Find the length of the missing side
10
11
12
13

(a) T is one-to-one if and only if T carries linearly independent subsets of V onto linearly independent subsets of W.

(b) If T is one-to-one, then S is linearly independent if and only if T(S) is linearly independent.

(c) If β is a basis for V and T is one-to-one and onto, then T(β) is a basis for W.

(a) Assume T is one-to-one. Let S be a linearly independent subset of V, and suppose T(S) is linearly dependent. Then there exist distinct vectors s1, s2, ..., sn in S such that T(s1), T(s2), ..., T(sn) are linearly dependent. This means that there exist scalars c1, c2, ..., cn, not all zero, such that c1T(s1) + c2T(s2) + ... + cnT(sn) = 0. Since T is linear, we have T(c1s1 + c2s2 + ... + cnsn) = 0. But since T is one-to-one, this implies that c1s1 + c2s2 + ... + cnsn = 0, contradicting the assumption that S is linearly independent. Hence, T(S) must be linearly independent.

Conversely, assume that T carries linearly independent subsets of V onto linearly independent subsets of W. Let v1 and v2 be distinct vectors in V, and suppose T(v1) = T(v2). Then {v1, v2} is linearly dependent, which implies that there exist scalars c1 and c2, not both zero, such that c1v1 + c2v2 = 0. Applying T to both sides yields c1T(v1) + c2T(v2) = 0, which implies that T(v1) and T(v2) are linearly dependent. This contradicts the assumption that T carries linearly independent subsets of V onto linearly independent subsets of W. Hence, T must be one-to-one.

(b) Assume T is one-to-one and let S be a subset of V. Suppose S is linearly independent and that T(S) is linearly dependent. Then there exist distinct vectors s1, s2, ..., sn in S such that T(s1), T(s2), ..., T(sn) are linearly dependent. This means that there exist scalars c1, c2, ..., cn, not all zero, such that c1T(s1) + c2T(s2) + ... + cnT(sn) = 0. Since T is linear, we have T(c1s1 + c2s2 + ... + cnsn) = 0. But since T is one-to-one, this implies that c1s1 + c2s2 + ... + cnsn = 0, contradicting the assumption that S is linearly independent. Hence, T(S) must be linearly independent.

Conversely, assume that T(S) is linearly independent whenever S is a linearly independent subset of V. Let v1 and v2 be distinct vectors in V, and suppose T(v1) = T(v2). Then {v1, v2} is linearly dependent, which implies that there exist scalars c1 and c2, not both zero, such that c1v1 + c2v2 = 0. Since {v1, v2} is linearly dependent, we have either v1 = 0 or v2 = 0. Without loss of generality, assume v1 = 0. Then T(v1) = 0 = T(v2), and hence T({v1, v2}) = {0} is linearly dependent. This contradicts the assumption that T carries linearly independent subsets of V onto linearly independent subsets of W. Hence, S must be linearly independent.

(c) First, we will show that T(β) spans W. Let w be an arbitrary vector in W. Since T is onto, there exists some vector v in V such that T(v) = w. Since β is a basis for V, there exist scalars c1, c2, ..., cn such that v = c1v1 + c2v2 + ... + cnvn. Applying T to both sides, we have w = T(v) = T(c1v1 + c2v2 + ... + cnvn) = c1T(v1) + c2T(v2) + ... + cnT(vn), which implies that T(β) spans W.

Next, we will show that T(β) is linearly independent. Suppose there exist scalars c1, c2, ..., cn such that c1T(v1) + c2T(v2) + ... + cnT(vn) = 0. Applying T to both sides, we have T(c1v1 + c2v2 + ... + cnvn) = 0. But since T is one-to-one, this implies that c1v1 + c2v2 + ... + cnvn = 0, which implies that c1 = c2 = ... = cn = 0, since β is a basis for V. Hence, T(β) is linearly independent.

Since T(β) spans W and is linearly independent, it is a basis for W. Therefore, if β is a basis for V and T is one-to-one and onto, then T(β) is a basis for W.

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Answer:

Triangle XYZ is similar to Triangle RQS

Step-by-step explanation:

Triangle XYZ is similar to Triangle RQS

XZ/SR = 12/3 = 4

ZY/QS = 16/4 = 4

YX/RQ = 14/3.5 = 4

similar figures have all corresponding sides in the same ratio

The radius of the circle defined by the equation (x-x₁)² + (y-y₁)² is given by the formula √((x - x₁)² + (y - y₁)²).

You have been asked to find the radius of a circle that is defined by the equation (x-₁)² + (y-₁)². This equation represents all the points that are a certain distance away from the point (₁,₁).

To find the radius, we need to determine the distance from the center (₁,₁) to any point on the circle. Since the circle is defined by the equation (x- x₁)² + (y- y₁)², we can use the distance formula to find the radius.

The distance formula states that the distance between two points (x₁, y₁) and (x₂, y₂) is given by:

distance = √((x₂ - x₁)² + (y₂ - y₁)²)

In this case, we want to find the distance from the center (₁,₁) to a point on the circle (x,y). Using the distance formula, we can write:

radius = √((x - x₁)² + (y - y₁)²)

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By the multiplication principle of counting, the total number of positive perfect square factors of the product is the product of the number of choices for each prime factor, which is $6\times7\times8=336$. Therefore, 336 positive perfect square integers are factors of(2¹⁰ )(3¹² )(5¹⁵)

We can find the number of positive perfect square integers that are factors of the given product by first considering the prime factorization of the product, which is (2¹⁰ )(3¹² )(5¹⁵)

To get a perfect square factor, we need each prime factor to have an even exponent. Therefore, we can choose any even exponent for the prime factors of 2, 3, and 5, respectively, as long as the exponents are not greater than 10, 12, and 15, respectively.

For the factor of 2, we can choose any even exponent from 0 to 10, so there are 6 choices. Similarly, for the factor of 3, we can choose any even exponent from 0 to 12, so there are 7 choices. Finally, for the factor of 5, we can choose any even exponent from 0 to 15, so there are 8 choices.

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Sea "x" la longitud de cada lado del jardín original en metros.

La superficie del jardín original es x^2 m².

Si cada lado del jardín original se aumenta en 5 m, el nuevo lado del jardín será de (x+5) metros, y la nueva superficie será (x+5)^2 m².

Según el problema, la nueva superficie total es de 121 m²:

(x+5)^2 = 121

Tomando la raíz cuadrada en ambos lados:

x+5 = 11

Restando 5 en ambos lados:

x = 6

Por lo tanto, la longitud de cada lado del jardín original es de 6 metros.

Por lo tanto, la respuesta correcta es O 6 metros.

x - the length of each side of the original garden

A = x²

( x + 5 )² = 121  /√

x + 5 = 11,  or  x + 5 = - 11;

x = 11 - 5

x = 6  ( another solution in negative )

Answer:

A ) 6 m

Answer:

[-1,inf) and [-2,inf)

Step-by-step explanation:

Domain is all the X coordinates that the function will pass through. it starts at -1 and goes to inf so [-1,inf) is the domain. The range is all the Y coordinates the function will pass through. It starts at -2 and goes up to inf so [-2,inf) and that is your answer

Answer:

a+b>0, a>0, b>0, a-b<0

Step-by-step explanation:

well, a>0 and b>0 since they're both positive

Answer:

sample answer

Step-by-step explanation:

The number further left on a number line is the smaller number. For positive numbers, the number closest to zero is smaller. For negative numbers, the number closest to zero is larger. If a is less than b, and they are both positive, then a is closer to 0 than b. The opposite of a is also closer to zero than the opposite of b, so the opposite of a must be larger than the opposite of b.

The area of a sector of a circle is equal to the central angle in radians multiplied by the square of the radius.

Therefore, we can use this formula to solve for the radius:

Area = (Central Angle) × (Radius)²

Therefore, we can rearrange this formula to solve for the radius:

Radius = √(Area / Central Angle)

Plugging in the values given, we get:

Radius = √(15 cm² / 5 rad)

Simplifying, we get:

Radius = 3 cm

Therefore, the radius of the circle is 3 cm, rounded to the nearest tenth.

Answer:

Step-by-step explanation:

We can use the formula for the future value of an annuity to determine how much Marcia needs to invest today to meet her retirement goal of $90,000. The formula for the future value of an annuity is:

FV = PMT x [(1 + r/n)^(n*t) - 1] / (r/n)

where:

FV = future value of the annuity

PMT = payment (or deposit) made at the end of each compounding period

r = annual interest rate

n = number of compounding periods per year

t = number of years

In this case, we want to solve for the PMT (the amount Marcia needs to invest today). We know that:

Marcia wants to retire in 15 years (when she is 65), so t = 15

The interest rate is 10% per year, compounded semiannually, so r = 0.10/2 = 0.05 and n = 2

Marcia wants to have $90,000 in her retirement account

Substituting these values into the formula, we get:

$90,000 = PMT x [(1 + 0.05/2)^(2*15) - 1] / (0.05/2)

Simplifying the formula, we get:

PMT = $90,000 / [(1.025)^30 - 1] / 0.025

PMT = $90,000 / 19.7588

PMT = $4,553.39 (rounded to the nearest cent)

Therefore, Marcia needs to invest $4,553.39 today in order to meet her retirement goal of $90,000, assuming an interest rate of 10% per year, compounded semiannually.

The formula for the z-statistic is (x - μ) / (σm) or (x - μ) / (s/√n) depending on known or unknown population standard deviation.

The formula for the z-statistic, which is used in hypothesis testing for a sample mean when the population standard deviation is known, is:

z = (x - μ) / (σm)

where x is the sample mean, μ is the population mean, and σm is the standard error of the mean (SEM), which is the standard deviation of the sampling distribution of the mean.

Alternatively, we can use the estimated standard error of the mean (s/√n) when the population standard deviation is unknown, and the formula for the z-statistic becomes:

z = (x - μ) / (s/√n)

where s is the sample standard deviation and n is the sample size.

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The chi-squared value [tex](0.032)[/tex] is smaller than for the crucial value [tex](3.84)[/tex], thus do not reject the null hypotheses that there is no connection between the kind of art and the style.

The significance level, alpha, which establishes the test's sensitivity, and the test statistic, which really is unique to each type of test, both influence the significance level for a hypothesis test.

Critical values in hypothesis testing indicate whether the outcomes are statistically significant. They aid in determining the confidence intervals' upper and lower bounds. In both circumstances, crucial values accommodate for ambiguity in sample you're using it to make conclusions about a population.

                       modern    classical    total

paintings            0.38         0.62             1

sculptures          0.41          0.59             1

total                    0.79          1.21              2

The degrees of freedom for the test is [tex](r - 1) *(c - 1)[/tex]

[tex](2 - 1) * (2 - 1) = 1[/tex]

The chi-squared statistic [tex]0.032[/tex]

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The quadratic function's zeros are therefore  [tex]x = 1[/tex] and [tex]x = 0.2[/tex] . A degree two polynomial in one or so more variables that is a quadratic function.

Three points are used to determine a quadratic function, which has the form [tex]f(x) = ax2 Plus bx + c.[/tex]

[tex]Sqrt(b2 - 4ac) = [-b sqrt(b)][/tex] Where the quadratic function's coefficients are a, b, and c.

Here,  [tex]a = 20[/tex] , [tex]b = -19[/tex] , & [tex]c = 3[/tex]  . We obtain the quadratic formula by substituting these values: [tex]x = [-(-19) sqrt((-19)2 - 4(20)(3)] / 2(20) (20)[/tex]

When we condense this phrase, we get:

[tex]x = [19 +/- sqrt(361 - 240)] / 40 x = [19 +/- sqrt(121)] / 40\sx = [19 ± 11] / 40[/tex]

Therefore, The zeros of a quadratic equation   [tex]f(x) = 20x2 - 19x + 3[/tex] are as follows: [tex]x = (19 Plus 11) / 40 = 1 and x = (19 − 11) / 40 = 0.2.[/tex]

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To calculate the actual distance in km, we need to use the scale factor of 1:100 000. This means that 1 cm on the map is equivalent to 100 000 cm in real life.

Therefore, 12.3 cm on the map is equivalent to 12.3 x 100 000 cm in real life.

Now, 1 km is equivalent to 100 000 cm.

Therefore, 12.3 x 100 000 cm is equivalent to 1.23 km.

Hence, the actual distance in km is 1.23 km.

Correct Option is No matter what number you choose to substitute for the variable, the result will be a true statement.

The number of solutions of an equation based on the total number of variables contained in it. As a result, the equation system consists of two or more equations with two or more variables.

It can be any combination such as

There are three different forms of equation solutions, depending on the quantity of variables and equations. They are

The term “infinite” represents limitless or unboundedness.

For the infinite number of solution of the equation No matter what number you choose to substitute for the variable, the result will be a true statement.

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The value of standard deviation of the stock returns is 10.246%.

The variance or dispersion of a group of data points is measured by standard deviation. Finding the square root of the variance, which is the sum of the squared deviations between each data point and the mean, is how it is determined. In statistics, the term "standard deviation" is used to characterise the distribution of a data collection and to estimate the probability of certain outcomes or events.

The standard deviation is determined using the formula:

√(V).

The mean of the given data is:

(−12 + 10 + 5 − 7 + 3) / 5 = −0.2%

Now, the variance is:

Variance = [ (−12 − (−0.2))² + (10 − (−0.2))² + (5 − (−0.2))² + (−7 − (−0.2))² + (3 − (−0.2))² ] / 5

Variance = 104.96

Now, for standard deviation:

Standard deviation = √(104.96) = 10.246%

Hence, the value of standard deviation of the stock returns is 10.246%.

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The bearing of point D from C as required to be determined in the task content is; 315°.

By definition, the bearing of a point is the number of degrees in the angle measured in a clockwise direction from the north line to the line joining the centre of the compass with the point. A bearing is characteristically used to represent the direction of one point relative to another point.

The bearing of the point D from C in this case where the angle motion is clockwise starting from the north pole by convention;

The bearing of D from C is; 360 - 45 = 315°.

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Answer:

Step-by-step explanation:

central rectangle area

(A=LxW)

7x4=28 units^2,

congruent triangle area

(1/2bxh)

1/2( 7x2)= 7 units^2. (one triangle)

let's add the 3 areas 28 + 7 + 7 = 42 units^2 (your answer)

The length of side PR can be either 14 or 5.

In geometry, corresponding sides are the sides of two or more geometric figures that are in the same relative position to each other. When two or more figures are similar, their corresponding sides are in the same ratio or proportion, meaning that the ratio of the length of one side in the first figure to the length of the corresponding side in the second figure is constant. For example, if two triangles are similar, then their corresponding sides are proportional to each other. The side that corresponds to another side is typically identified using the same letter with a prime symbol (') added to it.

Since the triangles PQR and STV are similar, their corresponding sides are proportional. That is,

PR / SV = QR / TV

Substituting the given lengths, we get:

(3x-19) / (3x-9) = (x-4) / 12

Cross-multiplying and simplifying, we get:

36x - 228 = (3x-9)(x-4)

36x - 228 = 3x^2 - 21x + 36

3x^2 - 57x + 264 = 0

Dividing both sides by 3, we get:

x^2 - 19x + 88 = 0

Factoring the quadratic equation, we get:

(x-11)(x-8) = 0

Therefore, x = 11 or x = 8.

If x = 11, then PR = 3x-19 = 3(11) - 19 = 14.

If x = 8, then PR = 3x-19 = 3(8) - 19 = 5.

Therefore, the length of side PR can be either 14 or 5.

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Answer: The answer is D

Step-by-step explanation:

Pythagorean theorem: a²+b²=c²

x²+x²=14²

2x²=196

Evaluate...

x=7√2

the temperature in Australia at 11:30 would be 2°C.

How to  find?

We can use a simple formula to calculate the temperature at 11:30 based on the initial temperature and the rate of increase.

Between 8:00 and 12:00, the temperature increases by 2°C every hour, for a total of 4 hours. So the temperature at 12:00 will be:

-5°C + (4 hours x 2°C/hour) = -5°C + 8°C = 3°C

To find the temperature at 11:30, we need to calculate how many hours have passed since 8:00. From 8:00 to 11:00, it has been 3 hours. And from 11:00 to 11:30, it has been an additional 0.5 hours. So, the total time since 8:00 is 3.5 hours.

To find the temperature at 11:30, we can use the same formula as before, but substitute 3.5 for the number of hours:

-5°C + (3.5 hours x 2°C/hour) = -5°C + 7°C = 2°C

Therefore, the temperature in Australia at 11:30 would be 2°C.

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Answer: The boat is approximately 357.4 feet from the base of the cliff.

Step-by-step explanation:

Let x be the horizontal distance from the base of the cliff to the boat. Using the tangent function, we can write:

tan(38) = 275 / x

Solving for x, we have:

x = 275 / tan(38)

Using a calculator, we get:

x ≈ 357.4 feet

Therefore, the boat is approximately 357.4 feet from the base of the cliff.

Answer:

352m

Step-by-step explanation:

h = 275m

a = b (alternative angles)

.: b = 38°

Let the base from the boat to the cliff be d

Tan 38° = opposite ÷ adjacent

Tan 38° ° = 275 ÷d

d = 275 ÷ Tan 38 °

d = 352m

.: The boat is 352m away from the foot of the cliff

By answering the presented question, we may conclude that Sales Rep: expression This entity maintains information about sales reps such as SalesRepID and SalesRepName.

An expression in mathematics is a collection of representations, numbers, and conglomerates that mimic a statistical correlation or regularity. A real number, a mutable, or a mix of the two can be used as an expression. Mathematical operators include addition, subtraction, fast spread, division, and exponentiation. Expressions are often used in arithmetic, mathematic, and form. They are used in the representation of mathematical formulas, the solution of equations, and the simplification of mathematical relationships.

The ER diagram above depicts the database entities and their connections. Here's a quick rundown of each entity and its characteristics:

Employee: This object contains information on employees such as EmployeeID, Name, DateOfBirth, and Dependents.

ProductID, ProductName, ManufacturingDate, ExpiryDate, and SupplierID are all stored in this object.

Supplier: This entity holds supplier-specific information such as SupplierID, SupplierName, ContactPerson, and ContactNumber.

CustomerID, CustomerName, ContactPerson, and ContactNumber are all stored in the Customer entity.

Sales Rep: This entity maintains information about sales reps such as SalesRepID and SalesRepName.

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Answer:

The function that models the number of flowers, F(t), is given as F(t) = cd, where c and d are constants. We need to find the values of c and d in order to write the equation for the number of flowers in the pond at time, t.

From the table, we know that when t=0, F(t) = 200. This means that:

F(0) = cd = 200

Similarly, when t=1, F(t) = 800. This means that:

F(1) = cd = 800

We can solve this system of equations for c and d by dividing the second equation by the first equation:

F(1)/F(0) = 800/200

4 = d/c

Now we can substitute the value of d/c into either equation to solve for one of the constants. Let's use the first equation:

cd = 200

c(d/c) = 200

d = 200/c

Substituting this into the equation d/c = 4, we get:

4 = d/c = (200/c) / c

4c = 200

c = 50

Now we can find the value of d using d = 200/c:

d = 200/50 = 4

Therefore, the equation for the number of flowers in the pond at time, t, is:

N(t) = cd = 50(4) = 200

So, the answer is N(t) = 200(1), which means that at any time t, the number of flowers in the pond is 200.

The population of the termite colony will be approximately 593 in 4 years.

Since the population of the termite colony will double in 6 years, we can use the formula:

P = P0 * 2^(t/k)

where P0 is the initial population, P is the population after time t, and k is the time it takes for the population to double (in this case, 6 years).

We know that the current population is 16000, so P0 = 16000. We also know that the population consists only of females, so we can assume that the population grows linearly.

To find the population of the termite colony in 4 years, we need to determine how much the population grows in that time. Since the population doubles in 6 years, we can assume that it grows by a factor of 2/6 per year. Therefore, the population after 4 years can be calculated as:

P = P0 * (2/6)^4

= 16000 * (1/27)

= 592.59 (rounded to the nearest whole number)

Therefore, the population of the termite colony will be approximately 593 in 4 years.

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I chose to explore the Angle Sum Theorem because it is a fundamental theorem in geometry, and it has important implications for many geometric proofs and applications. Moreover, the theorem can be easily visualized and understood, making it a great starting point for learning geometry.

The formal definition of the Angle Sum Theorem is: "In any triangle, the sum of the interior angles is equal to 180 degrees."

In simpler terms, the Angle Sum Theorem states that if you add up all the angles inside a triangle, you will always get a total of 180 degrees. This means that no matter how you move the sides of the triangle or what shape it takes, the total measure of the angles will always be the same.

Let's consider a triangle with angles of 30 degrees, 60 degrees, and 90 degrees. According to the Angle Sum Theorem, the sum of the interior angles of any triangle must be 180 degrees.

So, in this case, we have:

30 degrees + 60 degrees + 90 degrees = 180 degrees

Therefore, the Angle Sum Theorem holds true for this example.

I taught the Angle Sum Theorem to my younger brother, and he was fascinated by it. He said that he had never thought about triangles in this way and found it cool that the angles always added up to the same number, no matter what the triangle looked like. He then went on to ask me more questions about triangles and geometry, which I was happy to answer.

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The values of x for which the series converges is x ∈ (-1/5, 1/5). The sum of the series for those values of x is (-5x)/(1 + 5x).

The series is [tex]\Sigma^{\infty}_{n=1}(-5)^nx^n[/tex].

We can write this series as [tex]\Sigma^{\infty}_{n=1}(-5x)^n[/tex].

This is a infinite geometric series with first term a = -5x and common ration r = -5x.

It is convergent when

|r| < 1

|-5x| < 1

|-5| |x| < 1

5|x| < 1

Divide by 5 on both side, we get

|x| < 1/5

The series is convergent when x ∈ (-1/5, 1/5).

Sum of the series is

Sₙ = a/1 - n

Sₙ = (-5x)/{1 - (-5x)}

Sₙ = (-5x)/(1 + 5x)

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The complete question is:

Find the values of x for which the series converges. Find the sum of the series for those values of x.

[tex]\Sigma^{\infty}_{n=1}(-5)^nx^n[/tex]

IV individuals from different cultures

DV the belief that dreams have meaning.

Independent Variable: Culture

Belief in the meaning of dreams is a quasi-independent variable (since it cannot be manipulated or assigned randomly)

The response to whether or not dreams have meaning is the dependent variable.

An experimental investigation typically contains three types of variables: independent variables, dependent variables, and controlled variables.

A compared to the rest of the country. Because the variable levels are pre-existing, it is not possible to assign participants to groups at random.

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Answer: 2

Step-by-step explanation:

because the common range here is 7-13.

2 isnot in this common range so it could be identified as an outlier

The inventory turnover ratio can be calculated by dividing the cost of goods sold by the average inventory value. In this case, the inventory turnover ratio is 0.5, and the correct answer is (D) 0.5.

The inventory turnover ratio measures the number of times a company sells and replaces its inventory during a period. It can be calculated by dividing the cost of goods sold by the average inventory value.

The inventory turnover ratio = Cost of goods sold / Average inventory value

In this case, the cost of goods sold is $600,000 and the average inventory value is $1,200,000.

Inventory turnover ratio = $600,000 / $1,200,000 = 0.5

Therefore, the inventory turnover ratio is 0.5, and the correct answer is (D) 0.5.

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Complete question:

If the average aggregate inventory value is $1,200,000 and the cost of goods sold is $600,000, which of the following is inventory turnover?

A)60

B)10.4

C)2

D)0.5

E)None of these