1. Find any extrema or saddle points of f(x,y) = x^3 + 12xy - 3y^2 - 27x + 34 2. A company plans to manufacture closed rectangular boxes that have a volume of 16 ft? Without using Lagrange multipliers, find the dimensions that will minimize the cost if the material for the top and bottom costs twice as much as the material for the sides

To perform a hypothesis test on the proportion of people who subscribe to a cable television service versus others, we need to check the following conditions:

1. Random Sampling: Ensure that the sample is randomly selected, which means each person has an equal chance of being selected. This is important to avoid biases and ensure the sample is representative of the population.

2. Independence: If sampling without replacement, the sample size (n) should be less than 10% of the population size (N) to assume independence. This is called the 10% Rule. Independence is important because the outcome of one person's choice should not affect another person's choice.

3. Normality: We need to ensure that the sampling distribution of the sample proportion (p-hat) is approximately normal. This can be checked using the success-failure condition, which states that both np and n(1-p) should be greater than or equal to 10. This ensures that the distribution is sufficiently normal to apply the z-test.

To help the student, remind them to:

1. Verify that the sample is random.
2. Check the independence condition by confirming the sample size is less than 10% of the population size.
3. Check the normality condition by calculating np and n(1-p) and making sure both values are greater than or equal to 10.

It is crucial to check these conditions because if they are not met, the hypothesis test's results may not be valid or reliable. Meeting these conditions ensures that the statistical methods applied are appropriate and provide accurate results for making conclusions.

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The decision with the highest expected value should be chosen. In this case, the economics book has a higher expected value (32,000 copies) compared to the statistics book (30,500 copies). Therefore, the optimum decision alternative is to write the economics book.

a. Decision tree diagram:
2. Branch off two nodes from the root for each option.
3. From the economics book node, branch off two nodes for major and smaller publisher placement. Assign probabilities of 50% and 50% for each.
4. From the statistics book node, branch off two nodes for major and smaller publisher placement. Assign probabilities of 40% and 60% for each.
5. Assign ultimate sales to each end node (40,000 and 30,000 for economics; 50,000 and 35,000 for statistics).

b. The probability that the economics book would wind up being placed with a smaller publisher is 50% (1 - 50% chance of placing it with a major publisher).

c. The probability that the statistics book would wind up being placed with a smaller publisher is 60% (1 - 40% chance of placing it with a major publisher).

d. Expected value for the decision alternative to write the economics book:
(0.50 * 40,000) + (0.50 * 0.80 * 30,000) = 20,000 + 12,000 = 32,000 copies.

e. Expected value for the decision alternative to write the statistics book:
(0.40 * 50,000) + (0.60 * 0.50 * 35,000) = 20,000 + 10,500 = 30,500 copies.

f. Expected value for the optimum decision alternative:
The decision with the highest expected value should be chosen. In this case, the economics book has a higher expected value (32,000 copies) compared to the statistics book (30,500 copies). Therefore, the optimum decision alternative is to write the economics book.

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At a significance level of 0.01, the conclusion would be that there is not enough evidence to support the alternative hypothesis (H1: μ > 72).

To lea

a. To calculate the p-value, we can use the standard normal distribution and the test statistic formula:

Test statistic (Z) = (sample mean - population mean) / (population standard deviation / sqrt(sample size))

Z = (75.2 - 72) / (6 / sqrt(16))

Z = 3.2 / 1.5

Z = 2.13 (rounded to two decimal places)

To find the p-value, we need to calculate the area under the standard normal curve to the right of the test statistic (Z = 2.13). Using a standard normal distribution table or a calculator, we find that the area to the right of Z = 2.13 is approximately 0.016.

Since this is a one-sided test (H1: μ > 72), the p-value is the probability of observing a test statistic as extreme or more extreme than the one obtained. Therefore, the p-value is 0.016.

b. If the test were made at a significance level of 0.01 (1%), we would compare the p-value to the significance level. In this case, the p-value (0.016) is less than the significance level (0.01). Therefore, we would reject the null hypothesis.

c. To find the critical value at a significance level of 0.01, we need to determine the z-score that corresponds to an area of 0.01 in the upper tail of the standard normal distribution.

Using a standard normal distribution table or a calculator, we find that the critical value for a significance level of 0.01 is approximately 2.33.

Comparing the critical value (2.33) with the test statistic (Z = 2.13), we see that the test statistic is less than the critical value. In hypothesis testing, if the test statistic is less than the critical value, we fail to reject the null hypothesis.

Therefore, at a significance level of 0.01, the conclusion would be that there is not enough evidence to support the alternative hypothesis (H1: μ > 72).

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The Set A defined as A = { 0,3,8} is not a subset of which set is set rational numbers. So, option(a) is right one.

Set is defined as a well defined collection of objects. It is denoted by capital letters of alphabets. The set represented by clearly brackets. We have a set A = { 0,3,8}

Subset is a also a set. A set A is called subset of another set B iff all elements of the set A are elements of the set B. Now,

The counting set is defined as a set with finite number of elements. It has range set {-∞, ∞}. As we see set A is consists three elements that is counted set. Also, these elements belong to whole and integers set. Hence, set A is subset of whole numbers, natural numbers and counting set.

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Complete question :

Set A = 0,3,8 Set A is Not a subset of which set of numbers?

a) Rational numbers

b) Integers

c) Whole numbers

d) Counting numbers

The cost of the carpeting Gabriela will use in her living room is  $3,305.47 and Area of living room is 346.41 sq ft

Area of rectangular room = length x width = 25 ft x 16 ft = 400 sq ft

Area of quarter-circle = (1/4) x pi x r^2 = (1/4) x pi x 8^2 = 16 pi sq ft

So the area of the living room is:

Area of living room = Area of rectangular room - Area of quarter-circle

= 400 sq ft - 16 pi sq ft

= 346.41 sq ft

The cost of carpeting this area is:

Cost of carpeting = Area of living room x Cost per square foot

= 346.41 sq ft x $9.55/sq ft

≈ $3,305.47

Therefore, the cost of the carpeting Gabriela will use in her living room is $3,305.47.

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Answer:

The given expression can be simplified using the rule for multiplying powers with the same base by adding their exponents.

(9^-3)(9^12) can be written as (1/9^3)(9^12)

Now, using the rule of adding exponents, (1/9^3)(9^12) can be simplified as 9^(12-3) = 9^9.

Therefore, the expression (9^-3)(9^12) simplified as 9^9.

Step-by-step explanation:

The formula for standard error that applies to this problem is SE = sqrt[p(1-p)/n], where SE represents the standard error, p represents the probability of success, and n represents the sample size. In this case, N represents the population size, but it is not necessary for calculating the standard error.

Substituting the values given in the problem, we have:

SE = sqrt[0.45(1-0.45)/44] = 0.0717 (rounded to four decimal places)

Therefore, the standard error for this problem is 0.0717. This value represents the degree of variability or uncertainty in the sample proportion, or the degree to which the sample proportion is likely to deviate from the true population proportion. A larger sample size or a more extreme probability of success (closer to 0 or 1) would result in a smaller standard error, indicating greater precision in the estimate of the population proportion.

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The probability of getting heads or tails when flipping a coin is 0.5 or 50%. The final probability will give you the chance of getting between 4900 and 5100 heads when flipping a fair coin 10,000 times.

However, the probability of getting a certain number of heads when flipping a coin, a certain number of times can be calculated using probability theory. In this case, the probability of getting between 4900 and 5100 heads when flipping a fair coin 10000 times can be calculated using the binomial distribution formula.

The binomial distribution formula is P(X=k) = (n choose k) * p^k * (1-p)^(n-k), where P(X=k) is the probability of getting k heads, n is the number of coin flips, p is the probability of getting a head (0.5 in this case), and (n choose k) is the binomial coefficient.

Using this formula, the probability of getting between 4900 and 5100 heads when flipping a fair coin 10000 times is approximately 0.023 or 2.3%. This means that out of 1000 trials, you can expect to get between 4900 and 5100 heads around 23 times.

In summary, the probability of getting between 4900 and 5100 heads when flipping a fair coin 10000 times is around 2.3%. This probability can be calculated using the binomial distribution formula, which takes into account the number of coin flips and the probability of getting a head.

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The total number of fourth-grade students in the school is 144

From the question, we have the following parameters that can be used in our computation:

This means that

Fourth-grade students = 6 * Ms. Smyth's fourth-grade class.

Substitute the known values in the above equation, so, we have the following representation

Fourth-grade students = 6 * 24

Evaluate

Fourth-grade students = 144

Hence, the total number of students is 144

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The value of x is 88.22 units.

Given is right triangle, we need to find the value of x,

tan 60° = 85 / a

a = 85 / √3

a = 49

Now,

tan 30° = 85 / a + x

a+x = 85 ÷ 1/√3

a+x = 147.22

x = 147.2-49

x = 88.22

Hence, the value of x is 88.22 units.

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To decrease the width of a confidence interval while maintaining the same level of confidence, you can increase your sample size i.e., Option C is the correct answer.

A confidence interval is a range of values that is likely to contain the true population parameter with a certain level of confidence. The width of the confidence interval is influenced by several factors, including the sample size, level of confidence, and population standard deviation.

When the sample size is increased, the standard error of the sample mean decreases, which in turn decreases the width of the confidence interval. This means that a larger sample size provides more precise estimates of the population parameter and reduces the variability of the sample means. As a result, a larger sample size allows for a narrower confidence interval while maintaining the same level of confidence.

On the other hand, decreasing the sample size can widen the confidence interval, as there are less data available to estimate the population parameter. Increasing the significance level or decreasing the population standard deviation may also widen the confidence interval, as this increases the range of plausible values for the population parameter.

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Option E is incorrect because we have enough evidence to reject the null hypothesis.

The correct answer is (C) There is evidence that μ ≠ 70.

The null hypothesis is that the true mean tire pressure for the population of tires on city buses in this city is equal to 70psi, i.e., H0: μ = 70. The alternative hypothesis is that it is not equal to 70psi, i.e., Ha: μ ≠ 70.

The p-value of 0.043 is less than the significance level of 0.05. Therefore, we reject the null hypothesis and conclude that there is evidence that the true mean tire pressure is not equal to 70psi.

Option A is incorrect because we cannot conclude that the true mean tire pressure is less than 70psi. Option B is incorrect because we cannot conclude that the true mean tire pressure is equal to 70psi. Option D is incorrect because the p-value is not the probability that μ = 70. Option E is incorrect because we have enough evidence to reject the null hypothesis.

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The volume of the whole lime represented by a sphere is given by  12 cubic centimeters.

Volume of the lime in hemispherical shape is equal to

= 6 cubic centimeters

Let us consider 'r' be the radius of the sphere.

Volume of hemisphere = ( 2/3 ) πr³

Volume of a sphere = ( 4 /3) πr³

Relation between volume of hemisphere and volume of a whole lime sphere

Volume of a whole lime sphere = 2 times of volume of hemisphere

⇒Volume of a whole lime sphere =  2 × 6

⇒Volume of a whole lime sphere =  12 cubic centimeters.

Therefore, the volume of the whole lime is equal to  12 cubic centimeters.

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To test your hypothesis that the average time to deliver pizza, once the order is placed, is greater than 25 minutes in the population, you would perform a one-sample T-test.

1. Formulate the null hypothesis (H0) and the alternative hypothesis (H1).

In this case, H0: the average delivery time is equal to 25 minutes, and H1: the average delivery time is greater than 25 minutes.

2. Collect a sample of delivery times and calculate the sample mean and sample standard deviation.

3. Determine the appropriate T-distribution based on your sample size (degrees of freedom = sample size - 1).

4. Calculate the T-statistic using the sample mean, sample standard deviation, and sample size.

5. Determine the critical T-value based on your chosen level of significance (e.g., 0.05) and the one-tailed T-distribution.

6. Compare the calculated T-statistic to the critical T-value. If the T-statistic is greater than the critical T-value, you can reject the null hypothesis in favor of the alternative hypothesis.

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The argument is not valid.

The argument assumes that a flood occurs only if the global temperature has increased, but this is not necessarily true. A flood can occur due to other reasons such as heavy rain, land use changes, or other natural disasters. Therefore, the conclusion "a flood occurs" does not necessarily follow from the premises given.

This means that if either the ice melts or a flood occurs, then the global temperature must have increased. However, the converse of this statement is not necessarily true. That is, the global temperature can increase for reasons other than the ice melting or a flood occurring. Therefore, it is not valid to conclude that a flood must occur just because the global temperature has not increased.

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Answer:

.40x < 15

x < 37.5 copies

The maximum number of copies is 37.

37 × $.40 = $14.80

Answer:

14

Step-by-step explanation:

eere4

If a given certificate C is a legitimate answer to the instance, we can check in polynomial time if it is (B, S1, S2, ..., Sm, K) of SETCOVER. Hence, the language SETCOVER is in NP.

To prove that the language SETCOVER is in NP, we need to show that given an instance (B, S1, S2, ..., Sm, K) of SETCOVER and a certificate C, we can verify in polynomial time whether C is a valid solution to the instance.

The certificate C in this case is a subset IC of {1, 2, ..., m} of size K, which represents the indices of the sets that form a cover for B. To verify whether C is a valid solution, we need to check two things:

Verify that IC has size K: We can simply count the number of elements in IC and check if it equals K. This can be done in O(m) time, which is polynomial in the size of the input.

Verify that the sets S_i for i in IC form a cover for B: We can iterate through the elements in B and check whether each element is present in at least one of the sets S_i, where i is in IC. Since B has at most |B| elements, and each set S_i has at most |B| elements, this can be done in O(K |B|) time, which is polynomial in the size of the input.

Therefore, If a given certificate C is a legitimate answer to the instance, we can check in polynomial time if it is (B, S1, S2, ..., Sm, K) of SETCOVER. Hence, the language SETCOVER is in NP.

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In a perfectly competitive market, when the price is greater than the minimum average total cost for all firms, it indicates that the firms are experiencing economic profits. Here's a step-by-step explanation:

1. In a perfectly competitive market, there are numerous firms producing a homogeneous product with free entry and exit for new firms.

2. The demand curve for each firm is perfectly elastic (horizontal), which means that firms can sell any quantity of the product at the market price without affecting the price.

3. The average total cost (ATC) includes both the average variable cost (AVC) and the average fixed cost (AFC). The ATC curve is U-shaped, with the minimum ATC representing the lowest average cost per unit.

4. When the market price is greater than the minimum ATC, it means that the firms are covering all their variable and fixed costs, with additional revenue left over as profit.

5. In this scenario, the firms are earning economic profits, which can be calculated by subtracting the ATC from the market price and multiplying the result by the quantity produced.

6. Economic profits attract new firms to enter the market, leading to increased competition. This, in turn, drives down the market price, reducing economic profits.

7. Over time, as more firms enter and exit the market, the price will eventually reach a point where it is equal to the minimum ATC. This is the long-run equilibrium, where firms earn zero economic profits, covering all their costs but not earning any additional profit.

To summarize, in a perfectly competitive market, when the price is greater than the minimum average total cost for all firms, it indicates that the firms are experiencing economic profits. These profits will attract new firms to enter the market, which will eventually lead to the price falling to the minimum ATC in the long-run equilibrium.

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Isaac pay for the cost of racket is 39.9.

The cost of a pair of shoes is three times the cost of a bag, so we can write:

Cost of shoes = 3b

Similarly, the cost of the racket is twice the cost of the bag, so we can write:

Cost of racket = 2b

Now we can use the given information to set up an equation:

Cost of racket + Cost of bag + Cost of shoes = $119.70

Substituting the expressions we found above, we get:

2b + b + 3b = $119.70

Simplifying the equation:

6b = $119.70

Dividing both sides by 6:

b = $19.95

So the cost of the bag is $19.95.

We can use this to find the costs of the shoes and racket:

Cost of shoes = 3b = 3($19.95) = $59.85

Cost of racket = 2b = 2($19.95) = $39.90

Therefore, Isaac paid $39.90 for the racket.

A cost is an expenditure required to produce or sell a product or get an asset ready for normal use. In other words, it's the amount paid to manufacture a product, purchase inventory, sell merchandise, or get equipment ready to use in a business process.

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y=x(a+b)

Divide both sides by (a+b)

x=y/(a+b)

The points lie on a vertical line passing through the point x = -1.

We have,

The two points (-1, 6) and (-1, -6) have the same x-coordinate but different y-coordinates.

This means that they lie on a vertical line passing through the point x = -1.

When graphed on a coordinate plane, the line would appear as a vertical line at x = -1, with one point above the x-axis and the other point below the x-axis.

Thus,

The points lie on a vertical line passing through the point x = -1.

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It has been 16 hours since the time of death.

Stage 1: First 12 hours

12 hours * 1.4F per hour = 16.8F

Stage 2: After 12 hours

Since the body has already cooled 16.8F in the first 12 hours, we only need to find the remaining temperature drop:

19.6F - 16.8F = 2.8F

Now, we can calculate the time taken for the remaining temperature drop using the second cooling rate:

2.8F / 0.7F per hour = 4 hours

Adding the time taken for both stages, we get:

12 hours (Stage 1) + 4 hours (Stage 2) = 16 hours

So, it has been 16 hours since the time of death.

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Jane's total combined monthly cost for auto loan payment and insurance is $736.67 (rounded to the nearest dollar).

If Jane's monthly car insurance payment of $170 is same to 30% of her monthly car loan fee, then we are able to set up the subsequent equation:

0.3x = 170

Where x is the monthly auto loan fee. To solve for x, we will divide each facets by using 0.3:

x = 170 / 0.3 = $566.67

So, Jane's monthly auto loan charge is $566.67.

To discover her general combined monthly price for auto loan price and insurance, we simply upload her monthly car coverage charge to her monthly auto loan payment:

$566.67 + $170 = $736.67

Consequently, Jane's total combined monthly cost for auto loan payment and coverage is $736.67

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The radius of the circle is 24 units.

We know that for an arc defined by an angle θ on a circle of radius R, the area is:

A = (θ/360°)*π*R²

Here we can see that the area of the sector is 432π and the angle is 270°, then we can replace that in the formula above so we get:

432π = (270°/360°)*π*R²

432 = (3/4)*R²

(4/3)*432 = R²

√576 = R

24 = R

The radius is 24 units.

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The slope of the line is 3 which represents that the time is rising as the distance increases.

The slope of the line that passes through the pair of points (3, 6) and (1, 3) is calculated as follows:

slope = (6 - 3) / (3 - 2)

slope = (3) / (1)

slope = 3

In this case, the slope of 3 means that the time is rising as the distance increases.

Therefore, the slope of the line that passes through the pair of points  (3, 6) and (1, 3) is 3.

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The 95% confidence interval for the population mean is (54.70, 70.96).

To find the 95% confidence interval for the population mean, we need to first find the sample mean and sample standard deviation.

Sample Mean (x) = (60.3 + 65.2 + 60 + 70.6 + 62 + 55.9) / 6 = 62.33

Sample Standard Deviation (s) = sqrt([(60.3 - 62.33)^2 + (65.2 - 62.33)^2 + (60 - 62.33)^2 + (70.6 - 62.33)^2 + (62 - 62.33)^2 + (55.9 - 62.33)^2] / (6 - 1)) = 5.28

Next, we can use the formula for the confidence interval:

CI = x ± t*(s/sqrt(n))

where x is the sample mean, s is the sample standard deviation, n is the sample size, and t is the t-value for the 95% confidence level with (n-1) degrees of freedom.

From a t-distribution table with 5 degrees of freedom (n-1 = 6-1 = 5) and a 95% confidence level, we find that the t-value is 2.571.

Plugging in the values, we get:

CI = 62.33 ± 2.571*(5.28/sqrt(6))

CI = (54.70, 70.96)

Therefore, the 95% confidence interval for the population mean is (54.70, 70.96).

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Answer:

11

Step-by-step explanation:

If quilt has 8 rows of "quilt-blocks" and has an area of 7,744 square inches, then the "side-length" of one quilt blocks is 11 inches.

Let us assume that each quilt block has a side length of "x" inches.

Since the quilt has 8 rows of quilt blocks, the total number of quilt blocks in the quilt is 8 times the number of quilt blocks in each row, which is 8x.

The shape of a quilt is a square and has a "total-area' of 7744 square inches, we equate the area equation;

We get,

⇒ (x × 8)² = 7744,

Simplifying the equation,

We get,

⇒ 64x² = 7744,

Dividing both sides by 64,

We get,

⇒ x² = 121,

⇒ x = 11,

Therefore, each quilt-block has a side length of 11 inches.

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By the Comparison Test, we can conclude that Σ (4sin(n))/([tex]n^2[/tex] - 1) converges.

We can use the Limit Comparison Test to determine the convergence or divergence of the series:

∑(4 sin(n))/n

To do this, we need to find a series whose convergence is known and which can be compared to the given series. Let's choose the series ∑(1/n) since we know that it diverges.

We take the limit of the ratio of the nth term of each series as n approaches infinity:

lim n→∞ (4 sin(n)/n)/(1/n) = lim n→∞ 4 sin(n) = DNE

Since the limit does not exist, we cannot apply the Limit Comparison Test. Therefore, Therefore, by the Comparison Test, we can conclude that Σ (4sin(n))/([tex]n^2[/tex] - 1) converges.

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The solution is, the solutions using the Zero Product Property: is x =0 and 3/4.

The expression to be solved is:

4x² - 3x = 0

we know that,

The zero product property states that the solution to this equation is the values of each term equals to 0.

now, we have,

4x² - 3x = 0

or, x ( 4x - 3 ) = 0

i.e. we get,

x × ( 4x - 3 ) = 0

so, using the Zero Product Property:

we get,

x = 0

or,

( 4x - 3 ) = 0

so, we have,

x = 0 or, x = 3/4

The answers are 0 and 3/4.

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Answer:

A, C, E

Step-by-step explanation:

remember to law of sine :

a/sin(A) = b/sin(B) = c/sin(C)

or the "upside-down" version :

sin(A)/a = sin(B)/b = sin(C)/c

with a, b, c being the sides of the triangle, and A, B, C being the corresponding opposite angles in the triangle.

so, as you can clearly see, we always need at least one angle and one side (in fact either 2 angles one side or 1 angle 2 sides) to use the law of sine to solve the rest of the triangle.

therefore, the answer options A, C, E are correct.

for SSS (all 3 sides are known) we need the law of cosine to solve the angles (at least one of them, and then we could continue with either law).

remember :

c² = a² + b² - 2ab×cos(C)

again, a,b,c are the sides, and C is the opposite angle of whatever side we define as "c".

that's why I always call this the extended Pythagoras.

for AAA (all 3 angles are known) we cannot solve the triangle, because dilated triangles all have the same angles. and therefore there are infinitely many triangles with the same angles.