Question # 7
Multiple Choice
10 students were randomly sampled and asked their shoe size. Which line plot displays the data for this sample?

9, 7, 8, 10, 9, 10, 11, 8, 8, 9

The statements that are true about the triangle are

a) m∠5 + m∠6 = 180°

b) ∠ 2+ ∠ 3 = ∠ 6

c) m∠2 + m∠3 + m∠5 = 180°

Given data ,

Let the triangle be represented as ΔABC

Now , An exterior angle of a triangle is equal to the sum of the opposite interior angles.

For Exterior ∠ 1 we have

∠ 1 = ∠ 5 + ∠ 3 ( Exterior angle Property of Triangle )

Similarly,

For Exterior ∠ 4 we have

∠ 4 = ∠ 5 + ∠ 2 ( Exterior angle Property of Triangle )

Similarly,

For Exterior ∠ 6 we have

∠ 6 = ∠ 2 + ∠ 3 ( Exterior angle Property of Triangle )

From the triangle sum property , we get

Ina triangle sum of the measures of angles is equal to 180°

m∠2 + m∠3 + m∠5 = 180°

Hence , the triangle is solved

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The complete question is attached below :

A triangle is shown with its exterior angles. The interior angles of the triangle are angles 2, 3, 5. The exterior angle at angle 2 is angle 1. The exterior angle at angle 3 is angle 4. The exterior angle at angle 5 is angle 6. Which statements are always true regarding the diagram? Select three options.
m∠5 + m∠3 = m∠4
m∠3 + m∠4 + m∠5 = 180°
m∠5 + m∠6 =180°
m∠2 + m∠3 = m∠6
m∠2 + m∠3 + m∠5 = 180°

Answer:

In the given diagram, QP is tangent to a circle with centre O, RS is a straight line, T is a point on the circle, and PS bisects TPQ. We know that SPQ = 22°. Let's try to find out the value of the angle TPQ.

Since QP is tangent to the circle, the angle between RS and QP (i.e., angle RQP) is equal to the angle between QP and the radius drawn to the point of tangency (i.e., angle QOT). So, we can say that:

angle RQP = angle QOT

Also, since PS bisects TPQ, we can say that:

angle TPS = angle TPQ / 2

Now, let's consider the triangle TPQ. We know that:

angle TPQ + angle TQP + angle PTQ = 180° [Sum of angles in a triangle]

Substituting the values we have:

angle TPQ + angle TQP + (angle TPS + angle SPQ) = 180°

angle TPQ + angle TQP + (angle TPQ/2 + 22°) = 180°

Multiplying both sides by 2 to eliminate the fraction:

2(angle TPQ) + 2(angle TQP) + angle TPQ + 44° = 360°

Simplifying:

3(angle TPQ) + 2(angle TQP) = 316°

We don't know the values of angle TPQ and angle TQP, so we can't solve this equation exactly. However, we do know that these angles are both less than 180° (since they are angles in a triangle). Therefore, we can try some values for angle TPQ (let's call it x) and see if we can find a corresponding value for angle TQP that satisfies the equation.

If we take x = 40°, then we get:

3(40°) + 2(angle TQP) = 316°

120° + 2(angle TQP) = 316°

2(angle TQP) = 196°

angle TQP = 98°

Now, we can use the fact that angle TPS = angle TPQ / 2 to find angle TPS:

angle TPS = x/2 = 20°

Finally, we can use the fact that PS bisects TPQ to find angle PQT:

angle PQT = angle TPS

Step-by-step explanation:

Answer:

x = 5√3, and y = 10.

In a 30°-60°-90° triangle, the length of the longer leg is √3 times the length of the shorter leg, and the length of the hypotenuse is twice the length of the shorter leg.

Answer:

a first

Step-by-step explanation:

because I just don't

Estimating the square root of the number -√7 gives -2 and -2.6

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

-√7

To estimate the number is to approximate the number

When the square root of 7 is evaluated, we have

-√7 = -2.64575131106

Approximate to the nearest integer

-√7 = -2

Approximate to the nearest tenth.

-√7 = -2.6

Hence, the estimates are -2 and -2.6

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Answer:

The correct answer is option B: 2x^2 + 11x + 12 cubic inches

Step-by-step explanation:

The volume of a rectangular prism is given by the formula V = lwh, where l is the length, w is the width (or base), and h is the height of the prism.

Given that the area of the base is (x + 4) square inches and the height is (2x + 3) inches, we can substitute these values into the formula to find the volume:

V = (x + 4)(2x + 3)

Now, we can multiply the binomials using the distributive property:

V = 2x^2 + 3x + 8x + 12

V = 2x^2 + 11x + 12

So, the correct answer is option B: 2x^2 + 11x + 12 cubic inches.

The PM of the circle is 6 in.

Since the diameter of the circle divide the circle into two equal parts. We can say:

PM = MQ

Applying the Intersecting Chord Theorem (When two chords intersect each other inside a circle, the products of their segments are equal). That is:

SM * RM = PM * MQ

SM * RM = PM²

SM = 13 - 4

SM = 9 in

RM = 4 in

Substituting:

SM * RM = PM²

9 * 4 = PM²

PM² = 36

PM = √36

PM = 6 in

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The word "sleepless" has 8 letters. To find the number of distinguishable permutations, we can use the formula for permutations of a set with no repeated elements, which is n!, where n is the number of elements.

Therefore, the number of permutations for the word "sleepless" can be calculated as 8!, which is equal to 40,320. This means that there are 40,320 different ways we can arrange the letters in the word "sleepless" while keeping all the letters distinct.

Note that if the word had repeated letters, we would have to divide the result by the factorials of the number of times each letter was repeated.

Answer:

-3x^2 + 12x + 5x - 20 = -3x^2 + 17x - 20

Step-by-step explanation:

(-3x - 5)(-x + 4) is a binomial expression, where (-3x - 5) is one expression and (-x + 4) is the other.

As the text eludes to, we can multiply binomial expressions using the FOIL method, where you multiply the first terms (3x and -x), outer terms (3x and 4), the inner terms (-5 and -x), and the last terms (-5 and 4)

This is how you get

(3x)(-x) + (3x)(4) + (-5)(-x) + (-5)(4)

Now, multiply the terms and combine like terms:

[tex](3x)(-x)+(3x)(4)+(-5)(-x)+(-5)(4)\\-3x^2+12x+5x-20\\-3x^2+17x-20[/tex]

(1) The two triangles are similar because they have equal angles.

(2)  Triangle QRS is similar to triangle QLM because they have equal angles.

(3) Both triangles are similar and the value of x is 21.

Two triangles are said to be similar if they have equal sides, equal angles or both.

The missing angles of the triangles for the question is calculated as;

Bigger triangle; missing angle = 180 - (44 + 46) = 90

Smaller triangle; missing angle = 90 - 46 = 44⁰

Both triangles are similar.

For the second question; triangle QRS is similar to triangle QLM  because angle R is equal to angle L, and also they have common angle Q, which implies that angle S must be equal to angle L.

For third question, the triangles are similar because their corresponding angles are equal.

The value of x is calculated as;

48 + 4x + (180 - (56 + 76)) = 180 (sum of angles on a straight line)

48 + 4x + 48 = 180

4x = 84

x = 84/4

x = 21

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A line starts at 0 with a constant positive slope.

The graph that shows the distance Sam travels over time is given below.

Here the graph starts at the point (0, 0).

So the line starts at 0.

Now the line is moving in such a way that as time increases, the distance travelled also increases.

So the slope is positive.

Hence the complete description is that line starts at 0 with a constant positive slope.

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The upper limit of the control chart such that it will include roughly 97 percent of the sample means when the process is in control is 1.008.

For a process with a normal distribution, the mean of the sample means is equal to the population mean and the standard deviation of the sample means is equal to the population standard deviation divided by the square root of the sample size. In this case, the mean of the output is 1.0 liter and the standard deviation is 0.01 liter, so the standard deviation of the sample means is 0.01 / √25 = 0.002.

To construct a control chart for the sample means, we need to determine the upper and lower control limits such that the process is in control when the sample means fall within these limits. Assuming the process is in control, we want to find the upper limit such that roughly 97% of the sample means fall below this limit.

Using the standard normal distribution, the Z-score corresponding to the 97th percentile is approximately 1.88.

Therefore, the upper control limit is 1.0 + 1.88(0.002) = 1.008. Any sample mean above this limit should be investigated for potential process issues.

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Lets take the variable x for the son.

Son: x

Dad: 3x

Mom: 3x-4

THREE years ago:

Son: x-3

Dad: 3x-3

Mom: 3x-4 -3

so, 3x-7

SUM=54

(x-3)+(3x-3)+(3x-7)=54

x-3+3x-3+3x-7=54

7x-13=54

7x=54+13

7x=67

so , x=67/7

x= 9.5

now lets see for the dad:

3x= 3*9.5

=28.5

Finally for the mom:

3x-4= 3*9.5 -4

= 28.5-4

= 24.5

The man's age is 32, his wife's age is 28.

Let's use algebra to solve this problem.

Let's represent the man's age as "M", his wife's age as "W", and their child's age as "C".

From the first sentence of the problem, we know that:

M = W + 4

From the second sentence, we know that:

M = 3C

Finally, from the third sentence, we know that the sum of their ages three years ago was 54:

(M-3) + (W-3) + (C-3) = 54

Substituting M = W + 4 and M = 3C into the third equation, we get:

(W+4-3) + (W-3-3) + (1/3M - 3) = 54

Simplifying this equation, we get:

2W + (1/3)(W+4) - 12 = 54

Multiplying both sides by 3 to eliminate the fraction, we get:

6W + W + 4 - 36 = 162

Combining like terms, we get:

7W - 32 = 162

Adding 32 to both sides, we get:

7W = 194

Dividing both sides by 7, we get:

W = 28

Substituting W = 28 into M = W + 4, we get:

M = 32

Finally, substituting M = 3C into the equation, we get:

32 = 3C

C = 32/3

Therefore, the man's age is 32, his wife's age is 28.

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The given  statement pattern is a tautology.

To determine whether the following statement pattern is a tautology, a contradiction, or a contingency: (p→q)∨(q→p), follow these steps:

1. Write out the truth table for p and q:
  p | q
  -----
  T | T
  T | F
  F | T
  F | F

2. Calculate the truth values for (p→q) and (q→p) using the implication rule (p→q is false only when p is true and q is false):
  (p→q) | (q→p)
  -----------
   T     |   T
   F     |   T
   T     |   F
   T     |   T

3. Finally, calculate the truth values for the given statement pattern (p→q)∨(q→p) using the disjunction rule (a disjunction is true if at least one of the statements is true):
  (p→q)∨(q→p)
  ---------
     T
     T
     T
     T

Since the statement pattern (p→q)∨(q→p) is true for all possible truth values of p and q, it is a tautology.

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Based on the information provided, it appears that there are 4 independent variables included in the regression model.

This is indicated by the "regression" row in the ANOVA table, which shows that there are 4 degrees of freedom (df) for the regression. This is determined by the degrees of freedom (df) for the regression, which is 4. The df for regression represents the number of independent variables in the model.

A statistical method called regression links a dependent variable to one or more independent (explanatory) variables. A regression model can demonstrate whether changes in one or more of the explanatory variables are related to changes in the dependent variable.

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Answer:

The answer to your problem is, A. 10

Step-by-step explanation:

So first add up all the total number of ‘ X ‘

Which is:

4 + 3 + 2 + 1 = 10

Technically 10 is our answer.

Thus the answer to your problem is, A. 10

Not a 100% sure

Answer:

5, 8, 8, 8, 8, 9, 9, 9, 10, 10

The median of this data set is 8.5, so the correct answer is B.

Step-by-step explanation:

Since there are 10 observations, we are looking for the number halfway between the two middle observations (observations #5 and #6) when the data are arranged in order. Here, observation #5 is 8, and observation #6 is 9, so the median of this data set is (8 + 9)/2 = 8.5. B is the correct answer.

The number or expression underneath the top line of the symbol is called the radicand. The cube root symbol is a grouping symbol, meaning that all operations in the radicand are grouped as if they were in parentheses.

To write the radicand as the product of a perfect cube (first) and a factor that does not contain a perfect cube (second), follow these steps:

1. Identify the radicand in the given expression. The radicand is the number or expression inside the cube root symbol.
2. Determine the prime factors of the radicand by breaking it down into its smallest prime factors.
3. Group the prime factors into sets of three identical factors. These sets will form the perfect cube factors.
4. Multiply the sets of three factors together to form the perfect cube part of the product.
5. Multiply any remaining factors together to form the factor that does not contain a perfect cube.
6. Write the radicand as the product of the perfect cube (step 4 result) and the factor that does not contain a perfect cube (step 5 result).

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The value of the test statistics is 113.42.

To test the claim that the standard deviation of pulse rates of adult males is less than 10 bpm, we will use a chi-square test. Given a random sample of 135 adult males with a standard deviation of 9.2 bpm, let's find the value of the test statistic.

Step 1: State the null and alternative hypotheses.
H0: σ = 10 bpm (null hypothesis)
H1: σ < 10 bpm (alternative hypothesis)

Step 2: Determine the appropriate test statistic.
In this case, we will use the chi-square test statistic: χ² = (n-1)(s²/σ²), where n is the sample size, s is the sample standard deviation, and σ is the population standard deviation.

Step 3: Calculate the test statistic.
χ² = (135-1)(9.2²/10²) = (134)(84.64/100) = 134 * 0.8464 ≈ 113.42

The value of the test statistic is approximately 113.42. This value can be compared to the critical value from the chi-square distribution table with (n-1) degrees of freedom to determine whether to reject or fail to reject the null hypothesis.

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Without additional evidence that shows Stew-topia's pricing strategy was intended to harm competition, it cannot be conclusively determined whether Stew-topia is guilty of predatory pricing.

To prove that Stew-topia engaged in predatory pricing, Larry would need to provide evidence that Stew-topia priced stew below average variable cost with the specific intention of driving 2 Live Stew out of business, or that the owner of Stew-topia threatened the owner of 2 Live Stew to keep him from matching the price cut. Simply observing that both 2 Live Stew and Stew-topia lowered their prices to $2.00 for a bowl of stew and that Stew-topia always lowered its price first is not enough to conclusively determine whether Stew-topia is engaging in predatory pricing.

Predatory pricing is a strategy used by a dominant firm in a market to deliberately set prices below the average variable cost (AVC) of production in order to drive competitors out of business and reduce competition in the long run. To prove that Stew-topia engaged in predatory pricing, Larry would need to show evidence that Stew-topia priced its stew below its average variable cost with the specific intent of driving 2 Live Stew out of business. This would require detailed financial data and documentation from Stew-topia, such as production costs, pricing decisions, and internal communications that demonstrate their predatory intent.

Alternatively, Larry would need to provide evidence that Stew-topia used anti-competitive tactics, such as threatening the owner of 2 Live Stew to prevent them from matching the price cut or engaging in anti-competitive agreements or collusion. This would require concrete proof, such as emails, recordings, or witnesses, that demonstrate Stew-topia's anti-competitive behavior.

Simply observing that both 2 Live Stew and Stew-topia lowered their prices and that Stew-topia always lowered its price first is not enough to prove predatory pricing. There could be other reasons for the price cuts, such as seasonal promotions, cost reductions, or competitive pressures.

Therefore, without additional evidence that shows Stew-topia's pricing strategy was intended to harm competition, it cannot be conclusively determined whether Stew-topia is guilty of predatory pricing.

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The probability that a randomly chosen voter from the survey  is under 50 years old is 0.75

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

The table of values

Where we have

Voters under 50 years old = 847 + 804 + 773

Total = 3228

So, the required probability is

P = (847 + 804 + 773)/3228

Evaluate

P = 0.75

Hence, the probability is 0.75

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Answer:

74% is about 75%, or 3/4.

57 is about 60.

So 74% of 57 is about 3/4 of 60, or 45.

74% of 57 is .74 × 57 = 42.18, so the estimate seems reasonable

Answer:

42.18

Step-by-step explanation:

[tex]\frac{57*74}{100} = 42.18[/tex]

The student solved 12 problems correctly.

To find out how many problems the student solved correctly, we can use the given information and set up an equation using the terms "correct problems" and "incorrect problems."

Let x represent the number of correct problems and y represent the number of incorrect problems. We know the following:

1. The total number of problems completed is 33, so x + y = 33.
2. The teacher pays 70 cents for correct problems and takes 40 cents for incorrect problems, and neither owes anything to each other. So, 70x - 40y = 0.

Now, we'll solve this system of equations step-by-step:

Step 1: Solve the first equation for x: x = 33 - y.
Step 2: Substitute the expression for x in the second equation: 70(33 - y) - 40y = 0.
Step 3: Simplify and solve for y: 2310 - 70y - 40y = 0 => 2310 - 110y = 0 => y = 21.
Step 4: Substitute the value of y back into the equation for x: x = 33 - 21 => x = 12.

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To solve for I and Q in this electric circuit, we can use the equations for the charge and current in a series RL circuit with a capacitor:

Q = CV(1 - e^(-t/RC))
I = (E/R)e^(-t/tau) + (Q/R) where tau = L/R

Plugging in the given values, we have:

Q = (100 micro farads)(100 volts)(1 - e^(-t/(20 ohms)(0.05 henry)))
I = (100 volts/20 ohms)e^(-t/(0.05 henry/20 ohms)) + Q/20 ohms

Using the initial conditions Q = 0 and I = 0 at t = 0, we can simplify the equations to:

Q = 100 micro farads * 100 volts * (1 - e^(-t/1 millisecond))
I = (100 volts/20 ohms)e^(-t/1 millisecond)

So at t = 1 millisecond, we have:

Q = 100 micro farads * 100 volts * (1 - e^(-1))
 ≈ 42.36 microcoulombs
I = (100 volts/20 ohms)e^(-1)
 ≈ 1.831 amperes

Therefore, at t = 1 millisecond, the charge on the capacitor is about 42.36 microcoulombs and the current in the circuit is about 1.831 amperes.

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The series after solving the following term f(x) = [tex]e^{-2x}[/tex] is given as:

[tex]e^{-2x}[/tex] = 1 - 2x + 2x² - 4/3x³ + 2/3x⁴ - 4/15x⁵+....|x| < ∞.

The fundamental concepts in mathematics are series and sequence. A series is the total of all components, but a sequence is an ordered group of items in which repeats of any kind are permitted. One of the typical examples of a series or a sequence is a mathematical progression.

By using the formulae to solve issues, one may gain a deeper understanding of the principles. The main distinction between them and sets is that in a sequence, certain terms might appear again in different locations. A series can be finite or infinite in length and has length equal to the number of terms.

Given f(x) = [tex]e^{-2x}[/tex], center x = 0

f(x) = [tex]e^{-2x}[/tex] = [tex]1 - 2x + \frac{(2x)^2}{2!} +\frac{(2x)^3}{3!} +\frac{(2x)^4}{4!} +\frac{(2x)^5}{5!} +.....|x|[/tex]

[tex]e^{-2x}[/tex] = 1 - 2x + 4([tex]\frac{x^2}{2}[/tex]) - 8([tex]\frac{x^3}{6}[/tex]) + 16([tex]\frac{x^4}{24}[/tex]) - 35([tex]\frac{x^5}{100}[/tex])+.....|x| < ∞

[tex]e^{-2x}[/tex] = 1 - 2x + 2x² - 4/3x³ + 2/3x⁴ - 4/15x⁵+....|x| < ∞

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a) The equilibrium point is x = 2 or x = 8

b) The consumer surplus equilibrium point is $6,062.67

c) The producer surplus equilibrium point is $13,208.67

a) To find the equilibrium point, we need to set the demand function equal to the supply function and solve for x:

D(x) = S(x)

[tex](x - 5)^2 = x^2 + x + 3[/tex]

Expanding the left side and simplifying, we get:

[tex]x^2 - 10x + 22 = 0[/tex]

Using the quadratic formula, we get:

[tex]x = (10[/tex] ± [tex]\sqrt{36})/ 2[/tex]

[tex]x = 5[/tex] ± [tex]3[/tex]

[tex]x = 2[/tex] or [tex]x = 8.[/tex]

b) To find the consumer surplus at the equilibrium point, we need to calculate the area under the demand curve and above the equilibrium price, which is given by the supply curve. Since we have two possible equilibrium points, we need to check both of them to see which one gives us a positive consumer surplus.

For [tex]x = 2[/tex], the equilibrium price is given by [tex]S(2) = 11[/tex], which is above the demand curve. Therefore, there is no consumer surplus at this equilibrium point.

For [tex]x = 8[/tex], the equilibrium price is given by [tex]S(8) = 75[/tex], which is below the demand curve. Therefore, the consumer surplus is given by the area under the demand curve and above the price of 75:

[tex]∫[75, 8] (x - 5)^2 dx = [(x - 5)^3 / 3][/tex] from 8 to 75

≈[tex]6,062.67[/tex]

This means that at the equilibrium point x = 8, consumers are willing to pay a total of approximately $6,062.67 more than what they actually pay.

c) To find the producer surplus at the equilibrium point, we need to calculate the area under the equilibrium price and above the su

For x =2 supply curve. Again, since we have two possible equilibrium points, we need to check both of them to see which one gives us a positive producer surplus.

2, the equilibrium price is given by [tex]S(2) = 11,[/tex] which is above the demand curve. Therefore, there is no producer surplus at this equilibrium point.

For x = 8, the equilibrium price is given by[tex]S(8) = 75[/tex], which is below the demand curve. Therefore, the producer surplus is given by the area above the supply curve and below the price of 75:

∫[tex][8, 75] (75 - x^2 - x - 3) dx = [(75x - x^3/ 3 - x^2 / 2 - 3x)][/tex] from 8 to 75

≈ [tex]13,208.67[/tex]

This means that at the equilibrium point x = 8, producers receive a total of approximately $13,208.67 more than their costs.

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The final value is 4 (mod5)

Hence, 17^342 ≡ 2^342 ≡ 4 (mod5).

To find 5^903 (mod60), we can use Euler's totient function. Since 60 = 2^2 × 3 × 5, we have φ(60) = 2^1 × 3^1 × 4 = 24. Therefore, we can use Euler's theorem to write:

5^24 ≡ 1 (mod60)

Raising both sides to the power of 37, we get:

5^(24*37) ≡ 1^37 ≡ 1 (mod60)

So 5^888 ≡ 1 (mod60).

Now, we can write:

5^903 = 5^888 * 5^15

Since 5^888 ≡ 1 (mod60), we just need to find 5^15 (mod60).

To do this, we can use the repeated squaring method. Writing 15 in binary form, we have:

15 = 1111 (in binary)

So we can compute:

5^1 ≡ 5 (mod60)

5^2 ≡ 25 (mod60)

5^4 ≡ 25^2 ≡ 25 (mod60)

5^8 ≡ 25^2 ≡ 25 (mod60)

Therefore:

5^15 ≡ 5^8 * 5^4 * 5^2 * 5^1 ≡ 25 * 25 * 25 * 5 ≡ 25 (mod60)

Hence, 5^903 ≡ 5^15 ≡ 25 (mod60).

To find 17^342 (mod5), we can use the fact that 17 ≡ 2 (mod5). Therefore:

17^342 ≡ 2^342 (mod5)

Using the repeated squaring method again, we can compute:

2^1 ≡ 2 (mod5)

2^2 ≡ 4 (mod5)

2^4 ≡ 1 (mod5)

Therefore:

2^342 ≡ 2^2 * (2^4)^85 ≡ 4 * 1^85 ≡ 4 (mod5)

Hence, 17^342 ≡ 2^342 ≡ 4 (mod5).

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The unknown angle of the parallelogram is as follows:

m∠T = 63 degrees

A parallelogram is a quadrilateral with opposite sides parallel to each other and opposite congruent to each other.

Therefore, the opposite angles of a parallelogram are equal. Consecutive angles are supplementary angles to each other.

Hence,

10w + 53 + 17w + 100 = 180

27w + 153 = 180

27w = 180 - 153

27w = 27

divide both sides by 27

w = 27 / 27

w = 1

Therefore,

m∠T = 10(1) + 53 = 63 degrees

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The area of the pizza box measuring 2/3 feet wide by 4/5 feet long is 8/15 square feet.

The shape of the pizza box is a rectangle. The rectangle is a quadrilateral with opposite sides parallel and equal with an equal angle and of 90°.

The area of a rectangle is considered as:

A = L * B

where L is the length

B is the breadth

Given in the question,

L = 4/5 feet

B = 2/3 feet

The area is calculated by multiplying the fractions. For the multiplication of fractions, we multiply the numerators and denominators separately. And final answer is calculated by simplifying the resulting fraction.

A = 4/5 * 2/3

= 4*2 / 5*3

= 8/15 square feet

Thus, the pizza box has an area of 8/15 square feet

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The break-even x-value for Year-round-Recreation is approximately 26.19.

To find the break-even points for Year-round-Recreation, we need to set the total costs equal to total revenues and solve for x. In this case, the total costs are given by C(x) = 2750 + 30x + 2, and the total revenues are given by R(x) = 135x.

The break-even point is when C(x) = R(x), so:

2750 + 30x + 2 = 135x

Now, we need to solve for x:

1. Subtract 30x from both sides:
2750 + 2 = 105x

2. Subtract 2 from both sides:
2750 = 105x

3. Divide both sides by 105:
x = 2750 / 105
x ≈ 26.19

The break-even x-value for Year-round-Recreation is approximately 26.19.

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Substituting n = -1 in the given expression, we get:

n² + n² + n² for n = -1

= (-1)² + (-1)² + (-1)²

= 1 + 1 + 1

= 3

Therefore, n² + n² + n² for n = -1 is equal to 3.