BENCHMARK ASSESSMENT - 2/20

Which of the ratios below has a unit rate of 8? Select all that apply.

A) 40:5
B) 24:16
C) 96:12
D) 48:6
E) 2:24

So, the solution is x = 1, y = -1, z = 3.

One way to solve the system is using Gaussian elimination with back-substitution. The steps are:

1. Rewrite the system in matrix form, called the augmented matrix:

[3 3 6 | 9]

[1 1 2 | 3]

[2 5 10| 15]

[-1 2 4 | 6]

2. Perform row operations to convert the matrix to an upper triangular matrix.

[1 1 2 | 3]

[0 2 4 | 3]

[0 2 6 | 6]

[0 0 -2 | -6]

3. Use back-substitution to find the values of the variables. Starting with the last equation:

-2z = -6 => z = 3

4y + 6(3) = 6 => y = -1

x + 2(-1) + 2(3) = 3 => x = 1

So, the solution is x = 1, y = -1, z = 3.

Another way is Gauss-Jordan elimination, it's similar with Gaussian elimination but it will change the matrix to a row echelon matrix and then to a reduced row echelon matrix.

So, the solution is x = 1, y = -1, z = 3.

Complete Question: solve the system using either gaussian elimination with back-substitution or gauss-jordan elimination. (if there is no solution, enter no solution. if the system has an infinite number of solutions, express x, y, and z in terms of the parameter t.)                                                                                                              3x + 3y + 6z = 9

x + y + 2z = 3

2x + 5y + 10z = 15

−x + 2y + 4z = 6

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Answer:

The equation 49^2 + 16^2 = 784 can be rewritten as (x/24.5)^2 + (y/8)^2 = 1.

The standard form of an ellipse is (x/a)^2 + (y/b)^2 = 1, where a and b are the lengths of the major and minor axes, respectively. In this case, a = 24.5 and b = 8.

To graph the ellipse, we can use the equation in the standard form and plot points that satisfy the equation. The center of the ellipse is (0,0) and it has a major axis of 24.5 and minor axis of 8.

The graph of this ellipse will look like an oval, with the center at the origin, major axis along the x-axis and minor axis along the y-axis.

The equation y = x² represents a parabola as a curve

Two or more expressions with an Equal sign is called as Equation.

The equation y = x² represents a parabola as a curve

The equation is given as:

y = x²

The above equation has a degree of 2

Polynomials that have a degree of 2 are referred to as quadratic equations

The curve of a quadratic equations are called parabola

Hence, the equation y = x² represents a parabola as a curve

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The amount in the account after 15 years is, $6,671.67

A loan or deposit amount is subject to compound interest as interest. In everyday life, it is the most prevalent idea. Both the principal and the interest accrued over time affect the compound interest for a given amount. Between compound and simple interest, this is the primary distinction.

[tex]A = P[1 + r/n]^{nt}[/tex]

A  = P + I

Where,

A = compound interest

P = Principle Amount

r = Interest rate

n = number of period for which interest is calculated

t = time

Given that,

A equal to P quantity 1 plus 0.054 over 2 end quantity all raised to the power of 2 times t

an annual interest rate of 5.4% compounded semiannually

the initial investment P is $3,000

the amount in the account after 15 years = ?

[tex]A = 3000[1 + 0.054/2]^{2t}[/tex]

after 15 years compound interest,

[tex]A = 3000[1 + 0.054/2]^{30}[/tex]

A = $6,671.67

A = P + I

P (principal) = $3,000.00

I (interest) = $3,671.67

Hence, the amount in the account after 15 years is, $6,671.67

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The amount loaned at 6% was $9200. The solution has been obtained by making and solving the equations.

What is an equation?

A claim that two expressions are equal and connected by the equals sign "=" is the mathematical definition of an equation. The two expressions need not be entirely composed of variables; one expression may have variables while the other may not.

We are given that a bank loaned out $12,000 and a part of it at the rate of 6% per year and the rest at the rate of 16% per year.

Let the amount loaned at 6% be 'x'

So, amount loaned at 16% will be '12,000 - x'

The total interest received was $1000.

Therefore,

⇒6%x + 16%(12000-x) = 1000

⇒0.06x + 1920 - 0.16x = 1000

⇒920 = 0.1x

⇒9200 = x

Hence, the amount loaned at 6% was $9200.

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Answer:  Total area = 40 cm² + 25.12 cm² = 65.12 cm²

Step-by-step explanation:

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Answer:65.1

Step-by-step explanation:

If Canada specializes in producing wheat and the US specializes in producing TV sets, both countries will benefit from trade.

To determine this if Canada produces wheat, the opportunity cost is 1/2 TV set. If the US produces wheat, the opportunity cost is 2 TV sets. So, Canada has a comparative advantage in producing wheat. Similarly, the US has a comparative advantage in producing TV sets.

Let's say Canada produces 100 bushels of wheat, and the US produces 100 TV sets. If they trade with each other, Canada can exchange 50 bushels of wheat for 50 TV sets, and the US can exchange 50 TV sets for 50 bushels of wheat. Both countries would benefit from this trade.

Canada would end up with 50 TV sets and 50 bushels of wheat, while the US would end up with 50 bushels of wheat and 50 TV sets. Both countries would be better off than if they had not traded.

Opportunity cost in mathematics is a term that is used to describe the cost of one alternative in terms of the benefits forgone by the best alternative. It can also be represented in terms of mathematical equations.

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The probability that one player receives all the 13 hearts is 1/2598960.

How likely something is to occur can be determined using probability. For many events, it is challenging to make an accurate prediction.

The number of ways to distribute the 13 hearts among the 4 players is 4!. The number of ways to distribute the remaining 39 cards among the 4 players is 39!/(13!13!13!13!). Hence, the total number of ways to distribute the 52 cards among the 4 players is (52!)/(13!13!13!13!).

Therefore, the probability that one player receives all the 13 hearts is (4!)/((52!)/(13!13!13!13!)) = 1/((52515049)/(13121110)) = 1/2,598,960.

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Carmen scored 50%of the total number of items in second test, the number of items in the second test is, 75

The percentage is a mathematical term, which means a number or a ratio which is represented in fractions of 100. We use the '%' symbol to represent the percentage.

Formula for percentage;

Percentage = (present quantity/whole quantity)×100

Given that,

In the first Summative Test,

Carmen scored 20 points which is 80% of the number of items

In the second test,

Carmen scored 30 points,

Her total score is 50% of the number of items

The total number of the items in the second test = ?

Suppose, total number of items in the first test = x

                                               in the second test = y

⇒  20 points =  80x/100

⇒  20 points =  0.80x

⇒  x   =  25

⇒  0.5(x + y)  = 20 + 50

⇒     x + y = 100

⇒     y = 75

Hence, the total number of items in the second test is 75

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The quantity which represents the 0.031 in the given formula of compound interest, the value represents the annual compound interest rate 3.1%, So option A is correct

A loan or deposit amount is subject to compound interest as interest. In everyday life, it is the most prevalent idea. Both the principal and the interest accrued over time affect the compound interest for a given amount. Between compound and simple interest, this is the primary distinction.

[tex]A = P[1 + r/n]^{nt}[/tex]

Where,

A = compound interest

P = Principle Amount

r  =  Interest rate

n = number of period for which interest is calculated

t = time

Given that,

A equal to 2,400 plus quantity 1 plus 0.031 over 4 end quantity all raised to the power of 4 times t

[tex]A = 2400[1 + 0.031/4]^{4}[/tex]

So, here we can see that r is equal to 0.031

It represents the annual compound interest rate of the compound interest

Hence, it is a annual compound interest which has value of 3.1%

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The height of the tree is 39 ft from the ground.

In a right-angled triangle, the square of the hypotenuse side is equal to the sum of the squares of the other two sides, according to Pythagoras's Theorem. These triangle's three sides are known as the Perpendicular, Base, and Hypotenuse.

Let us suppose the height of the tree = l

The distance from the point on the ground to the top of the tree is thus:

h = 11 + 2(l)

Given that the base = 80 ft.

To find the height of the tree we can use the Pythagorean theorem.

The Pythagorean theorem is given as follows:

a^2 + b^2 = c^2

(80)^2 + (l)^2 = (11 + 2l)^2

Using the identity (a + b) ^2 = a^2 + 2ab + b^2 we can write the equation as:

6400 + l^2 = 121 + 44l + 4l^2

Take all the terms on a single side of the equation:

4l^2 - l^2 + 44l + 121 - 6400= 0

3l^2 + 44l - 6279 = 0

Solve the following quadratic equation using the equation of the quadratic function:

l = 39

Hence, the height of the tree is 39 feet.

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Answer:

n< -1

Step-by-step explanation:

1. Rewrite so n is on the left side of the inequality.

−5n+2n>3

2.Add 5n and 2n

-3n >3

3. Divide each term then simplify.

The expression that does not equal the length of one of the pieces in feet is C) 4/20

The expression that show the length of one of the pieces of feet would be the expressions that lead to the solution:

= Length of board cut by David / Number of equal pieces

= 20 / 4

= 5 ft each

Option A mirrors the equation above and so is equal to the length of one of the pieces. Option B also equals the equation above as the signs ÷ and / are both for division.

Option C does not equal the length of one of the pieces as it gives:

= 4 / 20

= 1 / 5 ft

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Options for this question are:

A)20/4

B)20 ÷ 4

C)4/20

Using expression {River=x(race)} River's length is 7.38 times the race's length, approximately.

What are expressions?

Expressions in math are mathematical statements that have a minimum of two terms containing numbers or variables, or both, connected by an operator in between. The mathematical operators can be of addition, subtraction, multiplication, or division. For example, x + y is an expression, where x and y are terms having an addition operator in between. In math, there are two types of expressions, numerical expressions - that contain only numbers; and algebraic expressions- that contain both numbers and variables.

e.g. A number is 6 more than half the other number, and the other number is x. This statement is written as x/2 + 6 in a mathematical expression. Mathematical expressions are used to solve complicated puzzles.

We know that

The river is 310 miles long.

The race is 42 miles long.

So, if we want to know know how many times as long as the race is the river, we just need to use this expression

River=x(race)

Where x represents the factor that expresses the ratio between both, solving for x we have

x=river/race=310/42≈7.38

Therefore,

            River's length is 7.38 times the race's length, approximately.

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Right question:-

The Annual Mug Race 42 mile is the

longest river sailboat race in the world. The

event is run along the St. Johns River, which

is 310 miles long. About how many times as

long as the race is the river?

There are 15120 integers greater than 5400 that have distinct digits and do not contain 2 and 7.

The number of required integers is the number of integers greater than 5400 which have distinct digits and do not contain 2 and 7. This can be calculated with the formula:

Number of Required Integers = (9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) - (2 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) = 30240 - 15120 = 15120.

This is because for the first part of the equation, we must calculate the total number of possible permutations of distinct digits from 0-9 (9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1). For the second part, we must calculate the number of permutations of digits from 0-9 excluding 2 and 7 (2 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1). We then subtract the second part from the first to get the required answer.

Therefore, there are 15120 integers greater than 5400 that have distinct digits and do not contain 2 and 7.

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Answer: 1/2n = 1/4n - 3n+3/2n

Start by multiplying both sides by 2n to get rid of the fractions:

1 = 1/2 - 3n+3/n

Then by multiplying both sides by n to get rid of the fraction on the right side:

n = -3n+3

Then by adding 3n to both sides:

4n = 3

Then by dividing both sides by 4:

n = 3/4

There is no extraneous solutions in this case, because no denominators are equal to zero and all the operations are valid. Therefore, the solution is n = 3/4

Step-by-step explanation:

x is the probability of people who receive their original hats back out of the n people. All possible re-assignments are equally likely.

x is the number of people out of the n people who are lucky enough to receive their original hats back from the hat-check attendant. All possible re-assignments of hats to people are equally likely. This means that each person has an equal chance of getting their own hat back, and the probability of any particular person getting their own hat back is 1/n. The expected value of x is therefore n/2, since the average of all possible outcomes is the expected value. The variance of x is (n-1)/12, since the variance is the average of the squared differences from the expected value.

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To lose the least amount of money in the sales promotion, you should order the minimum number of hats and pairs of sunglasses while still satisfying the constraints given.

Let x be the number of hats you order and y be the number of pairs of sunglasses you order.

From the information given, we can set up the following constraints:

y = 2x (because you will sell at least twice as many hats as pairs of sunglasses)

y <= 100 (because your supplier tells you that your order of sunglasses cannot exceed 100 pairs)

x + y >= 210 (because you estimate that you should order at least 210 items in all)

We also know that you will lose $3 on every hat and $2 on every pair of sunglasses sold. So the objective is to minimize the cost:

Cost = 3x + 2y

Now we can use the constraints to set up a linear optimization problem. We can use a solver to minimize the cost and find the optimal values of x and y that meet the constraints.

Solving the problem gives us x = 140, and y = 280, so you should order 140 hats and 280 pairs of sunglasses to lose the least amount of money in the sales promotion.

The transformation with respect to dilation will only preserve the angle measure i.e. B.

What is dilation in mathematics?

Dilation is a process of changing the size of an object or shape by decreasing or increasing its dimensions by some scaling factors. For example, a circle with radius 10 unit is reduced to a circle of radius 5 unit. The application of this method is used in photography, arts and crafts, to create logos, etc. In Geometry, there are four basic types of transformations. They are:

Some features of shapes that remain unchanged during the dilation transformations are:

Now,

As in dilation Each angle of the figure remains the same.

Hence,

              The transformation with respect to dilation will only preserve the angle measure

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The graphical summary of salaries is an example of b. descriptive statistics.

Descriptive statistics is the branch of statistics that is used to summarize, organize, and describe data. It is used to present data in a clear and meaningful way, so that patterns, relationships, and trends can be easily understood. A graphical summary of all employees' salaries, such as a histogram or a bar chart, is an example of descriptive statistics.

This graphical summary represents the entire population of employees and not a subset of it, so it's not a sample. The factory owner is not making any inferences about the population based on the data, so it's not an example of statistical inference. And this is not an experiment because no manipulation of the data was made.

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This method of observing the behaviour of each group of children is best characterized as experimental.

Experimental research is a study worked with a scientific approach using two sets of variables. The first set works as a constant, which we use to measure the differences of the second set.

This can be regarded as something that depends on other factors, such as a test score, which can be considered a dependent variable because it could change depending on several factors, such as how much you studied.

It should be noted that dependent and independent variables are variables in mathematical modelling and experimental sciences; however, the dependent variables can be seen as ones whose values are studied under the supposition.

Hence the correct option is a.

--The given question is incomplete; the complete question is

"A researcher randomly assigned boys and girls to each of two groups. One group watched a violent television program, while the other watched a nonviolent one. The children were then observed during free play, and the incidence of aggressive behaviour was recorded for each group. This method is best characterized as"--

a. Experimental

b. Case study

c. Correlational

d. Observation"--

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The green triangle has an area of 6 sq units. The purple triangle has an area of 6 sq units. If the orange triangle is placed directly next to the green triangle such that the two triangles together make a rectangle, the total area of this rectangle is 12 sq units, which is half the area of the green triangle.

We know that the area of a triangle is given by

1/2 X Base X height

Now, for the green triangle, the area  will be

1/2 X (5 - 1) X (9 - 6)

= 1/2 X 4 X 3 sq units

= 6 sq units

The distance between the highest vertex and the base's y coordinates for the purple triangle will make up the height. hence we get

1/2 X (9 - 5) X (4 - 1)

= 1/2 X 4 X 3 sq units

= 6 sq units

Now if the orange triangle is placed on the graph in such a way that it makes a rectangle with the green one, the length and width of the rectangle will be the same as the base and height of the triangle

Hence the area of the rectangle will be

4 X 3 sq units

= 12 sq units

We can see that it is double the area of the purple triangle.

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Trigonometry is best used in navigation to estimate where to point the compass to get a straight line. It will be simple to pinpoint a location as well as find distance and see the horizon using a compass and trigonometric functions in navigation.

The area of mathematics known as trigonometry examines the correlation between the ratios of a right-angled triangle's sides to its angles. Trigonometric ratios, such as sine, cosine, tangent, cotangent, secant, and cosecant, are used to study this relationship. The concept of trigonometry was developed by the Greek mathematician Hipparchus, and the word trigonometry is a derivative of Latin from the 16th century.

One of the most significant area of mathematic is rigometry. The words "Trigonon" and "Metron," which mean respectively "triangle" and "measure," are combined to form the word "trigonometry." It is the investigation of how a right-angled triangle's sides and angles relate to one another. Therefore, using formulas and identities based on this relationship aids in determining the measure of a right-angled triangle's unknown dimensions.

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Full question:

What is the best use of trigonometry?

Answer:

4 daysb

Step-by-step explanation:

Here the number of animals and the number of days are in inverse proportion. Hence the food will last 4 days.

The food will last for 6 days for 20 animals.

If there were 10 more animals in the cattle, the total number of animals would be 20+10 = 30.

The food will last for the same 6 days but now it would be divided among 30 animals.

Therefore, the food would last for 6 days / 30 animals = 0.2 days per animal.

So, the food will last for 0.2 days if there were 10 more animals in the farmer's cattle.

The volume of the common region between two spheres, each with radius r, is 4/3π(r^3 - 32r^6).

The volume common to two spheres, each with radius r, is given by the equation V = 4/3πr3.

The volume of the common region between two spheres, each with radius r, is given by the formula:

V = 4/3π(r^3 - (r^2 - d^2)^3/2),

where d is the distance between the centers of the spheres.

In this case, the distance between the centers is 2r, since the center of each sphere lies on the surface of the other sphere.

Thus, the volume of the common region is 4/3π(r^3 - (r^2 - (2r)^2)^3/2)

= 4/3π(r^3 - (r^2 - 4r^2)^3/2)

= 4/3π(r^3 - (r^2 - 4r^2)^3/2)

= 4/3π(r^3 - (4r^2)^3/2)

= 4/3π(r^3 - 64r^6/2)

= 4/3π(r^3 - 32r^6)

= 4/3π(r^3 - 32r^6)

= 4/3π(r^3 - 32r^6).

Therefore, the volume of the common region between two spheres, each with radius r, is 4/3π(r^3 - 32r^6).

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Complete question : What is the volume common to two spheres, each with radius r, if the center of each sphere lies on the surface of the other sphere?

The measure of the other acute angle of the right angled triangle is 54⁰.

For the stated question:

Length of one side = 8 ft.

Let angle A = 36⁰.

For right angled triangle, Angle B = 90⁰

Then,

∠A + ∠B + ∠C = 180⁰

36⁰ + 90⁰ + ∠C = 180⁰

∠C = 180⁰ - 126⁰

∠C = 54⁰

Thus, the other acute angle is found as 54⁰.

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The correct question is-

1. A right triangle has one leg that measures 8ft What is the measure of the acute triangle? Assume that the angle is 36⁰