The following relation is a function:


{(-2, 4), (3, 0), (-4, 3), (-2, -1), (0, -4)}



true



false

Verónica jogged 7/4 or 1 and 3/4 more miles in the first week than in the second week.

Verónica jogged 10 3/16 miles in the first week and 8 7/16 miles in the second week. To find how many more miles she jogged in the first week, we need to subtract the distance she jogged in the second week from the distance she jogged in the first week:

10 3/16 miles - 8 7/16 miles

We need to first convert both mixed numbers to improper fractions:

10 3/16 = (10 x 16 + 3) / 16 = 163 / 16

8 7/16 = (8 x 16 + 7) / 16 = 135 / 16

Now we can subtract the two fractions:

163 / 16 - 135 / 16 = (163 - 135) / 16 = 28 / 16

We can simplify the fraction by dividing both the numerator and the denominator by their greatest common factor (GCF), which is 4:

28 / 16 = (4 x 7) / (4 x 4) = 7 / 4

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In ∆DEF, DG−→− bisects ∠EDF. ∆FDG is similar to ∆EDG; ∆FDG and ∆EDG may not be congruent, but they are similar by the ASA Similarity Postulate. Therefore, the correct option is B.

1. Since DG bisects ∠EDF, it means that ∠EDG = ∠FDG. This is the Angle Bisector Theorem.

2. In triangles FDG and EDG, we know that ∠FDG = ∠EDG (from step 1) and ∠DFG = ∠DEG (both are vertical angles and therefore congruent).

3. Now we have two pairs of congruent angles: ∠FDG = ∠EDG and ∠DFG = ∠DEG.

4. According to the (Angle-Side-Angle) or  ASA Similarity Postulate, if two angles in one triangle are congruent to two angles in another triangle, then the triangles are similar. Therefore, ∆FDG is similar to ∆EDG.

Hence, the correct answer is option B: Yes; ∆FDG and ∆EDG may not be congruent, but they are similar by the ASA Similarity Postulate.

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well, according the Cavalieri's Principle, the volume of the oblique cone will be the same volume as the non-oblique cone, so

[tex]\textit{volume of a cone}\\\\ V=\cfrac{\pi r^2 h}{3}~~ \begin{cases} r=radius\\ h=height\\[-0.5em] \hrulefill\\ r=9\\ h=16 \end{cases}\implies V=\cfrac{\pi (9)^2(16)}{3}\implies V=432\pi ~cm^3[/tex]

Ranking of the events below in the correct order from least likely to most likely are:

Event B

Event A

Event C

The probability of an event is defined as a number that describes the chance that the event will eventually happen. An event that is sure to happen has a probability of 1. An event that can never possibly happen has a probability of zero. Finally, If there is a chance that an event will happen, then it will have a probability that is between zero and 1.

i) Arrow lands on a section labeled with an odd number: The odd numbers here are 1 and 3.

There are a total of four 1's, and two 3's. This tells us that there are 6 odd numbers on the spinner.

There are 8 numbers in total on the spinner. Thus, 6 out of the 8 numbers are seen as odd numbers. Therefore, the probability that the arrow lands on an odd number would be:

P(odd number) = 6/8 =  75%

ii) Arrow lands on a section labeled with the number 1: There are four 1's on the spinner, and there are seen to be 8 numbers in total on the spinner. Thus, the probability of the arrow landing on a 1 is:

P(Number 1) = 4/8 = 50%.

iii) Arrow lands on a section labeled with a number less than 4:

The numbers that are less than 4 are 3, 2, and 1.

There are two 3's.

There is one 2.

There are four 1's.

2 + 1 + 4 = 7.

The probability of the arrow landing on a number less than 4 is 7/8, which is 88.5%.

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To find the center of a circle when the circumference is not given, you still find it.

1. Determine the coordinates of at least three non-collinear points on the circle. Non-collinear points are points that do not lie on a straight line.
2. Using these points, create two line segments that are chords of the circle. A chord is a line segment connecting two points on the circle.
3. Find the midpoints of each chord. The midpoint formula is given as: Midpoint (M) = ((x1 + x2) / 2, (y1 + y2) / 2).
4. Calculate the slope of each chord using the slope formula: Slope (m) = (y2 - y1) / (x2 - x1).
5. Calculate the slope of the perpendicular bisectors of each chord. Since these lines are perpendicular to the chords, their slopes are the negative reciprocal of the chord slopes: m_perpendicular = -1 / m_chord.
6. Write the equation of each perpendicular bisector using the point-slope formula: y - y_midpoint = m_perpendicular * (x - x_midpoint).
7. Solve the system of equations formed by the two perpendicular bisectors. The solution is the coordinates of the center of the circle.

By following these steps, you can find the center of the circle even when the circumference is not given.

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The difference in masses is equal to 1,728 grams.

In Mathematics and Geometry, the volume of a rectangular prism can be calculated by using the following formula:

Volume of a rectangular prism = L × W × H

Where:

By substituting the given dimensions (parameters) into the formula for the volume of a rectangular prism, we have the following;

Volume of rectangular prism = 9 × 3 × 8

Volume of rectangular prism = 216 cm³.

Mass of gold = density × volume

Mass of gold = 19.3 × 216

Mass of gold = 4,168.8 grams.

Mass of lead = 11.3 × 216

Mass of lead = 2,440.8 grams.

Difference in masses = 4,168.8 grams - 2,440.8 grams.

Difference in masses = 1,728 grams.

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The sales price divided by total cost is 123%.

Based on the information provided, I can help you complete the table:

Purchase Price | Load | Total Cost | Sales Price | Sales Price ÷ Total Cost (nearest 1%)
ABC = $425     |   0  |   $425     |   $850      |        200%
DEF = $600     | $375 |   $975     |   $1,200    |        123%

For mutual fund ABC, there was no load, so the total cost is equal to the purchase price. The sales price ÷ total cost is 200% (nearest 1%). For mutual fund DEF, the total cost includes the $375 load, resulting in a total cost of $975. The sales price ÷ total cost is 123% (nearest 1%).

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Answer:

1) 1/2 + 1/4 = 2/4 + 1/4 = 3/4

2) To add these fractions, you need to find a common denominator. The smallest common multiple of 7 and 9 is 63, so we can write:

3/7 * 9/9 + 2/9 * 7/7 = 27/63 + 14/63 = 41/63

3) To add these fractions, you need to find a common denominator. The smallest common multiple of 5 and 15 is 15, so we can write:

3/5 * 3/3 + 1/15 * 1/1 = 9/15 + 1/15 = 10/15

But we can simplify this fraction by dividing both the numerator and denominator by 5:

10/15 = 2/3

4) To add these fractions, you need to find a common denominator. The smallest common multiple of 9 and 8 is 72, so we can write:

1/9 * 8/8 + 7/8 * 9/9 = 8/72 + 63/72 = 71/72

5) To add these fractions, you need to find a common denominator. The smallest common multiple of 7 and 21 is 21, so we can write:

6/7 * 3/3 + 2/21 * 1/1 = 18/21 + 2/21 = 20/21

6) To add these fractions, we need to find a common denominator first. The smallest number that both 6 and 10 divide into is 30. So, we convert 4/6 to 20/30 by multiplying both the numerator and denominator by 5, and we convert 2/10 to 3/15 by multiplying both the numerator and denominator by 3. Now we have:

20/30 + 3/15 = (20x1 + 3x2)/(30x2) = 23/60

Therefore, 4/6 + 2/10 = 23/60.

7) To add these fractions, we need to find a common denominator first. The smallest number that both 11 and 22 divide into is 22. So, we convert 1/11 to 2/22 by multiplying both the numerator and denominator by 2, and we convert 3/22 to 3/22 (it is already in terms of 22). Now we have:

2/22 + 3/22 = (2 + 3)/22 = 5/22

Therefore, 1/11 + 3/22 = 5/22.

8) To add these fractions, we need to find a common denominator first. The smallest number that both 4 and 20 divide into is 20. So, we convert 1/4 to 5/20 by multiplying both the numerator and denominator by 5, and we convert 8/20 to 8/20 (it is already in terms of 20). Now we have:

5/20 + 8/20 = (5 + 8)/20 = 13/20

Therefore, 1/4 + 8/20 = 13/20.

9) To add these fractions, we need to find a common denominator first. The smallest number that both 7 and 9 divide into is 63. So, we convert 4/7 to 24/63 by multiplying both the numerator and denominator by 3, and we convert 2/9 to 14/63 by multiplying both the numerator and denominator by 7. Now we have:

24/63 + 14/63 = (24 + 14)/63 = 38/63

Therefore, 4/7 + 2/9 = 38/63.

10) To add these fractions, we need to find a common denominator first. The smallest number that both 10 and 30 divide into is 30. So, we convert 6/7 to 18/30 by multiplying both the numerator and denominator by 3, and we convert 2/30 to 1/15 by multiplying both the numerator and denominator by 15. Now we have:

18/30 + 1/15 = (18x1 + 1x2)/(30x2) = 37/30

Therefore, 6/7 + 2/21 = 37/30.

Answer: V = 2527.2 in^3

Step-by-step explanation:

V = Bh

that is, Volume = base area x height

the base area is the hexagon, and the height is given as 12.

Think of dividing the hexagon into 6 equal triangles, with height 7.8

so the area of all 6 triangles, (effectively the area of the hexagon), will be:

6(0.5 x 9 x 7.8) = 210.6 in^2

multiply this by the height to get the volume:

210.6 x 12 = 2527.2 in^3

thats it!

V = 2527.2 in^3

Answer:

I believe it is 38

Step-by-step explanation:

The monthly rent for the apartment would be $1238.84 in the fifth year.

The problem states that the monthly rent for an apartment is $925. To calculate the rent for the second year, we need to increase this amount by 6%.

To do this, we first need to calculate 6% of $925. We can do this by multiplying 0.06 (which is equivalent to 6%) by $925:

6% of $925 = 0.06 × $925 = $55.50

So, the rent for the second year would be:

$925 + $55.50 = $980.50

To find the rent for the third year, we need to increase the rent for the second year by 6%. We can follow the same process:

6% of $980.50 = 0.06 × $980.50 = $58.83

So, the rent for the third year would be:

$980.50 + $58.83 = $1039.33

To find the rent for any year n, we use the formula:

Rent for year n = $925 × (1 + 0.06)^n

In this formula, (1 + 0.06) represents the multiplier used to calculate the new rent each year.

For example, the multiplier for the second year is 1 + 0.06 = 1.06, and the multiplier for the third year is 1.06 × 1.06 = 1.1236 (rounded to four decimal places).

To find the rent for the fifth year, we plug in n = 5:

Rent for year 5 = $925 × (1 + 0.06)^5 = $925 × 1.3382 = $1238.84 (rounded to the nearest cent)

So, the monthly rent for the apartment would be $1238.84 in the fifth year.

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You need to add 2125 milliliters of water to fill the square aquarium completely.

We need to find the volume of the square aquarium and then determine the additional water needed to fill it completely. Here are the steps:

1. Convert the given length to meters: 15 cm = 0.15 m
2. Calculate the volume of the square aquarium: Volume = length × width × height. Since it's a square aquarium, all sides are equal, so Volume = 0.15 m × 0.15 m × 0.15 m = 0.003375 cubic meters.
3. Convert the volume to milliliters: 0.003375 cubic meters × 1,000,000 mL/cubic meter = 3375 mL.
4. Calculate the additional water needed: Total volume - Current volume = 3375 mL - 1250 mL = 2125 mL.

You need to add 2125 milliliters of water to fill the square aquarium completely.

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Answer: The median of the players' heights is not affected.

Step-by-step explanation: B

The median of the players' heights is increased.

we need to consider the current arrangement of heights and the positions

of the new players in relation to the existing players' heights.

If we assume that the heights of the players are sorted in ascending order,

adding two additional players can affect the median in the following ways:

If both new players have heights lower than the current median:

In this case, adding the new players would not change the median.

The median would remain the same because the new players would be

added below the existing median, and the position of the median would

not shift.

If one new player has a height lower than the current median and the other

has a height higher than the current median:

In this case, the median would be increased.

Adding a taller player would shift the median towards the higher end of the data set.

If both new players have heights higher than the current median:

In this case, the  would be increased.

Both new players would be taller than the current median, causing the

median to shift towards the higher end of the data set.

Based on these possibilities, the answer is C.

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(a) To find a non-zero vector orthogonal to the plane through the points P, Q, and R, we need to take the cross product of two vectors in the plane. One way to do this is to subtract one point from another to get a vector, and then take the cross product of the two resulting vectors. For example, we could subtract point Q from point P to get the vector PQ, and subtract point R from point P to get the vector PR. Then, taking the cross product of PQ and PR will give us a vector orthogonal to the plane:PQ = <-3, 0, -9>PR = <-3, -6, 0>PQ x PR = <54, 27, 18>Therefore, the vector <54, 27, 18> is orthogonal to the plane through the points P, Q, and R.(b) To find the area of the parallelogram with sides PQ and PR, we need to find the length of the projection of PQ onto PR, and then multiply by the length of PR. The projection of PQ onto PR is given by:proj_PR(PQ) = (PQ · u) uwhere u is a unit vector in the direction of PR, and · denotes the dot product. Since PR = <-3, -6, 0>, we can take u = <-1/sqrt(10), -3/sqrt(10), 0>, which is a unit vector in the direction of PR. Then:proj_PR(PQ) = (PQ · u) u = (-3/sqrt(10)) <-1/sqrt(10), -3/sqrt(10), 0> = <9/10, 27/10, 0>The length of this vector is sqrt((9/10)^2 + (27/10)^2 + 0^2) = 3sqrt(10), so the area of the parallelogram is:A = |PQ| |proj_PR(PQ)| = sqrt((-3)^2 + 0^2 + (-9)^2) * 3sqrt(10) = 27sqrt(10)

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A. Orthogonal vector = (0, -27, 18)

B. The area of the parallelogram with sides PQ and PR is 1053 square units.

(a) To find a non-zero vector orthogonal to the plane through points P, Q, and R, we need to compute the cross product of vectors PQ and PR.

Vector PQ = Q - P = (-3, 0, -9)
Vector PR = R - P = (-3, -6, 0)

Cross product PQ x PR = (i, j, k) × ((-3, 0, -9), (-3, -6, 0))
= i(0 * 0 - (-9) * (-6)) - j(-3 * 0 - (-3) * (-9)) + k(-3 * -6 - 0 * (-3))
= i(0) - j(27) + k(18)
Orthogonal vector = (0, -27, 18)

(b) To find the area of the parallelogram with sides PQ and PR, we can use the magnitude of the cross product of PQ and PR.

Magnitude of PQ x PR = ||(0, -27, 18)||
= √(0^2 + (-27)^2 + 18^2)
= √(729 + 324)
= √1053

The area of the parallelogram with sides PQ and PR is 1053 square units.

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The pattern of the data indicates a linear relationship or strong positive correlation between the average house prices and average monthly payments.

The monthly payment for a house costing $450,000 is $3,600.

A formulate connecting the average monthly payment with the average house price in slope-intercept form is y = 0.008x.

In Mathematics, a proportional relationship can be represented by this equation:

y = kx

Where:

Next, we would determine the constant of proportionality (k) as follows:

Constant of proportionality (k) = y/x

Constant of proportionality (k) = 8/1000

Constant of proportionality (k) = 1/125 or 0.008.

Therefore, a formula that connects the two variables is given by;

y = kx

y = 0.008x

When average house price (x) = $450,000, the average monthly payment (y) is given by:

y = 0.008(450,000)

y = $3,600.

In conclusion, we can logically deduce that the pattern of the data shows a linear relationship or strong positive correlation between the average house prices (x) and average monthly payments (y) because as the average house prices (x) increases, the average monthly payments (y) also increases.

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There are 20 different ways the students can be chosen for the positions of editor-in-chief and assistant editor-in-chief.

There are 5 students in the newspaper staff, and two positions to fill i.e. editor-in-chief and assistant editor-in-chief. We need to find the number of different ways the students can be chosen for these positions.

To solve this problem, we can use the formula for permutation

We know the formula for Permutation is

[tex]P(n,r)=\frac{n!}{(n-r)!}[/tex]

Here, n=5 and r=2

So, P(5,2) = 5!/(5-2)!

= 5!/3!

= 120/6

= 20

Therefore, there are 20 different ways the students can be chosen for the positions of editor-in-chief and assistant editor-in-chief.

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The area of ΔSTU is approximately 6.7 square centimeters.

To find the area of a triangle, we can use the formula A = (1/2)bh, where b is the base of the triangle and h is the height. In this case, we know that u is the base of the triangle, so we need to find the height.

To do this, we can use the sine function, which tells us that sin(6°) = h/u. Rearranging this equation, we get h = usin(6°). We can then substitute u and sin(6°) into the formula for the area to get

A = (1/2)(3.4)(3.4sin(6°)) ≈ 6.7 square centimeters.

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The Paige walks at a speed of approximately 0.04167 miles per minute to get to the park.

To calculate Paige's speed, we used the formula:

Speed = Distance / Time

Given that Paige walks to the park 2/3 mile away, we substitute Distance with 2/3 mile and Time with 16 minutes. We get:

Speed = 2/3 mile / 16 minutes

Simplifying the expression by converting minutes to hours, we get:

Speed = 2/3 mile / (16/60) hours

Simplifying further by multiplying both the numerator and denominator by 60, we get:

Speed = [tex](2/3) * (60/1)[/tex] mile/hour / (16/1) minutes

Speed = 0.04167 mile/minute (rounded to 5 decimal places)

"Paige walks to the park 2/3 mile away it takes her 16 minutes to get there how many miles per minutes" is that Paige walks at a speed of approximately 0.04167 miles per minute.

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Answer:

B) Unique

Step-by-step explanation:

J coordinates = (0,6)

K coordinates = ( 9, -9)

L coordinates = ( -9, -9)

Unique rule: 1/3x , 1/3y

Which means 1\3 times the x coordinate and 1/3 times the y coordinate

J,K,L after applying the unique rule equals

J= (0,2)

K= (3,-3)

L= (-3,-3)

Which lines up with J’ and K’ and L’.

Making the “unique” statement true, dilation= (1/3x, 1/3y)

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The limit of A. x^(2^x) as x approaches 0 is 1, and the limit of B. 2^x - 1 as x approaches 0 is ln 2.

A. To find the limit of A. x^(2^x) as x approaches 0, we can take the natural logarithm of both sides and use the fact that ln(1 + a) is approximately equal to a for small values of a. This gives us:

ln(A. x^(2^x)) = 2^x ln x

ln(A. x^(2^x)) / ln x = 2^x

Taking the limit as x approaches 0, the right-hand side goes to 1, and using the continuity of the natural logarithm, we have:

ln(A) = 0

A = 1

Therefore, the limit of A. x^(2^x) as x approaches 0 is 1.

B. To find the limit of B. 2^x - 1 as x approaches 0, we can use L'Hopital's Rule:

lim x→0 (2^x - 1)

= lim x→0 (ln 2 * 2^x / ln 2)

= ln 2 * lim x→0 (2^x / ln 2)

= ln 2 * (lim x→0 e^(x ln 2) / ln 2)

= ln 2 * (lim x→0 e^(x ln 2 - ln 2) / (ln 2 - ln 2))

= ln 2 * (lim x→0 e^(ln 2 * (x - 1)) / 1)

= ln 2 * e^0

= ln 2

Therefore, the limit of B. 2^x - 1 as x approaches 0 is ln 2.

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Jerry pays $279.40 this year in property taxes.

To calculate Jerry's property taxes for the year, we need to first decide how many of his month-to-month maintenance fee is going toward property taxes.

The problem states that 20% of the price is for Jerry's assets taxes, which means we will calculate the amount of his belongings taxes with the aid of finding 20% of his monthly charge.

To do this, we multiply the price through 0.20 such as this:

20% of $1,397 = 0.20 x $1,397 = $279.40

Therefore, Jerry pays $279.40 this year in property taxes.

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The Pythagorean Theorem with regards to the relationships between the lengths of the sides of a right triangle indicates that we get;

2. x = 51

3. x = 50

4. x = 82

8. x = 2·√(77)

9. x = √(39)

10. x = 2·√(19)

11. x = 2·√(154)

12. x = 3·√3

13. x = 6·√(13)

16. A right triangle

17. A right triangle

18. The triangle is not a right triangle

19. An obtuse triangle

20. An obtuse triangle

The Pythagorean Theorem states that the square of the length of the hypotenuse side of a right triangle is equivalent to the sum of the squares of the lengths of the other two sides.  

2. x² = 45² + 24² = 2601

x = √(2601) = 51

3. x² = 30² + 40² = 2500

x = √(2500) = 50

x = 50

4. x² = 80² + 18² = 6724

x = √(6724) = 82

x = 82

8. According to the Pythagorean Theorem, in the right triangle we get;

x² = 18² - 4² = 308

x = 2·√(77)

9. x² = 8² - 5² = 39

x = √(39)

10. x² = 20² - 18²

x² = 76

x = √(76) = 2·√(19)

x = 2·√(19)

11. x² = 25² - 3² = 616

x = √(616) = 2·√(154)

x = 2·√(154)

12. x² = 6² - 3² = 27

x = √(27)

x = 3·√3

13. x² = 22² - 4² = 468

x = √(468) = 6·√(13)

x = 6·√(13)

16. A triangle is a right triangle if the square of the side that is the longest is equivalent to the square of the other two sides, therefore;

17² = 289

15² + 8² = 289

Therefore, the triangle is a right triangle

17. 45² = 2025

27² + 36² = 2025

Therefore, the triangle is a right triangle

18. 11² = 121

9² + 4² = 97

Therefore, the triangle is not a right triangle

19. 6² = 36

4² + 3² = 25

The square of the side that is longest is larger than the sum of the squares of the other two sides, which indicates that the angle facing the longest side is lar1ger than 90°, and the triangle is an obtuse triangle.

20. 16² = 256

9² + 11² = 202

16² > 9² + 11²; Therefore, the triangle is an obtuse triangle

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Ansley's cousin is 12 years old, and Ansley's age can be found by plugging in 12 for Cousin's age.

The problem tells us that Ansley's age is 5 years less than 3 times her cousin's age. We can write this as an equation:

Ansley's age = 3 × Cousin's age - 5

We also know that Ansley is 31 years old. So we can substitute 31 for Ansley's age in the equation:

31 = 3 × Cousin's age - 5

Now we solve for Cousin's age. First, we add 5 to both sides of the equation:

31 + 5 = 3 × Cousin's age

Simplifying:

36 = 3 × Cousin's age

Finally, we divide both sides by 3:

Cousin's age = 12

So Ansley's cousin is 12 years old, and Ansley's age can be found by plugging in 12 for Cousin's age in the expression we found earlier:

Ansley's age = 3 × Cousin's age - 5 = 3 × 12 - 5 = 31

So Ansley is indeed 31 years old.

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There are 4.097 x [tex]10^43[/tex] paths from (0,0) to (90,160) that pass through (80,70).

To calculate the number of paths from (0,0) to (90,160) while passing through (80,70), we need to break down the problem into smaller steps.

First, we can calculate the number of paths from (0,0) to (80,70) and then

multiply that by the number of paths from (80,70) to (90,160).

To go from (0,0) to (80,70), we need to take 80 steps to the right and 70 steps up, which gives us a total of 150 steps. The order in which we take these steps doesn't matter, so we can think of it as choosing 70 steps out of 150 to be up. This can be calculated using the binomial coefficient, which gives us (150 choose 70) = 2.364 x  [tex]10^43[/tex]

To go from (80,70) to (90,160), we need to take 10 steps to the right and 90 steps up, which gives us a total of 100 steps. Using the same method as above, the number of paths from (80,70) to (90,160) is (100 choose 10) = 17,310,309.

Multiplying these two values together, we get the total number of paths from (0,0) to (90,160) that pass through (80,70):

(2.364 x 10^34) x (17,310,309) = 4.097 x  [tex]10^43[/tex]

Therefore, there are 4.097 x [tex]10^43[/tex] paths from (0,0) to (90,160) that pass through (80,70).

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Thus, the new coordinates for the image of translated Vertices of Square RSTU are-  R'(-6, 0), S'(-1, 3), T'(2, -2) and  U'(-3, -5).

In mathematics, a translation moves an object throughout the coordinate plane while preserving its dimensions and shape. After a translation, its area and orientation remain unchanged.

The vertical shift, horizontal shift, or perhaps a combination of the two can be referred to as a translation in mathematics.

Given that:

Vertices of Square RSTU.

R(-2, 1), S(3, 4), T(6, -1), and U(1, -4):

translation: (x, y) → (x-4, y − 1)

New vertices:

R(-2-4, 1 − 1) --> R'(-6, 0)

S(3-4, 4 − 1), ---> S'(-1, 3)

T(6-4, -1 − 1),  -> T'(2, -2)

U(1-4, -4 − 1) --> U'(-3, -5)

Thus, the new coordinates for the image of translated Vertices of Square RSTU are-  R'(-6, 0), S'(-1, 3), T'(2, -2) and  U'(-3, -5).

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The length of the diagonal is c = √269 feet.

To get the length of the diagonal of a rectangular patio, we can use the Pythagorean theorem, which states that for a right triangle with legs of length a and b, and hypotenuse of length c, a² + b² = c². In this case, the legs of the right triangle are the length and width of the rectangular patio, which are 10 feet and 13 feet, respectively. Let's use a and b to represent these lengths.
a = 10 feet
b = 13 feet
We want to find the length of the diagonal, which is the hypotenuse of the right triangle. Let's use c to represent this length.
a² + b² = c²
10² + 13² = c²
100 + 169 = c²
269 = c²
Now we need to find the square root of 269 to get the length of the diagonal.
c = √269
c ≈ 16.4 feet
So the length of the diagonal of the rectangular patio is approximately 16.4 feet. We can also find the ratio of the length, width, and diagonal of the rectangular patio.
length:width = 10:13
width:length = 13:10
length:diagonal = 10:√269
width:diagonal = 13:√269
diagonal:length = √269:10
diagonal:width = √269:13

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To determine what students want served in the cafeteria, the cook asks students in Ms. Andrew’s first period class. The sample used by the cook is known as Convenience.

The sample used is known as convenience sample. The cook only asks students in Ms. Andrews’ first period class which is a convenient and accessible group to ask but this method of sampling may not be representative of the entire student population as it only includes students in one class.

So, the results may not accurately reflect what all students want to be served in the cafeteria, hence, more representative sample could be obtained by using a simple random sample or systematic sample where  student in the population has an equal chance of being selected.

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The final result is  ∫ sin²(x)- cos²(x)/cos(x) dx = -3cos²(x)/2 + 2ln|cos(x)| + C.

To solve the indefinite integral ∫ sin²(x)- cos²(x)/cos(x) dx, we need to use trigonometric identities to simplify the integrand.

First, we use the identity sin²(x) + cos²(x) = 1 to write:

sin²(x) - cos²(x) = sin²(x) + cos²(x) - 2cos²(x) = 2sin²(x) - cos²(x)

Next, we use the identity sin²(x) = 1 - cos²(x) to write:

2sin²(x) - cos²(x) = 2(1-cos²(x)) - cos²(x) = 2 - 3cos²(x)

Substituting this into the original integral, we get:

∫ sin²(x)- cos²(x)/cos(x) dx = ∫ (2 - 3cos²(x))/cos(x) dx

Now, we use the substitution u = cos(x) and du/dx = -sin(x) dx to transform the integral into a simpler form:

∫ (2 - 3cos²(x))/cos(x) dx = ∫ (2 - 3u²)/u (-du/sin(x))

= -∫ (3u² - 2)/u du

= -3∫ u du + 2∫ du/u

= -3u²/2 + 2ln|u| + C

= -3cos²(x)/2 + 2ln|cos(x)| + C

where C is the constant of integration.

Substituting back u = cos(x), we obtain the final result

∫ sin²(x)- cos²(x)/cos(x) dx = -3cos²(x)/2 + 2ln|cos(x)| + C

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The  volume of the region U is 16π cubic units.

To find the volume of the region U, we can use cylindrical coordinates. In cylindrical coordinates, a point in space is represented by the coordinates (r, θ, z), where r is the distance from the z-axis, θ is the angle between the x-axis and the projection of the point onto the xy-plane, and z is the height above the xy-plane.

In this case, the region U is defined by 0 ≤ r ≤ 2, 0 ≤ θ ≤ 2π, and 0 ≤ z ≤ 12 - r sin(θ).

To find the volume of U, we can integrate over the cylindrical coordinates. The volume of U is given by the integral:

V = ∫∫∫_U dV

where dV = r dz dr dθ is the volume element in cylindrical coordinates.

Substituting in the limits of integration, we have:

V = ∫₀²π ∫₀² ∫₀^(12-rsinθ) r dz dr dθ

Integrating with respect to z, we get:

V = ∫₀²π ∫₀² r(12-rsinθ) dr dθ

Integrating with respect to r, we get:

V = ∫₀²π [(6r² - (1/3)r³sinθ)] from r=0 to r=2 dθ

Simplifying, we get:

V = ∫₀²π [(24 - 16/3 sinθ)] dθ

Integrating, we get:

V = [24θ + 16/3 cosθ] from θ=0 to θ=2π

Simplifying, we get:

V = 48π/3 = 16π

Therefore, the volume of the region U is 16π cubic units.
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