2) - 5.

- Horizontal shrink by a factor of two: This means that the x-value of each point on the graph will be divided by 2. To achieve this, we can multiply the x-value inside the function by 1/2, or divide it by 2. So, the function becomes

f(x) = 2^((x/2))

- Translation right by 10: This means that the entire graph will shift 10 units to the right. To achieve this, we can subtract 10 from the x-value inside the function. So, the function becomes

f(x) = 2^((x-10)/2)

- Translation 5 units down: This means that the entire graph will shift 5 units down. To achieve this, we can subtract 5 from the entire function. So, the function becomes

f(x) = 2^((x-10)/2) - 5

Therefore, the new function is f(x) = 2^((x-10)/2) - 5

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An equation that could describe the graph below include the following: A. y = (1/2)^x + 6.

In Mathematics, an exponential function can be modeled by using this mathematical expression:

f(x) = a(b)^x

Where:

Generally speaking, When the base value or y-intercept (a) is less than one (1), the graph of an exponential function increases exponentially to the left. Additionally, the smaller the value of y-intercept (a), the steeper the slope (b) of the line.

By critically observing the graph of this exponential function, we can logically deduce that it was vertically shifted up by 6 units.

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The table with order pairs (-3,2), (0, 2), (3, 7),(5, 8) (6, 8) represents the function.

A relation is a function if it has only One y-value for each x-value.

In the given tables the third table represents the function.

The left bottom table is the third table.

As we observe the other tables there are repeating values of x.

In a function, for every y value there should be one unique x value.

(-3,2), (0, 2), (3, 7),(5, 8) (6, 8) represents the function.

Hence, the table with order pairs (-3,2), (0, 2), (3, 7),(5, 8) (6, 8) represents the function.

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Answer:

See explanation below

Step-by-step explanation:

We have RO parallel to LF

Segments RF and LO are transversal segments

For two parallel lines intersected by a transversal , the alternate interior angles are equal

In this figure, there are two pairs of alternate interior angles

∠O and ∠L form one pair and are equal
∠R and ∠F form another pair and are equal

Since the point T is formed by the intersection of two straight line segments,  the vertically opposite angles must be equal
m∠RTO = m∠LTF

So in the two triangles, ΔRTO and Δ FTL we have

m∠R = m∠F
m∠O = m∠L
m∠RTO = m∠LTF

By the AAA theorem of similarity of angles the two triangles are similar

AAA Similarity Criterion for Two Triangles

The Angle-Angle-Angle (AAA) criterion for the similarity of triangles states that “If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar”.

The arrow travels a distance of 2145 meters into the bale of paper after 3 seconds.

The distance an arrow travels into a bale of paper is given by the equation \[ s(t)=2401-(7-t)^{4}, 0 \leq t \leq 7 \]. To find the distance the arrow travels at a specific time, we simply plug in the value of \( t \) into the equation and solve for \( s \).

For example, if we want to find the distance the arrow travels after 3 seconds, we would plug in \( t=3 \) into the equation:
\[ s(3)=2401-(7-3)^{4} \]
\[ s(3)=2401-(4)^{4} \]
\[ s(3)=2401-256 \]
\[ s(3)=2145 \]

Similarly, we can find the distance the arrow travels at any time \( t \) by plugging in the value of \( t \) into the equation and solving for \( s \). This will give us the distance the arrow travels horizontally into the bale of paper at that specific time.

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The total value of Claude's liabilities is $24,950.(option a).

In Claude's balance sheet, the liabilities are shown separately from the assets. The liabilities are the debts or obligations that Claude owes to others. To find the total value of Claude's liabilities, we need to add up all the amounts listed under liabilities.

However, before we do that, let's quickly review what assets are. Assets are things that have value and can be owned or controlled by a person or a company.

Next, we can eliminate the two larger options, $50,370 and $25,420, because they are higher than Claude's total assets, which are not given in the question. If the liabilities were greater than the assets, this would mean that Claude owes more than he owns, which would not be a good financial situation.

This leaves us with the remaining option, $24,950. This is the total value of Claude's liabilities based on the information given in the question.

Hence the option (a) is correct.

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In a linear equation, There is one more test this quarter 39.

What in mathematics is a linear equation?

An algebraic equation with simply a constant and a first-order (linear) term, such as y=mx+b, where m is the slope and b is the y-intercept, is known as a linear equation.

                    Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables. Equations with variables of power 1 are referred to as linear equations. One example with only one variable is where ax+b = 0, where a and b are real values and x is the variable.

lowest average score = 80% * 50  = 40

 suppose the final exam score she must get at least x.

   ( x + 45 + 38 + 36) = 4 ≥ 40

                                x ≥ 39

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Answer:

Below

Step-by-step explanation:

The graph of the linear inequality y ≥ −2x − 1 is the region _______ the graph of the line y= - 2x -1

≥  is greater than or equal to and translates to 'on or above'

Zach still has to save 1/12 of the total amount after this month.

To find out how much of the total amount Zach still has to save after this month, we need to calculate how much he has already saved and how much he will save from this month's paycheck.

First, we know that Zach has already saved 2/3 of the money he will need for his trip. So, if we let x be the total amount he needs for the trip, we can write this as:
Already saved = 2/3 * x

Next, we know that Zach plans to put aside 3/4 of what he has left to save from this month's paycheck. Since he has already saved 2/3 of the total amount, he has 1/3 of the total amount left to save. So, the amount he will save from this month's paycheck is:
Amount from paycheck = 3/4 * (1/3 * x) = 1/4 * x

Now, we can add these two amounts to find out how much Zach has saved in total:
Total saved = Already saved + Amount from paycheck
Total saved = (2/3 * x) + (1/4 * x) = (8/12 * x) + (3/12 * x) = 11/12 * x

Finally, we can subtract this amount from the total amount he needs to find out how much he still has to save:
Amount still to save = Total amount - Total saved
Amount still to save = x - (11/12 * x) = (12/12 * x) - (11/12 * x) = 1/12 * x

So, Zach still has to save 1/12 of the total amount after this month.  

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Answer:

Step-by-step explanation:

The experiment is repeating and recording the result of an event in order to determine the events probability. For example, tossing a coin is an experiment.

The set that is resulting of the union operation between set A and it's complementary A' is given as follows:

U = {x:6≤ x ≤40, x is a positive whole number}.

The union operation of two sets is a mathematical operation that combines all the elements from two sets into a single set, without any duplicates. The union of two sets A and B is denoted as A ∪ B, and is defined as the set that contains all elements that are in at least one of the sets A and B.

The set A is given as follows:

A = {1, 2, 3, 4, 5}.

The complement of it's set is the set containing all the elements that are in the universe set and are not in A.

The union operation between a set and it's complement is always the universe set, hence it is given as follows:

U = {x:6≤ x ≤40, x is a positive whole number}.

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Answer:

Q1    Value of  angle B = 50°

Q2   BC =  1.3

Q3    AB = 2.57

Step-by-step explanation:

For Part 1
The three angles of  a triangle must add up to 180°

Therefore m∠B + 30 + 100 = 180

m∠B + 130 = 180

m∠B = 180 - 30 = 50°

part 2
The law of sines states that, in  a triangle, the ratio of each side to the sine of the angle opposite that side must be the same for all sides

We have side AC opposite ∠B

AC = 2 and we found that m∠B = 50° from part 1

The side BC is opposite ∠A which is 30°

Therefore, applying the law of sines
[tex]\dfrac{{AC}}{\sin 50^\circ} = \dfrac{{BC}}{\sin 30^\circ}\\\\\\\dfrac{2}{\sin 50^\circ} = \dfrac{{BC}}{\sin 30^\circ}\\\\[/tex]

Multiplying both sides by sin 30° we get
[tex]\dfrac{2}{\sin 50^\circ} \times \sin 30^\circ =BC\\\\or\\\\BC = \dfrac{2}{\sin 50^\circ} \times \sin 30^\circ[/tex]

Using a calculator to compute the right side we get
BC = 1.30540 ≈ 1.3

part 3

Similarly
[tex]\dfrac{AB}{\sin 100} = \dfrac{2}{\sin50}\\\\AB = \sin 100 \times \dfrac{2}{\sin 50} = 2.57115 \approx 2.57[/tex]

The Synthetic divison is (2x^(3)-10x^(2)+14x-24) ÷ (x-4) = 2x^(2) - 2x + 6.

To divide (2x^3 - 10x^2 + 14x - 24) by (x - 4) using synthetic division, the following steps should be taken:

1. Write the coefficients of the dividend in a row, placing the divisor to the extreme left of the row.
2. Bring the first coefficient of the divisor down.
3. Multiply the divisor by the number directly below it in the row and write the answer to the right of the number.
4. Add the two numbers to the right of the divisor and write the answer below.
5. Repeat steps 3 and 4 until the last number in the row is reached.
6. The last number in the row is the remainder, while the numbers to its left form the quotient.

For the example given, the process is as follows:

Therefore, the quotient is

.
To divide (2x^(3)-10x^(2)+14x-24) by (x-4) using synthetic division, we can follow the steps below:

Step 1: Write down the coefficients of the dividend polynomial in descending order of the exponents. In this case, the coefficients are 2, -10, 14, and -24.

Step 2: Write down the constant term of the divisor polynomial with the opposite sign. In this case, the constant term is -4, so we write down 4.

Step 3: Bring down the first coefficient of the dividend polynomial, which is 2.

Step 4: Multiply the number brought down by the constant term of the divisor polynomial, and write the result under the next coefficient of the dividend polynomial. In this case, 2 * 4 = 8, so we write 8 under -10.

Step 5: Add the numbers in the column, and write the result below. In this case, -10 + 8 = -2.

Step 6: Repeat steps 4 and 5 until all the coefficients of the dividend polynomial have been used. In this case, we get:

-2 * 4 = -8, 14 + (-8) = 6

6 * 4 = 24, -24 + 24 = 0

Step 7: The numbers in the last row are the coefficients of the quotient polynomial, and the last number is the remainder. In this case, the quotient polynomial is 2x^(2) - 2x + 6, and the remainder is 0.

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a. The expected number of applications rejected is 3  

b. The probability that among the next 15 applications, none will be rejected is 0.042

c. The probability that among the next ten applications, seven will be rejected is 0.48

d. The probability that among the next ten applications, between 1 and 8 applications is 0.95

The binomial distribution formula is used to calculate the probability of getting a certain number of successes (x) in a fixed number of independent trials (n) with a known probability of success (p) for each trial. The formula is:

Here we have

30% of the applications received for a position in a graduate school are rejected.

a) The number of rejected applications among the next 10 applications follows a binomial distribution with parameters n = 10 and p = 0.3.

The expected number of rejected applications is:

E(X) = np = 10 * 0.3 = 3

Hence, the expected number of applications rejected is 3

b) The probability of being rejected is 0.3

The probability that none of the next 15 applications will be rejected is:

P(X = 0) = (1 - p)ⁿ = (1 - 0.3)¹⁵= 0.042

Therefore, the probability that none of the next 15 applications will be rejected is 0.042 or approximately 4.2%.

c) The probability that 7 of the next ten applications will be rejected is:

By using the binomial distribution formula

P(X = 7) = (10, 7) × 0.3⁷ × 0.7³ =  

=  6435 × 0.0002187 × 0.343 = 0.48

Therefore, the probability that 7 of the next 10 applications will be rejected is 0.48 or approximately 48%.

d) The probability that between 1 and 8 (inclusive) of the next ten applications will be rejected is:

P(1 ≤ X ≤ 8) = P(X ≤ 8) - P(X ≤ 0) = Σ P(X = i) for i = 1 to 8

We can use the complement rule and calculate the probability of having 0 or 9 rejected applications, and subtract that from 1:

=> P(1 ≤ X ≤ 8) = 1 - [P(X = 0) + P(X = 9) + P(X = 10)]

= 1 - [(1 - p)ⁿ + (n, 1) × p¹ × (1 - p)⁽ⁿ⁻¹⁾ + (n, 0) × p⁰ × (1 - p)ⁿ]

= 1 - [(0.7)¹⁰ + ((10,1) × 0.3 × 0.7⁹) + (10, 0) (0.3)¹⁰]

= 1 - [ 0.02824 + 0.01210 + 0.000006]

= 1 - [ 0.040346]

= 0.95

Hence, The probability that between 1 and 8 (inclusive) of the next ten applications will be rejected is 0.95 or approximately 95%  

Therefore,

a. The expected number of applications rejected is 3  

b. The probability that among the next 15 applications, none will be rejected is 0.042

c. The probability that among the next ten applications, seven will be rejected is 0.48

d. The probability that among the next ten applications, between 1 and 8 applications is 0.95

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Answer: y = x + 8

Step-by-step explanation:

The provide situation can be modeled as a linear equation, y = 50 + 20x, where y is total cost and x is number of person who will attend the party. The slope of line tells me that the additional cost per person attending the party is $20. Thus, write option is option (b).

Emmett wants to through a party at an escape room. There are two charges for

escape room, one is booking fee and other one is additional cost per person attending the party. Let the booking fee and additional cost per person who attended the party of escape room be 'a ' and 'b'. Also, consider total person who will attend the party are equal to 'x'. So, total cost = booking cost + additional cost per person × number of people's who will attend the party. Let total cost of escape room for x people be 'y'. Then,

y = a + bx --(1)

it is act as modeled linear relationship, see above graph, which represents it graphically. In the graph initial total cost is 50, when no additional cost occur, that is booking fee equals to $50 or a = $50. After that, number of people increase total cost also increases. When there are 10 people in party then total cost is $250 and the additional cost = $200 for 10 people. So, additional cost per person

= 200/10 = 20 , i.e., b = $20 and equation(1) is rewrite, y = 50 + 20x.

If we compare it with standard linear equation, y = mx + c

where m --> slope of line

c --> y-intercept

then slope of equation (1) is equals to 'b' and b = 20. In this situation, the slope of line tell me that except the booking fee, the additional cost per person who will attend the party is equals to $20. Hence, the required answer is option(b).

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Complete question:

Emmett is planning a party at an escape room. The escape room will charge Emmett a booking fee and an additional cost per person attending the party.

This situation can be modeled as a linear relationship, see above graph. What does the slope of the line tell you about the situation?

A. The booking fee is $50

B. After the booking fee the party will cost $20 per person

C. After the booking fee the party will cost $25 per person

D. Including the bookin fee it would cost emmett $250 total for 10 people to attend.

B. After the booking fee the party will cost $20 per person

The area of the rectangle is 56 square yards.

What is the area of rectangle?

The area of a rectangle is the amount of space or surface inside the shape, measured in square units. It is calculated by multiplying the length of the rectangle by its width.

The formula for the area of a rectangle is:

Area = length x width

In this case, the width is 7 yards and the length is 8 yards. Therefore, the area of the rectangle is:

Area = 7 yards x 8 yards = 56 square yards

So the area of the rectangle is 56 square yards.

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When the sum of the angles is exactly 360 degrees, there is no overlap or gap at the vertex.

What is a polygon ?

A polygon is a two-dimensional closed shape made up of line segments. It has three or more straight sides and angles. Some common examples of polygons are triangles, rectangles, squares, pentagons, hexagons, and octagons.

The properties of a polygon, such as the number of sides, angles, and vertices, depend on its type and shape.

The condition that will not create a gap or an overlap at a vertex of polygons is "The sum of the measures of the angles is equal to 360°."

When the sum of the angles is exactly 360 degrees, there is no overlap or gap at the vertex.

This condition is also known as the angle sum property of polygons, and it states that the sum of the interior angles of a polygon with n sides is equal to (n-2) times 180 degrees.

Therefore, When the sum of the angles is exactly 360 degrees, there is no overlap or gap at the vertex.

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Answer:

∠M = ∠P = 112

∠N = ∠Q = 68

Step-by-step explanation:

The interior angles of a quadrilateral always sum up to 360°.

since NP//MP and NM//PQ => ∠N = ∠Q and ∠M = ∠P

and ∠N + ∠Q + ∠M + ∠P = 360

∠M = ∠P

6x - 2 = 4x + 36

6x - 4x = 36 + 2

2x = 38

x = 38/2 = 19

=> ∠M = ∠P = 4(19) + 36 = 112

∠N + ∠Q + ∠M + ∠P = 360

2∠N + 2∠M = 360

∠N + ∠M = 180

∠N = 180 - ∠M = 180 - 112 = 68

∠N = ∠Q = 68

1 meter = 1000 millimeter

Answer: :)

Step-by-step explanation:

The metric system is based on the International System of Units (SI), which is a standardized system of measurement used worldwide. The SI unit for length is the meter, and all other units of length in the metric system are derived from it.

The relationship between millimeters and meters can be expressed mathematically as follows:

1 mm = 0.001 m

1 m = 1000 mm

This means that if you have a length measurement in millimeters, you can convert it to meters by dividing by 1000. Conversely, if you have a length measurement in meters, you can convert it to millimeters by multiplying by 1000.

For example, if you have a piece of wire that measures 500 mm long, you can convert this to meters as follows:

500 mm ÷ 1000 = 0.5 m

Similarly, if you have a piece of fabric that measures 2.5 m long, you can convert this to millimeters as follows:

2.5 m x 1000 = 2500 mm

In summary, the relationship between millimeters and meters is that they are both units of length in the metric system, with one meter being equal to 1000 millimeters.

As a result, the park's estimated area, calculated using a trapezoidal sum with three equally spaced subintervals, is 50 square miles.

A quadrilateral with at least one set of parallel edges is known as a trapezoid. The bases and legs of a trapezoid are referred to by their parallel and non-parallel edges, respectively. The space between the bases of a trapezoid measured perpendicularly is its height. A = (1/2)h(b1 + b2), where A is the area, h is the height, and b1 and b2 are the lengths of the two bases, is the formula for a trapezoid's area.

given

By calculating the function's average at the left and right ends of the subinterval, it is possible to determine the height of each trapezoid.

A1 = (1/2) * (base1 + base2) * height

= (1/2) * (2 + 2) * 9 = 18 is the area of the first trapezoid.

A2 = (1/2) * (base1 + base2) * height

= (1/2) * (2 + 2) * 11 = 22 is the formula for the area of the second trapezoid.

The third trapezoid's area is given by

A3 = (1/2) * (base1 + base2) * (1/2) * (2 + 2). * 10

= 10

The three trapezoids' combined areas make up the entire area under the curve, which roughly corresponds to the size of the park:

A total = A1 + A2 + A3 

= 50

As a result, the park's estimated area, calculated using a trapezoidal sum with three equally spaced subintervals, is 50 square miles.

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The solution of the exponential equation is x = 0.35

An exponential equation is an equation that contains exponents.

Since we have the exponential equation

4ˣ = 8ˣ - 1

We proceed to solve as follows

4ˣ = 8ˣ - 1

(2²)ˣ = (2⁴)ˣ - 1

(2ˣ)² = (2ˣ)⁴ - 1

Let 2ˣ = y

So, we have that

y² = y⁴ - 1

Re- arranging, we have that

y⁴ - y² - 1 = 0

Also, let y² = p. So, we have that

p² - p - 1 = 0

Now, we find p using the quadratic formula.

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

where a = 1 b = -1 and c = -1

So, [tex]p = \frac{-(-1) +/-\sqrt{(-1)^{2} - 4(1)(-1)} }{2(1)}\\= \frac{1 +/-\sqrt{1 + 4} }{2}\\= \frac{1 +/-\sqrt{5} }{2}\\p = \frac{1 + 2.24 }{2} or p = \frac{1 - 2.24 }{2}\\p = \frac{3.24 }{2} or p = \frac{-1.24 }{2}\\p = 1.62 or p = -0.62[/tex]

We ignore the negative value.

So, p = 1.62

y² = 1.62

y = ±√1.62

y = ±1.273

Since y = 2ˣ, we have that

2ˣ = ±1.273

We ignore the negative value since the value of y cannot be negative.

So, 2ˣ = 1.273

Taking natural logarithm of both sides, we have that

㏑2ˣ = ㏑1.273

x㏑2 = ㏑1.273

x = ㏑1.273/㏑2

= 0.2412/0.693

= 0.348

≅ 0.35

So, x = 0.35

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Response to the given question would be that Hence, the surface area rectangular box has a surface area of 592 square cm.

Surface area is a measure of how much space an object's surface takes up overall. The total area of a three-dimensional shape's surroundings is its surface area. Surface area refers to the total surface area of a three-dimensional form. You may compute the surface area of a cuboid with six rectangular sides by adding together their individual areas. Instead, you may use the following formula to name the box's dimensions: Surface (SA) = 2lh plus 2lw plus 2hw. A three-dimensional shape's surface area is calculated as the total amount of space it occupies (a three-dimensional shape is a shape that has height, width, and depth).

Size (l) equals 10 cm

Height (h) = 8 cm

Size (h) equals 12 cm

We may use the following formula to get the box's surface area:

Surface Area = 2(lh, lw, and wh)

Inputting the values provided yields:

Surface Area equals 2 (10 x 8, 10 x 12, and 8 x 12) square centimetres.

Surface Area is equal to 2 (80, 120, and 96 square cm).

Surface Area: 2(296) square centimetres

592 square cm is the surface area.

Hence, the rectangular box has a surface area of 592 square cm.

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Answer:

5/33

Step-by-step explanation:

5/6 divided by 11/2

Invert the divider

5/6 x 2/ 11

10/66

Reduce to smallest fraction

5/33

Answer:

5/33

Step-by-step explanation:

Apply the fraction rule: a/b ÷ c/d = (a × d) ÷ (b × c) for 5/6 ÷ 11/2

a = 5

b = 6

c = 11

d = 2

(5 × 2) ÷ (6 × 11)

For "(6 × 11)", break 6 down into "2 × 3", so that you can cancel out the common factor, because there is also a 2 in "5 × 2".

= (5 × 2) ÷ (2 × 3 × 11)         -- cancel out the 2's

= 5 ÷ (3 × 11)     *****3 × 11 = 33*****

= 5/33

A trinomial with the leading coefficient -10 in terms of x in simplest form is -10x^2.

A trinomial with a leading coefficient of -10 in terms of x can be written as:
-10x^2 + bx + c

Where b and c are any real numbers. The simplest form of this trinomial would be:
-10x^2 + 0x + 0

Which can be simplified to:
-10x^2

A trinomial is an algebraic expression that has three non-zero terms and has more than one variable in the expression. This expression has three terms. Therefore, this expression is called a trinomial.

A trinomial is a type of polynomial but with three terms.

The examples of trinomials are: x + y + 7. ab + a +b,

3x+5y+8z with x, y, z variables

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