suppose u is m×n. explain why if u has orthonormal columns, then we must have m ≥n

The given series is conditionally convergent.

To see why, note that the terms of the series alternate in sign, which suggests using the alternating series test. Furthermore, the absolute value of the terms is given by 6n/ n, which approaches infinity as n goes to infinity. Therefore, the series fails the criteria for absolute convergence.

However, by the alternating series test, the series converges conditionally. To see this, note that the terms of the series decrease in absolute value and approach zero as n goes to infinity. Furthermore, the partial sums of the series alternate in sign and converge to a limit as n goes to infinity. Therefore, the series is conditionally convergent.

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The fraction of tomatoe recipe greater than 1 used is 7/4

The question requires that we Subtract the required value of tomato recipe from 1

The subtraction expression can be written thus ;

Amount of tomatoe recipe - 1

[tex]2 \frac{3}{4} - 1[/tex]

Converting to a proper fraction;

11/4 - 1/1

Take L.C.M of the denominator

L.C.M of 4 and 1 is 4

11/4 - 1/1 = (11 - 4)/4

11/4 - 1/1 = 7/4

Therefore, the fraction of tomatoe greater than 1 used is 7/4

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"B: degrees of freedom > 10"  is not a way to check for the nearly normal condition.

To check the nearly normal condition, we can use several methods such as histogram, normal probability plot, and goodness of fit test. The central limit theorem can also be used when the sample size is large enough. However, checking the degrees of freedom is not a way to check for the nearly normal condition. Degrees of freedom refer to the number of independent pieces of information that are used to estimate a statistic. While it is important in hypothesis testing and confidence intervals, it is not related to checking for the nearly normal condition.

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Given that heart rate before an exam for STA 100 students is normally distributed with mean 100 bpm and standard deviation 20.2 bpm, the probability that a randomly selected student's heart rate is under 125 bpm given that it is over 100 bpm is approximately 0.6306.

We are given that the heart rate before an exam for STA 100 students follows a normal distribution with mean 100 bpm and standard deviation 20.2 bpm.

We want to find the conditional probability that the heart rate is under 125 bpm given that it is over 100 bpm.

Let A be the event that the heart rate is over 100 bpm, and let B be the event that the heart rate is under 125 bpm. Then we want to find P(B | A).

We can use Bayes' theorem to find P(B | A):

P(B | A) = P(A | B) * P(B) / P(A)

We know that P(A) = 1 - P(B), since the two events are complements. We also know that P(B) can be found using the standard normal distribution as follows:

P(B) = P(Z < (125 - 100) / 20.2) = P(Z < 1.2376) = 0.8917,

where Z is a standard normal random variable. To find P(A | B), we need to compute the conditional probability that the heart rate is over 100 bpm given that it is under 125 bpm:

P(A | B) = P(B | A) * P(A) / P(B) = P(B | A) * (1 - P(B)) / P(B).

Since the heart rate follows a normal distribution, we can standardize it by subtracting the mean and dividing by the standard deviation:

P(A | B) = P(Z > (100 - 100) / 20.2 | Z < (125 - 100) / 20.2)

= P(Z > 0 | Z < 1.2376)

= P(Z < 1.2376)

= 0.8917.

Finally, we can plug in the values we have found to obtain the probability we are looking for:

P(B | A) = P(A | B) * P(B) / P(A)

= 0.8917 * 0.8917 / (1 - 0.8917)

= 0.6306 (approximately).

Therefore, the probability that a randomly selected student's heart rate is under 125 bpm given that it is over 100 bpm is approximately 0.6306 or 63.06%.

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Complete Question

6. Assume that heart rate (in beats per minute, or bpm) before an exam for STA 100 students is distributed nor- mal, with a mean of 100 bpm and a standard deviation of 20.2 bpm. Assume all students in the following problem are selected from this population.

If we know a randomly selected students heart rate is over 100 (it is given), what is the probability that it is under 125?

Of the four thermodynamic quantities mentioned, δg (the change in Gibbs free energy) is the only state function. This means that δg depends only on the initial and final states of the system, regardless of the path taken to get there.

In contrast, δs (the change in entropy), δssurr (the change in entropy of the surroundings), and δsuniv (the change in total entropy) are all path-dependent quantities, meaning that they depend on the specific process or path taken between the initial and final states. This is because entropy is a measure of the degree of disorder or randomness in a system, and this randomness can depend on factors like temperature and pressure, which can change during a process. As a result, the change in entropy of a system depends on the specific conditions under which the process occurs, rather than just the initial and final states. In contrast, Gibbs free energy incorporates both the enthalpy and entropy of a system, and is defined as ΔG = ΔH - TΔS. Because enthalpy and entropy are also state functions, the change in Gibbs free energy is also a state function. This makes it a useful quantity for predicting the spontaneity and direction of chemical reactions, as reactions that result in a decrease in Gibbs free energy (ΔG < 0) are typically thermodynamically favored.

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Elena's yearly cash inflows from the project after the payback period is $15,000. A correct answer is an option (b).

The payback period is the time it takes for the project's cash inflows to equal the initial investment. In this case, the payback period is 6 years, meaning that after 6 years, Elena will have received enough cash inflows to recover her initial investment of $150,000.

Since the project pays Elena an even amount per year for 10 years, and the payback period is 6 years, she will receive cash inflows for an additional 4 years after the payback period. Therefore, to calculate Elena's yearly cash inflows, we divide the remaining cash inflows by the number of years remaining:

Remaining cash inflows = $150,000 (initial investment) - cash inflows received during the payback period

= $150,000 - ($15,000 x 6)

= $60,000

Yearly cash inflows = Remaining cash inflows/number of years remaining

= $60,000 / 4

= $15,000

Hence, B is the correct option.

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 The vector equation is (x,y,z) = x = -4t + 6

y = 2t + 6

z = 12t + 2

To find the vector equation and the parametric equations of the line through (6,6,2) that is perpendicular to the given lines, we can use the cross product of the direction vectors of the two lines as the direction vector of the new line.

First, we find the direction vectors of the two given lines:

Line 1: r1(t) = (2-2t, 5+6t, 3-2t), direction vector d1 = (-2, 6, -2)

Line 2: r2(t) = (-2, 5+t, 3-t), direction vector d2 = (0, 1, -1)

Next, we take the cross product of d1 and d2 to find a vector that is perpendicular to both lines:

d1 x d2 = (-4, 2, 12)

This vector is parallel to the new line we want to find, so we can use it as the direction vector of the line. To get the vector equation, we can use the point-slope form of the equation of a line:

(x,y,z) = (6,6,2) + t(-4,2,12)

This simplifies to:

x = -4t + 6

y = 2t + 6

z = 12t + 2

These are the parametric equations of the line.

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Shape R and Shape S were put in the same group because they are the shapes with perpendicular line segments. Therefore the correct option is option D.

Shape S, a trapezium with one vertical side, and Shape R, a square, both have perpendicular line segments. All of the sides of a square are perpendicular, while one of the non-parallel sides of a trapezium with a vertical side is perpendicular to the base.

Reasons for ruling out other options

Option A, "Shapes without any right angles," is incorrect because Shape R (a square) has four right angles.

Option B, "Shapes with exactly one pair of parallel sides," is incorrect because Shape S (a trapezoid with one vertical side) has only one pair of parallel sides, while Shape R(a square) has two pairs of parallel sides.

Option C, "Shapes without parallel line segments," is also incorrect because all the other shapes P, Q, and T have parallel line segments.

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The town's population in 10 years will be approximately 47.8% of its current population.

What is linear combinations?

In mathematics, a linear combination is a sum of scalar multiples of one or more variables. More formally, given a set of variables x1, x2, ..., xn and a set of constants a1, a2, ..., an, their linear combination is given by the expression:

a1x1 + a2x2 + ... + anxn

Let P be the current population of the town.

In the first year, the population will decrease by 7%, which means that the population after the first year will be:

P1 = P - 0.07P = 0.93P

In the second year, the population will decrease by another 7%, so the population after the second year will be:

P2 = 0.93P - 0.07(0.93P) = 0.8649P

We can continue this process for 10 years:

P10 = [tex]0.93^{10}[/tex]P ≈ 0.478P

Therefore, the town's population in 10 years will be approximately 47.8% of its current population.

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The exact value of the trigonometric function at the given real number is: 1. 0      2. -1      3. undefined.

To find the exact value of the trigonometric function at the given real numbers, we can use the unit circle and the properties of trigonometric functions.

(a) sin(34π):

Since the unit circle repeats every 2π radians, we can subtract multiples of 2π to find the equivalent angle within one full revolution.

34π = 17π + π

The angle π is equivalent to 180 degrees, so sin(π) = 0.

Therefore, sin(34π) = sin(17π + π) = sin(π) = 0.

(b) sec(67π):

Similar to the previous case, we can rewrite 67π as:

67π = 33π + π

The angle π is equivalent to 180 degrees, and the secant function is the reciprocal of the cosine function. Since the cosine function has a value of -1 at π, the reciprocal, sec(π), is -1.

Therefore, sec(67π) = sec(33π + π) = sec(π) = -1.

(c) cot(-3π):

To find the cotangent function, we need to determine the tangent function at -3π.

The tangent function has a period of π, so we can rewrite -3π as:

-3π = -2π - π

The angle -π is equivalent to -180 degrees, and the tangent function is the sine divided by the cosine. Since sin(-π) = 0 and cos(-π) = -1, the tangent function is undefined at -π.

Therefore, cot(-3π) is also undefined.

In summary:

(a) sin(34π) = 0

(b) sec(67π) = -1

(c) cot(-3π) is undefined.

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Answer:

V = 795 1/5 yd³

Step-by-step explanation:

The volume of a rectangular prism can be solved using the formula,

V = lwh

Given:

l = 14 1/5 yd or 71/5 yd

w = 7yd

h = 8yd

Substitute the values and solve

V = 71/5yd(7yd)(8yd)

V = 3976/5 yd³

V = 795 1/5 yd³

Five and three hundred seven thousandths.

Or do you want it in number-expanded form...?

Based on the characteristics of the line and parabola , the correct answer is: A. [tex]\(y = \begin{cases} x^2 + 2, ; x \leq 1 \\ -x + 2, ; x > 1 \end{cases}\)[/tex]

Based on the given information, let's analyze the characteristics of the line and parabola to determine the correct representation:

1. Line: In the context of graphing, a line appears as a straight line that can extend in any direction across the coordinate plane. It can have a positive or negative slope, or be horizontal or vertical.

- The line passes through the points [tex](1,1),(2,0) , (4, -2) , ( 8 , -6)[/tex]

- It extends along the first and fourth quadrants.

- A closed dot is shown at the point (1,1).

2. Parabola: In the context of graphing, a parabola appears as a curved line. It can open upward or downward and can be concave or convex. The vertex of the parabola represents the lowest or highest point on the curve, and the axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetric halves.

- The parabola passes through the points [tex](1,3) , (-2,6), (10,-3)[/tex]

- It extends along the first and second quadrants.

- An open dot is shown at the point (1,3).

- The vertex of the parabola lies at (0,2).

Given these characteristics, we can determine the correct representation:

The correct answer is:

A. [tex]\(y = \begin{cases} x^2 + 2, ; x \leq 1 \\ -x + 2, ; x > 1 \end{cases}\)[/tex]

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The equation of the attached trigonometric graph is

y = cos (0.4x)

The cos graph in the problem starts at (0, 1)

The equation is represented by y = A cos (Bx + C) + D

The amplitude of the graph is A and this is solved by

= (1 - (-1) / 2

= 2 / 2

= 1

then B is calculated by

B = 2π / period, and the period, form the graph is 5π

B = 2π / 5π = 2/5 = 0.4

The phase shift, C is 0

The vertical shift, D is 0

putting in the values as in the equation above, we have

y = 1 cos (0.4x + 0) + 0

y = cos (0.4x)

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The measure of the angle m∠EOF subtended by the arc EF at the center is equal to 76°

The angle subtended by an arc of a circle at it's center is twice the angle it substends anywhere on the circles circumference. Also the arc measure and the angle it subtends at the center of the circle are directly proportional.

The chords EH and FJ intersect at the center of the circle O, hence the arc measure 76° is equal to the angle EOF

So since;

arc angle FJ = arc angle EH

then;

arc angle EF = m∠EOF = 76°

Therefore, the measure of the angle m∠EOF subtended by the arc EF at the center is equal to 76°.

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(1)Elena should purchase a rainproof tarp that is at least 140 ft² in size. Option E (2)it would take 140 ft³ of water to fill the tent completely. Option A.

(1) To determine the size of the rainproof tarp Elena should purchase, we need to calculate the total surface area of the tent.

The tent has 4 sides, and each side has the same dimensions. Given that the height of the tent is 8.75 ft and the length and breadth are both 4 ft, the surface area of each side can be calculated as:

Side area = Length × Height = 4 ft × 8.75 ft = 35 ft²

Since there are 4 sides, the total surface area of the tent is:

Total tent surface area = Side area × 4 = 35 ft² × 4 = 140 ft²

Therefore, Elena should purchase a rainproof tarp that is at least 140 ft² in size.

The correct option would be e) X-Large: 144 ft², as it is the only size that is equal to or greater than the required size.

(2) If the tent fills up completely with water, we can calculate the volume of water it would hold using the formula:

Volume = Length × Breadth × Height

Given that the length and breadth of the tent are both 4 ft and the height is 8.75 ft, we can substitute these values into the formula:

Volume = 4 ft × 4 ft × 8.75 ft = 140 ft³

Therefore, it would take 140 ft³ of water to fill the tent completely.

Hence, the correct answer is 140 ft³. So Option E is correct  for 1. and Option A is correct for 2

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The margin of error is 0.1415, where z is the critical value for the desired confidence level,

The margin of error can be calculated using the formula: Margin of error = z * (standard deviation / sqrt(n))

where z is the critical value for the desired confidence level, standard deviation is the population standard deviation (which can be estimated using the sample standard deviation), and n is the sample size.

For a 99.8% confidence level, the critical value is 2.967. Using the given values of x and n, we can calculate the sample proportion as 46/98 = 0.4694.

To estimate the population standard deviation, we can assume that the sample proportion is a good estimate of the population proportion, and use the formula:

standard deviation = sqrt(p*(1-p)/n)

where p is the sample proportion. Substituting the values, we get: standard deviation = sqrt(0.4694*(1-0.4694)/98) = 0.0519

Now we can plug in the values into the margin of error formula to get: Margin of error = 2.967 * (0.0519 / sqrt(98)) = 0.1415

Therefore, the margin of error is 0.1415.

It is important to note that the margin of error represents the amount by which the sample proportion may differ from the population proportion with a certain level of confidence.

It is also important to remember that the margin of error is not the same as the sampling error, which is the difference between the sample mean and the population mean.

The margin of error can be used to determine the sample size required for a given level of precision, or to compare different sample sizes to determine which is more likely to yield a representative sample.

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The inverse of α=(123)(456) is α^−1 = (456)(123). Option (B) is the correct answer.

The question requires finding the inverse of the given permutation α=(123)(456). To get the inverse of any permutation, the order of the cycles should be reversed, and the order of elements in each cycle should be reversed. Following this rule, the inverse of α is found to be α^−1 = (456)(123). Therefore, option (B) is the correct answer. In other words, if we apply α and then α^−1 to any set, we get back the original set. The inverse permutation of a permutation has useful applications in algebra and cryptography, among other fields.

Therefore, the inverse of α=(123)(456) is α^−1 = (456)(123). Thus, option (B) is the correct answer.

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To estimate the average weekly sales of a new line of athletic footwear to within $300 with 99% confidence, the company needs to sample sales data for 87 weeks.

To estimate the average weekly sales of a new line of athletic footwear to within $300 with 99% confidence, we need to determine the sample size required. The standard deviation of the weekly sales figures is given as $1300.

To calculate the sample size, we can use the formula:

n = (z * σ / E)^2

where n is the required sample size, z is the critical value for the 99% confidence level (which is 2.576), σ is the standard deviation of the weekly sales figures, and E is the maximum allowable error or the desired margin of error (which is $300).

Substituting the given values in the formula, we get:

n = (2.576 * 1300 / 300)^2

n = 86.50

Since we cannot have a fractional number of weeks, we round up the value of n to 87. Therefore, the company needs to sample sales data for 87 weeks to estimate the average weekly sales of the new footwear to within $300 with 99% confidence.

To estimate the average weekly sales of a new line of athletic footwear to within $300 with 99% confidence, the company needs to sample sales data for 87 weeks. This can be calculated using the formula n = (z * σ / E)^2, where z is the critical value, σ is the standard deviation, and E is the maximum allowable error.

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The correct formula for the revenue generated by the item is Revenue = 200p - 5p^2.

The revenue generated by the item can be calculated by multiplying the price (p) by the unit sales (200 - 5p). Therefore, the correct formula for the revenue generated by the item is:

Revenue = Price x Unit Sales
Revenue = p(200 - 5p)
Revenue = 200p - 5p^2

The revenue generated by the item is a quadratic function of the price (p). To find the maximum revenue, we need to differentiate the function with respect to p and set it equal to zero:

dRevenue/dp = 200 - 10p = 0
10p = 200
p = 20

Therefore, the maximum revenue is generated when the price of the item is $20. Substituting p = 20 in the revenue formula, we get:

Revenue = 200(20) - 5(20^2) = $2000

Hence, the correct formula for the revenue generated by the item is Revenue = 200p - 5p^2, and the maximum revenue is achieved when the price of the item is $20, generating a revenue of $2000.

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The independent samples t-test is the appropriate statistic to use when comparing the mean number of hours worked per week for males and females.

To compare the mean number of hours worked per week for males and females, you would use the independent samples t-test. The independent samples t-test is a statistical test used to determine if there is a significant difference between the means of two independent groups. In this case, the independent groups are males and females.

The t-test allows you to compare the means of the two groups and determine if any observed difference is statistically significant or simply due to chance. It takes into account the sample means, sample sizes, and sample variances of both groups.

By conducting the independent samples t-test, you can assess whether there is evidence to suggest that the mean number of hours worked per week differs significantly between males and females. If the p-value associated with the t-test is below a predetermined significance level (commonly 0.05), it suggests that there is a statistically significant difference in the mean number of hours worked per week between the two groups.

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Because the sum of the interior angles must be 180°, we can see that the value of the third angle is 18°36'

Remember that for any triangle, the sym of the interior angles must always be equal to 180°.

So if the 3 angles are A, B, and C, we can write:

A + B + C = 180°.

In this case we can say taht:

A = 43° 36'

B =  117° 48'

Remember that 60 minutes are equal to one degree.

Now we can write:

43° 36' +  117° 48' + C = 180°

(43° + 117°) + (36' + 48') + C = 180°

160° + 84' + C  = 180°

161° + 24' + C = 180°

C = 180° - 161° - 24'

C = 19° - 24'

C = 18° + 36'

The measure of the third angle is 18°36'

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The t-statistic distribution is similar to the normal distribution in shape, but it has heavier tails due to its smaller sample size and greater variability. This means that the t-statistic distribution has more extreme values (both positive and negative) than the normal distribution.

The shape of the t-statistic distribution, compared to the normal distribution, can be described as having thicker tails and a more peaked center. This means that the t-distribution has a higher probability of extreme values than the normal distribution. As the sample size increases, the t-distribution approaches the normal distribution in shape.

The t-statistic distribution is a probability distribution used in hypothesis testing when the sample size is small and the population standard deviation is unknown. It is a family of continuous probability distributions that depend on the sample size and the degrees of freedom.

The t-distribution is similar in shape to the standard normal distribution, but it has heavier tails and is more spread out. The shape of the t-distribution changes as the degrees of freedom change. As the sample size increases, the t-distribution approaches the standard normal distribution.

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The correct equation to find the nth term in the arithmetic sequence is -171.

The equation that can be used to find the nth term in an arithmetic sequence is: an = a1 + (n-1)d,

Where an is the nth term,

a1 is the first term and

d is the common difference between consecutive terms.
In this case, we can see that the common difference is -7, since each term is decreasing by 7.

We also know that the first term is -3,

So we can plug these values into the formula: an = -3 + (n-1)(-7).
Simplifying this equation gives us: an = -7n + 4.
To find the 25th term in the sequence, we can plug in n = 25: a25 = -7(25) + 4 = -171.
Therefore, the 25th term in the sequence is -171.
The nth term in the arithmetic sequence is an = -7(n-1) + (-3).

To find the 25th term, simply substitute n with 25:

a25 = -7(25-1) + (-3)

= -7(24) - 3

= -168 - 3

= -171.

The 25th term in the sequence is -171.

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[tex]f(s) = ℒ{(4t+1)(t-1)} = 8/s^3 - 3/s^2 - 1/s[/tex] by using laplace transform for the function.

A mathematical technique for studying linear, time-invariant systems is the Laplace transform. It transforms a time-domain function into a frequency-domain function, making it simpler to analyse the behaviour of the system.

The transform is defined as the function's integral times an exponential that decays, with the integral being assessed from 0 to infinity. The Laplace transform can convert a differential equation into an algebraic equation, which can subsequently be solved more quickly, making it useful for solving differential equations.

Along with other disciplines, the Laplace transform has uses in signal processing, electrical engineering, and control theory. It is an effective technique for deconstructing complicated systems and comprehending their evolution.

To find f(s), we first need to apply the Laplace transform to the function (4t+1)(t-1), denoted by ℒ{(4t+1)(t-1)}. Using the linearity property of the Laplace transform, we can split this into two separate transforms:

[tex]ℒ{(4t+1)(t-1)} = ℒ{4t^2 - 3t - 1} = 4ℒ{t^2} - 3ℒ{t} - ℒ{1}[/tex]

Next, we use the table of Laplace transforms to find the transforms of each term:

[tex]ℒ{t^n} = n!/s^(n+1)[/tex] for any integer n greater than or equal to 0
[tex]ℒ{1} = 1/s[/tex]

Using these formulas, we get:

[tex]ℒ{t^2} = 2!/s^3 = 2/(s^3)\\ℒ{t} = 1!/s^2 = 1/s^2\\ℒ{1} = 1/s[/tex]

Substituting these results back into the original equation, we get:

[tex]ℒ{(4t+1)(t-1)} = 4(2/(s^3)) - 3(1/s^2) - (1/s)\\= 8/s^3 - 3/s^2 - 1/s[/tex]

Therefore, [tex]f(s) = ℒ{(4t+1)(t-1)} = 8/s^3 - 3/s^2 - 1/s[/tex].

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Answer:

option B: (x+1)^2+(y+3)^2=5.

Step-by-step explanation:

Answer:

[tex](x-1)^{2} + (y+3)^2 = 25[/tex]

Step-by-step explanation:

[tex]-2x = -x^{2} - 6y -y^{2} + 15\\x^{2} - 2x + y^{2} + 6y = 15\\(x-1)^{2} - 1 + (y + 3)^{2} - 9 = 15\\(x-1)^{2} + (y+3)^2 = 25[/tex]

The solution to the vector-valued differential equation is: r(t) = [tex](8 * e^{(5t)}, 2 * e^{(5t)}, 6 * e^{(5t)})[/tex]. The general solution can be found by using the exponential function.

To find the solution of the given differential equation r'(t) = 5r(t) with initial condition r(0) = (8, 2, 6), we first recognize that this is a first-order linear homogeneous differential equation.
Let r(t) = (x(t), y(t), z(t)). Then, we have the following three scalar differential equations:
1. x'(t) = 5x(t)
2. y'(t) = 5y(t)
3. z'(t) = 5z(t)
Now, we can solve each equation individually:
1. x(t) = x(0) * [tex]e^{(5t)}[/tex] = 8 * [tex]e^{(5t)}[/tex]
2. y(t) = y(0) * [tex]e^{(5t)}[/tex] = 2 * [tex]e^{(5t)}[/tex]
3. z(t) = z(0) * [tex]e^{(5t)}[/tex] = 6 * [tex]e^{(5t)}[/tex]
Thus, the solution to the vector-valued differential equation is:
r(t) = [tex](8 * e^{(5t)}, 2 * e^{(5t)}, 6 * e^{(5t)})[/tex]

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The surface area of the sphere is about 50.27 square feet.

The formula for the surface area of a sphere is:

Surface area = 4πr²

The circumference of the sphere as C = 4r ft.

The radius of the sphere as follows:

C = 2πr

4r = 2πr

r = 2 ft

The radius of the sphere can use the formula for surface area to find the answer:

Surface area = 4πr²

Surface area = 4π(2²)

Surface area ≈ 50.27 square feet

Rounding the answer to the nearest hundredth, we get:

Surface area ≈ 50.27 square feet (rounded to two decimal places)

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Answer: 1 mile

Step-by-step explanation:

Well it's 1 because half of 2 is 1, so Zachary ran 1 lap around the lake.

In a Cartesian join, also known as a cross join, each row from the first table is combined with each row from the second table.

If the first table has 5 rows and the second table has 3 rows, the result of the Cartesian join will consist of 5 * 3 = 15 rows.