Which step is the same when constructing an inscribed square and an inscribed equilateral triangle?

Answer:

it always means to multiply

Step-by-step explanation:

If there are 342 more trees bearing red apples than trees bearing green apples, then the number of trees with green apples are 1336.

The 3/5 of fruit trees are orange trees, then 2/5 of the fruit trees are apple trees.

Let the total number of fruit trees be = x. Then we can write two equations based on the given information that, there are 4515 orange trees and there are 342 more red apple trees than green apple trees,

The equations are :

⇒ 3/5 x = 4515         ...equation(1)

⇒ x - 4515 = (G + 342) + G   ...equation(2)

Where G is the number of trees bearing green apples.

From equation(1), we can solve for x:

x = (4515)/(3/5) = 7525

Substituting this value of "x" into equation(2),

We get,

⇒ 7525 - 4515 = (G + 342) + G

⇒ 3010 = 2G + 342

⇒ 2668 = 2G

⇒ G = 1334.

Therefore, there are 1336 trees with green apples.

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The given question is incomplete, the complete question is

A fruit plantation has two types of fruit trees: apple trees and orange trees. 3/5 of the fruit trees are orange trees. There are 4515 orange trees on the plantation. The apple trees bear apples of two different colors. There are 342 more trees bearing red apples than trees bearing green apples.

How many trees with green apples are there?

Express the rational function 2x⁴/(x² - 2x) as a sum or difference of two simpler rational expressions:

2x⁴/(x² - 2x) = 2x² + 4x + 8 + 16/(x - 2).

The given expression is,

2x⁴/(x² - 2x)

simplifying we get,

= 2x⁴/x(x - 2) [taking 'x' as common from both the term of denominator]

= 2x³/(x - 2) [Cancelling the same value from numerator and denominator]

Now if we divide '2x³' by (x-2) we get

Quotient = 2x² + 4x + 8

Remainder = 16

So from the division algorithm we know that,

Dividend = Divisor*Quotient + Remainder

Here Dividend = 2x³ and Divisor = x - 2

So, 2x³ = (x - 2)*(2x² + 4x + 8 ) + 16

Now, 2x⁴/(x² - 2x)

= 2x³/(x - 2)

= ((x - 2)*(2x² + 4x + 8 ) + 16)/(x - 2)

= ((x - 2)*(2x² + 4x + 8) )/(x - 2) + 16/(x - 2)

= 2x² + 4x + 8 + 16/(x - 2).

So the simpler rational functions are: (2x² + 4x + 8) and 16/(x - 2).

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The question is incomplete. Complete question will be -

"Express the rational function as a sum or difference of two simpler rational expressions: 2x⁴/(x² - 2x)"

a). The difference between the two population means is estimated at a location to be 2.7.

b). 49 different possible outcomes make up the t distribution. The margin of error at 95% confidence is 1.7.

c). The range of the difference between the two population means' 95% confidence interval is (0.0, 5.4).

d). The (0.0, 5.4) represents the 95% confidence interval for the difference among the two population means.

The variability or spread in a set of data is commonly measured by the standard deviation. The deviation between the values in the data set and the mean, or average, value, is measured. A low standard deviation, for instance, denotes a tendency for data values to be close to the mean, whereas a high standard deviation denotes a larger range of data values.

Using the equation \(x_1-x_2\), we can determine the point estimate of the difference between the two population means. In this instance, we calculate the point estimate as 2.7 by taking the mean of Sample

\(1(x_1=22.8)\) and deducting it from the mean of Sample \(2(x_2=20.1)\).

With the use of the equation \(df=n_1+n_2-2\), it is possible to determine the degrees of freedom for the t distribution. In this instance, the degrees of freedom are 49 because \(n_1\) = 20 and \(n_2\) = 30.

We must apply the formula to determine the margin of error at 95% confidence \(ME=t*\sqrt[s]{n}\).

The sample standard deviation (s) is equal to the average of \(s_1\) and \(s_2\) (3.4), the t value with 95% confidence is 1.67, and n is equal to the

average of \(n_1\) and \(n_2\) (25). When these values are entered into the formula, we get \(ME=1.67*\sqrt[3.4]{25}=1.7\).

Finally, we apply the procedure to determine the 95% confidence interval for the difference between the two population means \(CI=x_1-x_2+/-ME\).

The confidence interval's bottom limit in this instance is \(x_1-x_2-ME2.7-1.7=0.0\) and the upper limit is \(x_1+x_2+ME=2.7+1.7=5.4\).

As a result, the (0.0, 5.4) represents the 95% confidence interval for the difference among the two population means.

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Answer:

y = 0

Step-by-step explanation:

can you post the graph picture so this question can be answered better?

Answer:

The answer is 20%

Step-by-step explanation:

24/120 = 0.2= 20%

Hope that helps!

Answer:

3a² + ab + 4b²

Step-by-step explanation:

P - Q

= - 4a² - 2ab + 9b² - (- 7a² - 3ab + 5b² ) ← distribute parenthesis by - 1

= - 4a² - 2ab + 9b² + 7a² + 3ab - 5b² ← collect like terms

= 3a² + ab + 4b²

Answer:

3a² + ab + 4b²

Step-by-step explanation:

Combine like terms. Like terms have same variable with same power.

P - Q = -4a² - 2ab + 9b² - (-7a² - 3ab + 5b²)

To open the brackets, multiply each term of Q by (-1)

          = -4a² - 2ab + 9b² + 7a²+ 3ab - 5b²

-4a² and 7a² are like terms.

(-2ab) and  3ab  are like terms.

9b² and (-5b²) are like terms.

          = -4a² + 7a² -2ab + 3ab + 9b²- 5b²

          = 3a² + ab + 4b²

Answer:

1920

Step-by-step explanation:

What we can do is split this into two sections: the volume of the rectangle and the volume of the square.

Rectangle: We can follow the equation V = lbh, so we have 14x8x10, giving us 1120.

Square: We can use the same equation, V = lbh, giving us 10x10x8 which is 800.

Final Answer: To get the full volume, we have to add these together, so, 1120 + 800 = 1920.

To construct the matrix P that represents the orthogonal projection onto the subspace W spanned by the orthonormal basis β = {x1, ..., xk} in R^n, we can use the formula:

P = X (X^T X)^(-1) X^T

Here, X is the matrix with columns x1, ..., xk. The matrix P will have dimensions n x n.

To justify that Lp is the orthogonal projection with R(Lp) = W, we need to show two properties:

Lp is an orthogonal projection:

Lp is idempotent: (Lp)^2 = Lp

Lp is self-adjoint: (Lp)^T = Lp

R(Lp) = W:

Every vector in the range of Lp is in W

Every vector in W is in the range of Lp

Let's break down the steps to construct P and verify these properties:

Step 1: Compute the matrix X^T X

X^T is the transpose of X

Multiply X^T with X to obtain X^T X

Step 2: Invert the matrix (X^T X)^(-1)

If X^T X is invertible, calculate its inverse (if it is not invertible, the orthogonal projection does not exist)

Step 3: Compute the matrix P = X (X^T X)^(-1) X^T

Multiply X by (X^T X)^(-1)

Multiply the result by X^T

Step 4: Verify the properties of Lp

Check if (Lp)^2 = Lp

Check if (Lp)^T = Lp

Step 5: Verify R(Lp) = W

Check if every vector in W is in the range of Lp

Check if every vector in the range of Lp is in W

By following these steps and verifying the properties, we can rigorously show that Lp is indeed an orthogonal projection with R(Lp) = W.

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Answer:

Option B

Step-by-step explanation:

Slope of the line is calculated by the formula:

\(m=\frac{y_2-y_1}{x_2-x_1}\\\\\)

We have the points ( -2 , 4) and (3 , -5) so plug in the values in the formula:

\(m=\frac{y_2-y_1}{x_2-x_1}\\\\m=\frac{-5-4}{3-(-2)}\\\\m=\frac{-9}{5}\\\)

so the slope(m) equals -9/5

Answer:

The quadrilateral formed is a trapezoid

Step-by-step explanation:

In this case, it is necessary to know what points A, B, C, and D.

If the points are equal to the following coordinate points

A (-2, 1)

B (2, 1)

C (3, -2)

D (-3, -2)

As shown in the image the answer would be a trapezoid.

________________________________

The figure cannot be a square, a rhombus or a rectangle since by definition these are:

A square has all sides of equal length and all its right angles.

A rhombus is a parallelogram that has all sides of equal length and all its angles different from right angles.

A rectangle is a parallelogram that has all its right angles.

The figure that we obtain is neither a parallelogram nor does it have all its right angles.

Answer:

b) 5.8

Step-by-step explanation:

Explanation

Given diagram the angles are (9 x + 6)°, (10 x + 7)°, (10 x - 1)°

we know that the sum of the angles of the triangle is equal to 180°

∠A +∠B +∠C = 180°

(9 x + 6)°+(10 x + 7)°+ (10 x - 1)° = 180°

On simplification , we get

29 x + 12 = 180

29 x = 180 -12

29 x = 168

x = 5.79 ≅ 5.8

Answer:

b

Step-by-step explanation:

If each T-shirt costs Rs. 120 and Temoso receives a 15% commission, he needs to sell 20 of them in order to make Rs. 360.

Let's first find out how much commission Temoso earns on each T-shirt he sells. The cost of each T-shirt is Rs. 120, and Temoso earns a 15% commission on each T-shirt. This means that he earns 0.15 × 120 = Rs. 18 on each T-shirt he sells.

Now, let's find out how many T-shirts Temoso must sell to earn Rs. 360.

We can set up the following equation:

18x = 360

where x is the number of T-shirts that Temoso needs to sell.

Dividing both sides by 18, we get:

x = 20

Therefore, Temoso needs to sell 20 T-shirts to earn Rs. 360, assuming the cost of each T-shirt is Rs. 120 and he earns a 15% commission on each T-shirt he sells.

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The correct answer is:

**C) 0.28**

The sum of deviations refers to the sum of the differences between individual data points and the mean of the dataset. In any dataset, the sum of deviations from the mean will always equal 0.

This is because, when calculating the mean, you sum all the data points and then divide by the number of data points. The mean represents the "middle" or "balance" point of the dataset. When you add up the differences between each data point and the mean, the positive and negative deviations will cancel each other out, resulting in a sum of 0.

So, any non-zero value for the sum of deviations is impossible. In this case, 0.28 is the only non-zero value.

Answer:

ANSWER-) 3 litres =3000ml

capacity of 1 tube =75ml

NO. OF TOOTH PASTE REQUIRED = 3000/75=40 tubes.

An investor writes a s&p 100 Jan. 185 put at $6 on May 4. In this situation, his function maximum potential benefit is $600.

We could create a function with a codomain of sets of numbers and X being the set of individuals once more. For instance, create the function c so that its output c(x) is the number of offspring of each person x for which the codomain Y is the set of whole numbers.

A benefit is a positive outcome or advantage that results from something. It can refer to the positive aspects or advantages of a product, service, or action, and can be tangible or intangible. Benefits can include increased efficiency, cost savings, improved safety, increased customer satisfaction, and many other positive outcomes.

Here,

As a put buyer, the most you may lose is the premium you previously paid to purchase the put option.

The option price must be reduced to equal the strike price less any already-paid premium in order to break even.

The strike price less the premium is your maximum gain as a put buyer. The most bullish option strategy is thought to be purchasing call options.

This tactic allows the buyer of the call option the choice, but not the responsibility, to purchase the stock at a certain price at a precise moment.

Whenever the price of the security increases using this approach, the buyer benefits. A long position is one in which a call or put option is purchased or held.

Investors have the option of purchasing from or selling to the other investor at a predetermined price.

On the other hand, a short position is when you sell a call or put option. Writers must purchase from people holding long positions or buying options.

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On May 4th an investor writes on s&p 100 Jan 185 put at $6. his function maximum potential gain in this position is $600.

We could create a function with a codomain of sets of numbers and X being the set of individuals once more. For instance, create the function c so that its output c(x) is the number of offspring of each person x for which the codomain Y is the set of whole numbers.

Your maximum loss as a put buyer is the premium already paid to buy the put option. To break even the option price must be lowered to cover the strike price minus the premium already paid. Your maximum profit as a put buyer is the strike price minus the premium. Buying call options are considered the most bullish options strategy.

This strategy gives the call option buyer the right but no obligation to buy the security at a specific price at a specific time. With this strategy, the buyer benefits when the price of the security rises. Buying or holding a call or put option is a long position. Investors have the right to buy or sell to the writing investor at a specified price. Conversely selling or selling a call or put option is a short position. Writers must buy or sell from long position holders or option buyers.

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Dimensions 5in x 10in x 10in will yield a box with a max. volume of  500 cubic inches.

Volume =  height x length x width

Considering 'x' as the length of the square corners that has been cut out from the cardboard and also, that is height of the cardboard box.

Square corners are cut out so that the sides can be folded up to make a box, cardboard sides would reduce by 2x

Therefore,

V = x  (30-2x) (30-2x)                --------------------------------- (1)

V=( 30x - 2x²) (30-2x)

V= 900x- 60x² - 60x² + 4x³

V=  4x³ - 120 x²+ 900x

Taking derivative with respect to x:

dV /dx = 12x² - 240 x +900

dV/dx = 4 (3x² - 60x +225)

For maximum dV/dx, make it equal zero

    dV/dx = 0

⇒  4 (3x² - 60x +225)=0

⇒ 3x² - 60x+225=0   (taking 3 common)

⇒ x² - 20x + 75 =0

Solving this quadratic equation, we get,

x² - 15x -5x + 75 =0

x(x-15) - 5(x-15) =0

Either (x-15)=0

    x=15

Or,  x-5=0

   x = 5

If we substitute x = 15 in equation (1), volume becomes zero.

Therefore, x cannot be 15

When x= 5, putting in the equation (1)

V = 5  (30-2(5)) (30-2(5))

V= 5 (10) (10)

V= 500 cubic inches,

Therefore, Dimensions 5in x 10in x 10in will yield a box with a max. volume of  500 cubic inches

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Answer:

according to my calculations

Step-by-step explanation:

cheese

The solution to the given integral is 5ye10 - 5ye5 + c(y10 - y5).

To evaluate the given integral, we must first reverse the order of integration. This is done by switching the limits of integration for the two variables. In this case, the limits for the y variable are 10 and 5, and the limits for the x variable are 5 and 5. Thus, the integral can be rewritten as:

∫5∫10yex2dxdy

To evaluate this integral, we can use the following formula:

∫a∫bf(x,y)dxdy = ∫a∫bf(y,x)dydx

Using this formula, we can rewrite the integral as:

∫5∫10yex2dydx

We can now use the product rule to separate the integral into two separate integrals. First, we can integrate with respect to x:

∫5yex2dx

This integral can be solved using the fundamental theorem of calculus:

∫5yex2dx = 5ye5 - 5ye5 + c

Next, we can integrate with respect to y:

∫10(5ye5 - 5ye5 + c)dy

This integral can be solved by simply integrating the constants and the exponential:

∫10(5ye5 - 5ye5 + c)dy = 5ye10 - 5ye5 + c(y10 - y5)

Combining the two solutions, we get the following:

∫5∫10yex2dxdy = 5ye10 - 5ye5 + c(y10 - y5)

Therefore, the solution to the given integral is 5ye10 - 5ye5 + c(y10 - y5).

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The value of x in terms of the length of the chord BC is x = (1/2) * arcsin(BC / (2 * r)) - 30 degrees

Draw a diagram of the circle and label the given points and segments.

Identify the relevant circle properties: Recall the properties of circles such as the radius, diameter, circumference, and central angle. Identify any angles or lengths that can be determined from the given information.

Use the tangent-chord theorem: Since AD is tangent to the circle, we can use the tangent-chord theorem to find that angle ADB is equal to angle ABD.

This implies that triangle ABD is an isosceles triangle.

Use the properties of isosceles triangles: Since triangle ABD is isosceles, we know that angle BAD is equal to angle ADB.

We also know that the sum of the angles in a triangle is 180 degrees.

Therefore, angle ABD is equal to (180 - 2x) degrees.

Use the angle sum property: Since A, B, C, and D are points on the circle, angle BAC is equal to half of the central angle BOC.

Therefore, angle BOC is equal to 2 * angle BAC.

Use the angle subtraction property: Since angle ABD is an exterior angle of triangle BCD, we know that angle ABD is equal to the sum of angles BCD and BDC.

Therefore, (180 - 2x) degrees is equal to (2 * angle BAC) + angle BDC.

Use the angle sum property again: Since B, C, and D are points on the circle, angle BDC is equal to half of the central angle BOC.

Therefore, angle BDC is equal to angle BAC.

Substituting this value into the equation from step 6, we get (180 - 2x) degrees

= (2 * angle BDC) + angle BAC

= 2 * angle BAC + angle BAC

= 3 * angle BAC.

Therefore, angle BAC is equal to (180 - 2x) / 3 degrees.

2 * angle BAC = 2 * (180 - 2x) / 3 degrees.

Use the chord length formula: Using the chord length formula, we can find that BC

= 2 * r * sin(angle BOC / 2), where r is the radius of the circle.

Substituting the value of angle BOC from step 9, we get BC = 2 * r * sin[(2 * (180 - 2x) / 3) / 2]

= 2 * r * sin[(180 - 2x) / 3] .

Solve for x:

We need to express x in terms of BC, so we can use the formula from step 10 to get 2 * r * sin[(180 - 2x) / 3] = BC.

Solving for x gives x = (1/2) * arcsin(BC / (2 * r)) - 30 degrees.

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In order to advise the tire manufacturer on their advertising effectiveness and allocation, we need to calculate the marginal return to sales for each region, assess the impact of additional advertising expenditure, determine where the additional money should be spent, and consider the reallocation of current advertising expenditures.

To find the marginal return to sales for each region, we differentiate the sales equation with respect to x. For Region 1, ds_1/dx = 36 - 2.6x, and for Region 2, ds_2/dx = 19 - 1.2x. If $10 million is being spent on TV advertising in each region, we substitute x = 10 into the respective equations to find the marginal return to sales. In Region 1, ds_1/dx = 36 - 2.6(10) = 11, and in Region 2, ds_2/dx = 19 - 1.2(10) = 7. Hence, the marginal return to sales is $11 million in Region 1 and $7 million in Region 2.

To determine which region would benefit more from additional advertising expenditure, we compare the marginal returns. Since the marginal return in Region 1 is higher ($11 million) compared to Region 2 ($7 million), Region 1 would benefit more from additional advertising expenditure.

If additional money is made available for advertising, it should be spent in Region 1 as it has the higher marginal return. Increasing the advertising expenditure in Region 1 would likely result in a greater increase in sales revenue compared to Region 2.

To reallocate money already being spent to produce more sales revenue, we should compare the coefficients of the quadratic term (x^2) in the sales equations. In Region 1, the coefficient is -1.3, and in Region 2, it is -0.6. Since the coefficient in Region 1 is larger in absolute value, reallocating some advertising expenditure from Region 2 to Region 1 would likely lead to a larger increase in sales revenue overall.

Therefore, based on the analysis, the tire manufacturer should focus on increasing advertising expenditure in Region 1, as it has a higher marginal return, and consider reallocating some advertising funds from Region 2 to Region 1 to optimize sales revenue.

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Answer:

I got 270cm!

Hope this helped!

Train a total of 200 miles each week. 160 miles I bike, 32 miles I run, 8 miles I swim.

What do you mean by Simultaneous linear equations?

A system of simultaneous linear equations is any pair of linear equations or more that have the same unknown variables in common. To solve such a system, one must identify values for the unknown variables that simultaneously satisfy each of the equations. A system of simultaneous linear equations is any pair or group of linear equations where each equation has the same unknown variables. Finding values for the unknown variables that simultaneously satisfy each equation in such a system is the first step in its solution.

according to the question,

you bike five times as far as I run and you run four times as far as iswim. you train a total of 200 miles each week.

let, I swim x miles then I run 4x times and I bike 20x miles.

so, 4x+x+20x=200

by solving linear equations,

we get, x=8

hence,160 miles I bike.

32 miles I run.

8 miles I swim.

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Answer:

Step-by-step explanation:

according to the question

\( \frac{14}{2x + 7} = \frac{x - 3}{63} \)

\(882 = (x - 3)(2x + 7)\)

\(0 =882 - (x - 3)(2x + 7)\)

\( 882 - (2 {x}^{2} + x - 21) = 0\)

\(882 - {2x}^{2} - x + 21 = 0\)

\( - {2x}^{2} - x + 903 = 0\)

\(2x + x - 903 = 0\)

\((2x + 43)(x - 21) = 0\)

\(x = - \frac{43}{2} \\ x = 21\)

\( \huge \therefore x = 21\)

Using sampling concepts, it is found that the 100 students surveyed in this example would be the would be the sample.

It is a common statistics practice, when we want to study something from a population, we find a sample of this population, which is a group containing elements of a population. A sample has to be representative of the population, that is, it has to involve all segments of the population.

Hence, in this problem:

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Using a one-tailed hypothesis test at the 5% level with normal theory methods and a significance level of 0.05, the study provides significant evidence that the vaccine reduces the occurrence of ear infections in infants. The conditions for inference are met. However, the study does not establish causation.

To test the hypotheses, we can use the normal approximation to the binomial distribution since the sample size is large. We can assume that the number of ear infections in each group follows a binomial distribution with parameters n1 = 2455, p1 = unknown (probability of infection in the vaccinated group), n2 = 2452, p2 = unknown (probability of infection in the control group).

We want to test if p1 is less than p2, i.e., the one-tailed alternative hypothesis.

Let p1 and p2 be the sample proportions of ear infections in the vaccinated and control groups, respectively. Then

H0: p1 >= p2 (the vaccine does not have a significant effect)

Ha: p1 < p2 (the vaccine significantly reduces the occurrence of ear infections)

The significance level is 0.05, so we reject the null hypothesis if the p-value is less than 0.05.

Using the normal approximation, we can calculate the standard error of the difference between the sample proportions as:

SE = sqrt(p1(1-p1)/n1 + p2(1-p2)/n2) = 0.0107

The test statistic is

z = (p1 - p2) / SE = (333/2455 - 499/2452) / 0.0107 = -7.76

The p-value for a one-tailed test is P(Z < -7.76) < 0.0001.

Since the p-value is less than 0.05, we reject the null hypothesis in favor of the alternative hypothesis. This means that the evidence suggests that the vaccine significantly reduces the occurrence of ear infections in infants.

Conditions for inference

The sample size is large (n1 = 2455, n2 = 2452), so the normal approximation to the binomial distribution is valid.

The groups were randomly assigned to the vaccinated and control groups, which helps ensure that the groups are comparable.

The ear infection outcomes are independent for each infant.

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

" A new vaccine was recently tested to see if it course prevent the painful and recurrent ear infections that many infants suffer from. The Lancet, a medical journal, reported a study in which babies about a year old were randomly divided into two groups. One group received vaccinations the other did not. During the following year, 333 of 2455 vaccinated children had ear infections, compared to 499 of 2452 unvaccinated children in the control group.

Did this study provide significant evidence, at the 5% level, that the vaccination is effective? Do all the steps of a hypothesis test using normal theory methods, including finding a p-value, and write the conclusion in context. Use SE = 0.0107

Check conditions for inference

Does this study provide evidence of causation? Explain why or why not."--

A .The volume of the solid under the paraboloid z = 3x^2 + y^2 and above the region bounded by the curves x - y^2 and x - y - 2 can be found using a double integral. The answer cannot be provided in 15-20 words as it requires a detailed explanation.

To calculate the volume, we need to determine the limits of integration for both x and y. Let's find the intersection points of the two curves:

x - y^2 = x - y - 2

y^2 - y + 2 = 0

Solving this quadratic equation, we find that there are no real solutions for y. Therefore, the paraboloid does not intersect the region bounded by the curves x - y^2 and x - y - 2.

Since there is no intersection, the volume of the solid under the paraboloid above this region is zero.

B. The volume of the solid under the plane z = 2x + y and above the triangle with vertices (1, 0), (3, 1), and (4, 0) can also be determined using a double integral. The main answer is that the volume of the solid can be found by evaluating the appropriate integral, but the specific numerical value cannot be provided without performing the calculations.

To calculate the volume, we set up the double integral in terms of x and y. The limits of integration for x can be set from 1 to 4, as the triangle's base lies along the x-axis. For each value of x, the limits of integration for y can be determined by the equation of the lines that form the triangle's sides.

For the line passing through (1, 0) and (3, 1), the equation is given by y = 1/2 x - 1/2. For the line passing through (1, 0) and (4, 0), the equation is y = 0.

Thus, the volume can be calculated by evaluating the double integral ∫∫(2x + y) d A over the limits of integration: x = 1 to 4, and y = 0 to 1/2x - 1/2. The resulting value will provide the volume of the solid under the plane and above the given triangle.

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Answer:

2 x c chocolates.

Step-by-step explanation:

This is pretty much just a matter of interpreting the words, substituting "c" for chocolates, and x 2 for "2 times."