The american heart association is about to conduct an anti-smoking campaign and wants to know the fraction of americans over 26 who smoke. Suppose a sample of 897 americans over 26 is drawn. Of these people, 637 don't smoke. Using the data, estimate the proportion of americans over 26 who smoke.

Subtract adjacent terms:

10-2 = 8

50-10 = 40

The results are not the same, so we do not have a common difference.

But we do have a common ratio because dividing adjacent terms gets us the following

10/2 = 5

50/10 = 5

The common ratio is 5. It means we multiply each term by 5 to get the next term. The sequence is geometric.

The nth term formula is [tex]a_n = 2*5^{n-1}, \ \text{ where n = 1, 2, 3,}\ldots[/tex]

The side lengths of triangle ABC that are missing such as BC and AB would be = 4 and 5.7 respectively.

To calculate the missing lengths of the triangle ABC, the sine rule must be obeyed. That is;

a/sinA = b / sinB

Where a = line BC = ?

A = 45°

b = line AC = 4

B = 180-(90+45) = 45°

That is;

a/sin45° = 4/sin45°

Make a the subject of formula;

a = 4.

Using Pythagorean formula;

c² = a²+b²

= 4²+4²

= 16+16 = 32

c = √32

= 5.7

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The statement that best describes the meaning of f ( 250) is c. This is the average number of days the house stayed on the market before being sold for $250,000.

The function f(p) is a representation of the average amount of days that a house will remain on the market before being sold, in which p is stated in terms of $1,000s. Thus, when observing f(250) we can easily deduce that homes available for $250,000 typically stay listed for an average duration of 250 days prior to finding their rightful owner.

Option b, however, should not be considered correct; due to the fact that simply because it has been established that houses cost approximately 250 days for sale does not mean this same factor implies that each of the said houses are sold for $250,000.

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Full question is:

Let f(p) be the average number of days a house stays on the market before being sold for price p in $1,000s. Which statement best describes the meaning of f(250)? (2 points)

Select one:

a. The house sold for $250,000.

b. The house stayed on the market for an average of 250 days before being sold.

c. This is the average number of days the house stayed on the market before being sold for $250,000.

d. The house sold on the market for $250,000 and stayed on the market for an average of 250 days before being sold.

The volume V of a rectangular prism is given by the formula:

V = l x w x h

where l is the length, w is the width, and h is the height.

In this case, we have:

l = 12 inches

w = 25 inches

h = 20 inches

Substituting these values into the formula, we get:

V = 12 inches x 25 inches x 20 inches

Multiplying, we get:

V = 6,000 cubic inches

Therefore, the volume of the rectangular prism is 6,000 cubic inches.

The statement means that the observed relationship between the independent variable and the dependent variable is "very unlikely to occur by chance."

In statistical analysis, the concept of statistical significance helps determine if the results obtained from a study are reliable and not due to random chance. When an effect is deemed statistically significant, it suggests that the relationship between the independent variable (the variable being manipulated or studied) and the dependent variable (the variable being measured or observed) is likely to exist in the population from which the sample was drawn.

By reaching statistical significance, the researcher can confidently conclude that the observed effect is more than just a random occurrence and has practical implications in the real world, thus lending support to the underlying hypothesis or research question.

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Therefore, the tangent vectors at P are identical regardless of which parameterization is used.

Yes, this statement is true. Different parametrizations of the same curve will result in the same tangent vector at a given point on the curve because the tangent vector is independent of the choice of parameterization.

To see why this is the case, suppose we have two different parameterizations of the same curve: r(t) and s(u), where t and u are both variables that parameterize the curve. Let P be a point on the curve, and let a = r(t) = s(u) be the corresponding position vector of P.

The tangent vector to the curve at P is given by the derivative of the position vector with respect to the parameter:

r'(t) = lim Δt→0 [r(t+Δt) - r(t)]/Δt

s'(u) = lim Δu→0 [s(u+Δu) - s(u)]/Δu

Since both r(t) and s(u) correspond to the same point P on the curve, they must be equal:

r(t) = s(u) = a

Taking the derivative of both sides with respect to t and u respectively:

r'(t) = s'(u)

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-5, view below for what Maureen did wrong.

Given:

   -1.2+(-3.8)

A positive times a negative is a negative:

   -1.2-3.8

Subtract:

   -5

   Instead of distributing the positive sign into the -3.8 and subtracting, Maureen added 1.2 to -3.8, leading to -2.6 instead of -5.

To mitigate this risk, the algorithm is often run multiple times with different starting points, and the solution with the highest likelihood is chosen.

The EM algorithm is an iterative method for maximum likelihood estimation when some of the data is missing or unobserved. The algorithm works by alternately updating estimates of the unobserved data and the model parameters until convergence.

It can be proven that the EM algorithm monotonically increases the likelihood of the data at each iteration. This means that the likelihood of the data cannot decrease as the algorithm progresses, and therefore the algorithm is guaranteed to converge to a local maximum of the likelihood function.

To see why the likelihood of the data increases at each iteration, consider the E-step and the M-step of the algorithm. In the E-step, the algorithm estimates the expected value of the unobserved data given the observed data and the current estimates of the model parameters. In the M-step, the algorithm updates the estimates of the model parameters based on the expected values of the unobserved data.

The likelihood of the data is a function of the model parameters, and it can be shown that the expected value of the log-likelihood function increases at each iteration of the EM algorithm. This is because the E-step maximizes the expected value of the log-likelihood function with respect to the unobserved data, and the M-step maximizes the expected value of the log-likelihood function with respect to the model parameters. Since the expected value of the log-likelihood function is a lower bound on the actual log-likelihood function, the likelihood of the data increases at each iteration.

The monotonically increasing property of the EM algorithm has important implications for its convergence. Since the likelihood of the data is guaranteed to increase at each iteration, the algorithm is guaranteed to converge to a local maximum of the likelihood function. However, the algorithm may converge to a suboptimal local maximum, depending on the starting point of the algorithm and the shape of the likelihood function. To mitigate this risk, the algorithm is often run multiple times with different starting points, and the solution with the highest likelihood is chosen.

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Answer:

That function is f(x) = cos(πx) - 2.

Boolean algebra, truth tables, and logic diagrams are the three methods of Boolean functions.

The three methods to express the logical behavior of Boolean functions are:

1. Boolean algebra: This method uses algebraic expressions with binary variables and operators such as AND, OR, and NOT to represent the logical behavior of a Boolean function.

2. Truth tables: A truth table is a tabular representation of all possible input combinations and their corresponding output values for a given Boolean function. Each row in the table represents a unique input combination, and the corresponding output value is the result of applying the Boolean function to that input.

3. Logic diagrams: Also known as logic gates or circuit diagrams, these graphical representations use symbols to illustrate the logical relationships between input and output variables. Common symbols include AND, OR, and NOT gates, which represent the fundamental Boolean operations.

By utilizing these methods, you can effectively express and analyze the logical behavior of Boolean functions in a variety of scenarios.

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The diameter of the sphere with a volume of 2171 m3, to the nearest tenth of a meter, is 16.1 m.

To find the diameter of a sphere with a volume of 2171 m³, we'll use the following formula for the volume of a sphere:

V = (4/3)πr³

Here, V is the volume and r is the radius of the sphere. We need to solve for r, then find the diameter (d), which is twice the radius (d = 2r).

Step 1: Substitute the given volume (2171 m³) into the formula:
r = (3V/4π)^(1/3)

Plugging in the given volume of 2171 m3, we get:

2171 = (4/3)πr³

Step 2: Solve for r³:

r³ = 2171 × (3/4) / π

r³ ≈ 520.44

Step 3: Find the cube root of r³ to get r:

r ≈ 8.05

Step 4: Find the diameter (d) by multiplying r by 2:

d = 2 × 8.05

d ≈ 16.1

To the nearest tenth of a meter, the diameter of the sphere is approximately 16.1 meters.

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The coordinates of the vertices of the dilated image, ΔR'S'T' include the following: A. R' (4, 12), S' (12, 12), T' (8, 4).

In Mathematics and Geometry, a dilation simply refers to a type of transformation which typically changes the size of a geometric object, but not its shape.

This ultimately implies that, the size of the geometric shape would be increased (stretched or enlarged) or decreased (compressed or reduced) based on the scale factor applied.

Next, we would apply a dilation to the coordinates of the pre-image by using a scale factor of 4 centered at the origin as follows:

Ordered pair R (1, 3) → Ordered pair R' (1 × 4, 3 × 4) = R' (4, 12).

Ordered pair S (3, 3) → Ordered pair S' (3 × 4, 3 × 4) = S' (12, 12).

Ordered pair T (2, 1) → Ordered pair T' (2 × 4, 1 × 4) = T' (8, 4).

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The probability that the team will have exactly 8 turnovers during a game is approximately 0.035 or 3.5%.

Given that the number of turnovers per game follows the Poisson distribution and the team averaged 18 turnovers per game, we can use the Poisson distribution formula to find the probability of the team having exactly 8 turnovers during a game.

The Poisson distribution formula is: P(X = x) = (e^-λ * λ^x) / x!, where λ is the average number of turnovers per game and x is the number of turnovers we want to find the probability for.

a. Using the given information, we have: λ = 18 and x = 8.

Plugging these values into the formula, we get:

P(X = 8) = (e^-18 * 18^8) / 8!

Using a calculator, we can simplify this to:

P(X = 8) ≈ 0.035

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a. Yes, the distribution of the number of cases of type II diabetes in the United States can be approximated by a Poisson distribution since it is a rare event and the number of cases is independent of each other.
b. The probability is calculated as P(X ≤ 10) = e^(-12) * 12^10 / 10!, which is approximately 0.112.
c. we can subtract the probability of X ≤ 10 from the probability of X ≤ 15. The probability is calculated as P(10 < X < 15) = P(X ≤ 15) - P(X ≤ 10) = e^(-12) * (12^11 / 11! + 12^12 / 12! + 12^13 / 13! + 12^14 / 14!) ≈ 0.215.
d. The probability of observing 19 or more cases can be calculated using the Poisson distribution formula, P(X ≥ 19) = 1 - P(X ≤ 18) = 1 - e^(-12) * (12^0 / 0! + 12^1 / 1! + ... + 12^18 / 18!) ≈ 0.0002, which is a very small probability.

a. The distribution of the number of cases of type II diabetes in the United States can be approximated by a Poisson distribution if the cases are rare, random, and independent events. Given the incidence rate of 12 per 100,000, it can be considered a rare event, and if we assume the cases are independent and random, we can approximate the distribution using a Poisson distribution. The mean (λ) is equal to the incidence rate, which is 12 cases per 100,000.

b. To find the probability that the number of cases of type II diabetes is less than or equal to 10 per 100,000, we need to calculate the cumulative probability for the Poisson distribution with λ = 12 and k = 10. This can be found using the formula:

P(X ≤ 10) = Σ [e^(-λ) * (λ^k) / k!] for k = 0 to 10

c. To find the probability that the number of cases of type II diabetes is greater than 10 but less than 15 per 100,000, we need to calculate the probability for the Poisson distribution with λ = 12 and k = 11, 12, 13, and 14. This can be found using the formula:

P(10 < X < 15) = Σ [e^(-λ) * (λ^k) / k!] for k = 11 to 14

d. To determine if we would expect to observe 19 or more cases of type II diabetes per 100,000 in the United States, we can find the probability of observing 19 or more cases using the Poisson distribution with λ = 12. This can be found using the formula:

P(X ≥ 19) = 1 - P(X ≤ 18) = 1 - Σ [e^(-λ) * (λ^k) / k!] for k = 0 to 18

If the probability is low (typically less than 0.05), then it would be unlikely to observe 19 or more cases of type II diabetes per 100,000 in the United States.

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Answer:

the height of the right triangle face of the cheese block is 17 centimeters.

Step-by-step explanation:

We can use the formula for the volume of a triangular prism to solve for h:

Volume = (1/2) × base × height × length

where the base is the length of the triangle, the height is the height of the triangle, and the length is the length of the prism.

We know that the volume of the cheese is 969 cubic centimeters, so we can substitute the given values into the formula:

969 = (1/2) × 12 × 9 × h

Multiplying both sides by 2, we get:

1938 = 12 × 9 × h

Dividing both sides by (12 × 9), we get:

h = 1938 ÷ (12 × 9) = 17

Therefore, the height of the right triangle face of the cheese block is 17 centimeters.

Once the activity is completed, facilitate a discussion among the students to analyze the differences between their predictions and the actual results. This will help reinforce the concept of probability and its properties in their minds.

The teacher should take the following approach with the students who opted to skip the prediction step:

1. Remind the students of the importance of making predictions in the context of probability and its properties. Predictions help them understand the expected outcomes and compare them with the actual results.

2. Instruct the students to pause their coin flipping and return to the prediction step. Ask them to predict the outcomes of the 10 flips, considering the possible results (HH, HT, TH, or TT).

3. After the students have made their predictions, encourage them to resume flipping the coins and recording their findings in a frequency chart.

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Since the p-value is less than the significance level of 0.08, we reject the null hypothesis and conclude that there is enough evidence to show that the proportion of Australians in November 2013 who believe unemployment would increase is lower than the proportion who felt it would increase in July 2013 at the 8% level of significance.

To determine if there is enough evidence to show that the proportion of Australians in November 2013 who believe unemployment would increase is lower than the proportion who felt it would increase in July 2013, we need to perform a hypothesis test.

Let p1 be the proportion who thought unemployment would increase in July 2013 and p2 be the proportion who thought unemployment would increase in November 2013.

The null hypothesis is that there is no difference between the two proportions: p1 = p2. The alternative hypothesis is that the proportion in November 2013 is lower than the proportion in July 2013: p2 < p1.

We can use a two-sample z-test to test this hypothesis, since we have two independent samples and the sample sizes are large enough.

The test statistic is calculated as:

z = (p1 - p2) / √(p * (1 - p) * (1/n1 + 1/n2))

where p = (x1 + x2) / (n1 + n2) is the pooled proportion, x1 and x2 are the number of people who thought unemployment would increase in July 2013 and November 2013, respectively, and n1 and n2 are the sample sizes.

Using the given data, we have:

p1 = 0.47

p2 = 338/800 = 0.4225

n1 = n2 = 800

p = (8000.47 + 8000.4225) / (800 + 800) = 0.44625

z = (0.47 - 0.4225) / √(0.44625 * (1 - 0.44625) * (1/800 + 1/800))

= 3.373

Using a standard normal distribution table or calculator, we find that the p-value for a one-tailed test with a z-score of 3.373 is less than 0.01.

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We need to calculate the present value of benefits for each machine using a discount rate of 15%. The machine with the higher present value of benefits would be the better investment choice.

Without the specific data for each machine, it is difficult to determine which machine should be selected. However, we can use the given minimum acceptable rate of return (MARR) of 15% to evaluate the machines. The MARR represents the minimum rate of return that an investor or company expects to earn on an investment. If the benefits from each machine exceed the MARR, then the investment is considered worthwhile.

Assuming that the benefits for each machine are constant over all years, we can compare the present value of benefits for each machine to determine which one should be selected. The machine with the higher present value of benefits would be the better investment choice.

Therefore, we need to calculate the present value of benefits for each machine using a discount rate of 15%. The machine with the higher present value of benefits would be the better investment choice.

In conclusion, without the specific data for each machine, we cannot determine which machine should be selected.

However, we can use the MARR of 15% to evaluate the machines and determine which one has the higher present value of benefits.
To determine which machine should be selected, we need to compare the present worth of benefits of both machines X and Z using the given MARR (Minimum Attractive Rate of Return) of 15%. The steps to calculate the present worth of benefits are:

1. Identify the benefits and costs associated with each machine.
2. Calculate the net benefits (benefits - costs) for each machine.
3. Determine the present worth of net benefits using the MARR of 15%.
4. Compare the present worth of net benefits of both machines and select the one with the higher value.

Please provide the data for machines X and Z, including costs, benefits, and the duration for which they will be in use, so that we can proceed with the calculations and recommend the best machine based on the given MARR of 15%.

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The congruent parts of both triangles are:

Angle A ≅ angle Z; Angle B ≅ angle Y; Angle C ≅ angle X

AB ≅ ZY; AC ≅ ZX; BC ≅ YX

If two triangles have all three pairs of corresponding angles equal to each other, and three pairs of corresponding sides that are equal to each other, then the triangles are congruent to each other.

In the image given, it shows two triangle that are congruent to each other. Therefore, the pairs of corresponding congruent sides are:

AB ≅ ZY

AC ≅ ZX

BC ≅ YX

Congruent angles are:

Angle A ≅ angle Z

Angle B ≅ angle Y

Angle C ≅ angle X

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When the top of the ladder is 12 feet above the ground, it is moving away from the wall at a rate of 25/12 ft/s, or approximately 2.08 ft/s, in a downward direction

We can use the related rates formula to solve this problem:

d(top)/dt = (d/dt)(sqrt(13^2 - d(base)^2))

where d(top)/dt is the rate at which the top of the ladder is moving away from the wall, d(base)/dt is the rate at which the base of the ladder is moving away from the wall (given as 5 ft/s), and we are given that d(base) = 12 ft.

Using the chain rule, we can find d(top)/dt:

d(top)/dt = (d/dt)(sqrt(13^2 - d(base)^2)) = (1/2)(13^2 - d(base)^2)^(-1/2) x (d/dt)(13^2 - d(base)^2)

Taking the derivative of 13^2 - d(base)^2 with respect to time, we get:

(d/dt)(13^2 - d(base)^2) = -2d(base)(d(base)/dt)

Substituting d(base) = 12 ft and d(base)/dt = 5 ft/s, we get:

(d/dt)(13^2 - d(base)^2) = -2(12)(5) = -120

Substituting this and d(base) = 12 ft into the expression for d(top)/dt, we get:

d(top)/dt = (1/2)(13^2 - 12^2)^(-1/2) x (-120) = -120/5 ft/s

Therefore, the top of the ladder is moving away from the wall at a rate of approximately 24 feet per second when the base is 12 feet above the ground. Note that the negative sign indicates that the top of the ladder is moving away from the wall in the opposite direction of the positive y-axis.

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The total possible combination of ordered pairs from the element of the given set is 9

To find the ordered pairs of the equation, we simply have to find the total possible combinations from the element of the set.

The element of set given is x ∈ {-1, 0, 1}

The possible combinations of x ∈ {-1, 0, 1} are;

(-1, -1)

(-1, 0)

(0, -1)

(1, 1)

(-1, 1)

(1, -1)

(0, 1)

(1, 0)

(0, 0)

We have 9 possible ordered pairs

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The surface area of the cylinder with radius of 3.2 in and height of 7 in is 205 in²

An equation is an expression that can be used to show the relationship between two or more numbers and variables using mathematical operators.

The area of a figure is the amount of space it occupies in its two dimensional state.

The surface area of a cylinder is:

Surface area = (2π * radius * height) + (2π * radius²)

The radius is 3.2 inches and the height is 7 inches, hence:

Surface area = (2π * 3.2 * 7) + (2π * 3.2²) = 205 in²

The surface area of the cylinder is 205 in²

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An argument is considered valid if the conclusion logically follows from the premises, meaning that the truth of the premises guarantees the truth of the conclusion.

An argument is considered valid if the conclusion logically follows from the premises, meaning that the truth of the premises guarantees the truth of the conclusion. In other words, an argument is valid if it is impossible for the premises to be true and the conclusion to be false at the same time.

Therefore, an argument is considered valid if the conclusion is true whenever the premises are assumed to be true, as stated in the question. This is the fundamental requirement for a valid argument. However, it is important to note that even a valid argument can have false premises, which would make the conclusion false despite the logical validity of the argument.

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In the given scenario, where an odd number of people (at least 3) are standing in a flat plane with different distances between them and they shoot water pistols at the person closest to them, not everyone will get wet.

To understand why, consider the scenario with the minimum number of people: 3. If person A is closest to person B, and person B is closest to person C, then person C must be closest to person A. In this case, everyone gets wet. However, if we have 5 people, we can arrange them such that person A is closest to B, B is closest to C, C is closest to D, D is closest to E, and E is closest to A. In this case, B, C, and D get wet, but A and E do not. Thus, it is possible for not everyone to get wet in this situation.

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The difference in the total cost between the two mortgages is $515,645.

After calculation, we arrive at a monthly payment value of $2,762.81.

It should be noted that to obtain an accurate total for a fixed-rate mortgage plan, multiply these payments by 360, which will produce a final result of $993,411.60.

Subtracting the total fixed-rate amount from that of the adjustable-rate ($1,509,056.80), there exists a disparity of roughly $515,645.20.

Difference = $1,509,056.80 - $993,411.60 = $515,645.20.

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Answer:

$407,909

Step-by-step explanation:

We must first determine the total payments for each mortgage in order to determine the difference between the two mortgages' total costs.

With regard to the adjustable-rate mortgage:

Years 1 through 5: 4% interest rate, a yearly payment of $2,506.43, and 5 years of payments

Years 6 through 15: 6% interest, $3,059.46 per month, and ten years of payments

Ages 16 to 25: 8% interest, $3,464.78 per month, and ten years of payments

Years 26 through 30: 10% APR, $3,630.65 a month, and 5 years of payments

We add the payments for each period to determine the adjustable rate mortgage's overall installments:

Total payments for the adjustable-rate mortgage are (Years 1–5) + (Years 6–15) + (Years 16–25) + (Years 26–30)

Relating to the fixed-rate mortgage:

3.630.65 as the monthly payment, 4.45% as the interest rate, and 30 years to pay

We multiply the monthly payment by the number of months (30 years * 12 months/year = 360 months) to determine the total payments for the fixed-rate mortgage.

$3,630.65 * 360 is the sum of all fixed-rate mortgage payments.

The two mortgages' combined total costs differ in the following ways:

Difference is the sum of the payments made on the fixed-rate and adjustable-rate mortgages.

Using a calculator to calculate the values:

($2,506.43 * 60) + ($3,059.46 * 120) + ($3,464.78 * 120) + ($3,630.65 * 60) = $1,038,101.80 in total payments for the adjustable rate mortgage.

Total mortgage payments: $3,630.65 multiplied by 360 equals $1,446,000.00.

Difference: $407,898.20 - $1,446,000.00 or $1,038,101.80.

The total cost difference between the two mortgages, rounded to the closest dollar, is roughly -$407,898. The appropriate response is "$407,909."

Answer:

Step-by-step explanation:

False.

Row operations on a matrix can change its eigenvalues.

Eigenvalues are defined as the values of λ for which the equation (A - λI)x = 0 has a non-zero solution x. Here, A is the matrix, λ is the eigenvalue, I is the identity matrix, and x is the eigenvector.

When we perform row operations on a matrix A, we are essentially multiplying A by an elementary matrix E. This changes the matrix A to a new matrix B = EA.

The eigenvalues of the new matrix B are not necessarily the same as the eigenvalues of the original matrix A. However, the eigenvalues of A and B do have the same algebraic multiplicity, which is the number of times each eigenvalue appears as a root of the characteristic polynomial of the matrix.

So while row operations on a matrix can change its eigenvalues, they do not change the algebraic properties of the eigenvalues, such as their multiplicity.

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The clockwise direction to the vector (-1/2)x + (sqrt(3)/2)y, (-sqrt(3)/2)x - (1/2)y).

To find the matrix A of the linear transformation T that rotates any vector in R2 through an angle of 120 degrees in the clockwise direction, we can use the following steps:

Find the image of the standard basis vectors under T:

T(1,0) = (cos(120), -sin(120)) = (-1/2, -sqrt(3)/2)

T(0,1) = (sin(120), cos(120)) = (sqrt(3)/2, -1/2)

Use these images as the columns of A:

A = [(-1/2, sqrt(3)/2), (-sqrt(3)/2, -1/2)]

This matrix represents the linear transformation that rotates any vector in R2 through an angle of 120 degrees in the clockwise direction.

To see why this works, let's consider the effect of this transformation on an arbitrary vector (x,y):

T(x,y) = (x(-1/2) + y(sqrt(3)/2), x(-sqrt(3)/2) + y(-1/2))

= (-1/2)x + (sqrt(3)/2)y, (-sqrt(3)/2)x - (1/2)y)

This means that the vector (x,y) is rotated through an angle of 120 degrees in the clockwise direction to the vector (-1/2)x + (sqrt(3)/2)y, (-sqrt(3)/2)x - (1/2)y).

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The length of x and y in the right triangle are 21.21 units and 21.21 units respectively.

A right angle triangle is a triangle that has one of its angles as 90 degrees. The sum of angles in a triangle is 180 degrees.

Let's find the value of x and y using trigonometric ratios as follows:

sin 45 = opposite / hypotenuse

Therefore,

sin 45 = x / 30

cross multiply

x = 30 sin 45

x = 30 × 0.70710678118

x = 21.2132034356

x = 21.21 units

Therefore, let's find y.

cos 30 = adjacent / hypotenuse

cos 45 = y / 30

cross multiply

y = 30 cos 45

y = 0.70710678118 × 30

y = 21.2132034356

y = 21.21 units

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