Find the sum of the first 10 terms of the following series, to the nearest integer.


8,20/3,50/9

Answer:

m = -3

Step-by-step explanation:

We Know

Slope = rise/run or (y2 - y1) / (x2 - x1)

Pick 2 points (0,15) (2,9)

We see the y decrease by 6, and the x increase by 2, so the slope is

m = -6/2 = -3

So, the slope of the line representing the data is -3.

Note that the first coordinate of M is -1. (Option D)

Given :- coordinates of F = (-4,-2) = (x1,y1)

coordinates  of G = (2,-2) = (x2,y2)

coordinates of P = (2,-8) = (x3,y3)

and distance between point M and P is half of the distance between FG

To find :- first coordinate of point M

solution :- let the coordinate of M be (x4,y4)

as we know that distance between of the opposite point of parallel line segments are equal

so, second coordinate of M = -8

now by distance formula

FG = √(x2-x1)² + (y2-y1²)

= √[2-(-4)]² + [-2-(-2)]²

= √(2+4)² + (2-2)²

= √(6)² + (0)²

=√36

F G = 6

so, distance between point M and P = 1/2 × F G

= 1/2 × 6

= 3units

again, by distance formula

MP = √(x3-x⁴)² + (y3-y4)²

3 = √(2-x⁴)² + [-8-(-8)]²

squaring on both side

(3)² = (√(2-x⁴)² + [-8-(-8)]²)²

9 = (2-x⁴)² + [-8-(-8)]²

9 = (2-x⁴)²+(0)²

9 = (2-x⁴)²

√9 = 2-x⁴

3 = 2-x⁴

x⁴ = 2-3

x⁴ = -1

Hence the first coordinate of M is -1

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Full Question:

Although part of your question is missing, you might be referring to this full question:

Point M is located in the third quadrant.

The distance between point M and point P is half the distance between point F and point G.

• Segment MP is parallel to segment FG. What is the first coordinate of point M?

If the commission rate is ​%6.5 for the real estate agency, the real estate agency made $20,475 on the house. The agent earned $4,095 in commission.

First, let's determine the commission earned by the real estate agency. To do this, we'll multiply the house price ($315,000) by the commission rate (6.5%).

$315,000 * 6.5% = $20,475

The real estate agency made $20,475 on the house.

Next, let's determine the commission earned by the agent. We'll multiply the commission earned by the real estate agency ($20,475) by the agent's commission rate (20%).

$20,475 * 20% = $4,095

The agent earned $4,095 in commission.

So, the real estate agency made $20,475 on the house, and the agent earned $4,095 in commission.

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The prime number values of a, b and c that satisfy the equation a⁴ + b⁴ + c⁴ + 54 = 11abc are a = 3, b = 2, and c = 5.

Let's consider the equation a⁴ + b⁴ + c⁴ + 54 = 11abc. Due to the fact that the total of four even numbers is also even, the left-hand side is always even. As a result, since 2 is the only even prime, one of the factors a, b, or c must be 2 for 11abc to likewise be even.

Let's examine each instance,

Case 1: a = 2

Substituting a = 2 into the equation, we get,

16 + b⁴ + c⁴ + 54 = 22bc

b⁴ + c⁴ - 22bc + 38 = 0

Since b and c are primes, they must be odd. Let b = 3 and c = 5, we have,

3⁴ + 5⁴ - 2235 + 38 = 0

81 + 625 - 330 + 38 = 0

Case 2: b = 2

Substituting b = 2 into the equation, we get,

a⁴ + 16 + c⁴ + 54 = 22ac

a⁴ + c⁴ - 22ac + 70 = 0

Since a and c are primes, they must be odd. Let a = 3 and c = 5, we have,

3⁴ + 5⁴ - 2235 + 70 = 0

Case 3: c = 2

Substituting c = 2 into the equation, we get,

a⁴ + b⁴ + 16 + 54 = 22ab

a⁴ + b⁴ - 22ab + 70 = 0

However, for any odd number x, x⁴ mod 16 = 1, which means that a⁴ and b⁴ are both corrosponds to 1 mod 16. So, as the conclusion, a = 3, b = 2, and c = 5.

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The next model airplane will be 16 feet long.

To find the length of Aura's next model airplane, we need to multiply the length of her first model airplane by 4, since she wants the next one to be 4 times as long as the first. Therefore, the length of her next model airplane would be:

4 feet (length of first model airplane) x 4 = 16 feet

So, the length of her next model airplane would be 16 feet.

Note that this assumes that Aura's first model airplane is the baseline for measurement and that the "4" in the question refers to a factor of 4, not an additional 4 feet. If the question were interpreted differently, the answer may be different.

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a. If t represents time in hours and d represents distance in miles, and Caden walks at a steady rate of 3.6 miles per hour, the equation that models the relationship between these variables is:

d = 3.6t

b. To calculate the distance Caden will walk in 3/4 of an hour, plug 3/4 into the equation for t:

d = 3.6(3/4) = 2.7 miles

Caden will walk 2.7 miles in 3/4 of an hour.

c. To calculate how long it will take for Caden to walk 4.32 miles, plug 4.32 into the equation for d and solve for t:

4.32 = 3.6t
t = 4.32/3.6 ≈ 1.2 hours

It will take Caden approximately 1.2 hours to walk 4.32 miles.

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To the nearest tenth of an inch, the length of the shorter leg of each right triangle will be 2.7 inches.

A triangle is a polygon with three sides and three angles. The sum of the three angles in a triangle always adds up to 180 degrees. There are many different types of triangles, including equilateral triangles (where all three sides are equal in length and all three angles are 60 degrees), isosceles triangles (where two sides are equal in length and two angles are equal in measure), and scalene triangles (where no sides are equal in length and no angles are equal in measure).

Let x be the length of the shorter leg of each right triangle. Then, the longer leg will be twice as long, so its length will be 2x. Using the Pythagorean theorem, we can write:

x² + (2x)² = 6²

Simplifying and solving for x, we get:

x² + 4x² = 36

5x² = 36

x² = 7.2

x ≈ 2.7

Therefore, to the nearest tenth of an inch, the length of the shorter leg of each right triangle will be 2.7 inches.

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The expression equation y = sin x + cos x / csc x can be simplified to y = cos x sin^2 x + cos^2 x.

To simplify the expression, we can first replace csc x with 1/sin x. This gives us y = sin x cos x + cos^2 x / sin x.

Next, we can factor out cos x from the numerator of the second term to get y = cos x (sin x + cos x) / sin x.

Using the identity sin^2 x + cos^2 x = 1, we can replace sin^2 x with 1 - cos^2 x in the numerator of the first term. This gives us y = cos x (1 - cos^2 x) / sin x + cos x (sin x + cos x) / sin x.

Simplifying the expression further, we get y = cos x (1 - cos^2 x + sin x + cos x) / sin x.

Finally, we can combine the terms in the numerator to get y = cos x (sin^2 x + 2cos x sin x + 1) / sin x.

Using the identity sin^2 x = 1 - cos^2 x, we get y = cos x (3cos^2 x + 2cos x) / sin x.

Simplifying the expression, we arrive at y = cos x (cos x + 2) (3cos x + 2) / sin x.

Therefore, the simplified expression is y = cos x sin^2 x + cos^2 x, which can also be written as y = cos x (sin x)^2 + cos^2 x.

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If Shen drives fewer than 80 miles, the first plan will be cheaper. If he drives more than 80 miles, the second plan will be cheaper.

Let's denote the number of miles driven by "m".

Under the first plan, Shen will pay an initial fee of $40, and then an additional $0.30 for each mile driven. So the total cost, C1, can be expressed as:

C1 = 0.3m + 40

Under the second plan, Shen will not have to pay an initial fee, but he will be charged $0.80 for each mile driven. So the total cost, C2, can be expressed as:

C2 = 0.8m

To determine which plan is cheaper for a given number of miles driven, we can set the two expressions for cost equal to each other and solve for "m":

0.3m + 40 = 0.8m

Subtracting 0.3m from both sides, we get:

40 = 0.5m

Dividing both sides by 0.5, we get:

m = 80

So if Shen drives fewer than 80 miles, the first plan will be cheaper. If he drives more than 80 miles, the second plan will be cheaper.

It's worth noting that this assumes that Shen is only considering the cost of the rental when making his decision. If there are other factors he is considering, such as convenience or availability, he may choose a different plan even if it ends up being slightly more expensive.

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To find the percentage of students who collected between 49 and 98 kilograms of newspapers, we need to first count the number of students whose collection falls within this range. We can do this by sorting the data and counting the number of values that fall within this range.

Sorting the data, we get:

15, 23, 35, 45, 45, 49, 55, 56, 57, 60, 64, 75, 76, 87, 88, 90, 98, 100, 101, 105, 120, 122

We can see that there are 17 students whose collection falls within the range of 49 to 98 kilograms.

To find the percentage of students, we can divide the number of students whose collection falls within this range by the total number of students and then multiply by 100. The total number of students is the sum of the number of values in the two sets, which is 22 + 22 = 44.

Therefore, the percentage of students who collected between 49 and 98 kilograms of newspapers is:

17/44 * 100% ≈ 38.6%

So approximately 38.6% of the students collected between 49 and 98 kilograms of newspapers.

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Answer: 48 inches

Step-by-step explanation:

Answer:

134° + (2x)° = 180°

(2x)° = 46°

x = 23

So angles 1, 2, 3, and 4 all measure 23°.

Answer: 1/5

Step-by-step explanation:

There are 10 slips of paper.

The only numbers less than three are 1 and 2

The probability that he will pick up a slip of paper less than three is 2 since only 1 and 2 are less than three.

Therefore the probability is 2/10, and when simplified, it is 1/5.

Therefore the answer is 1/5.

If you have any more questions feel free to ask in the comments! I'd be happy to help!

The volume of the air in the container is: 3,585.125 in³

The volume of a box or cuboid is given by the formula:

V = L * W * H

Where:

L is length

W is width

H is height

We are given the external dimensions as:

Length = 22 inches

Width = 15 inches

Height = 12 inches

Since the glass is 1/2 inch thick, then the interior dimensions are:

New length = 22 - 1/2 = 21.5 inches

New Width = 15 - 1/2 = 14.5 inches

New height = 12 - 1/2 = 11.5 inches

Thus:

Volume of air in container = 21.5 * 14.5 * 11.5

= 3,585.125 in³

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To multiply -1(x^2-4x+8), we need to distribute the -1 to each term inside the parentheses:

-1(x^2-4x+8) = -x^2 + 4x - 8

So, the product of -1 and the expression (x^2-4x+8) is -x^2 + 4x - 8.

The customer can choose from 21 different gifts if they want to order only 1 item, 138 different gifts if they want to order 2 items of different kinds, and 280 different gifts if they want to order 3 items of different kinds.

To find the different number of gifts, we use the fundamental counting principle, which states that the total number of ways to choose a sequence of items is equal to the product of the number of choices available for each item.

⇒ If a customer wants to order only 1 item, they can choose from:

10 varieties of "greeting-cards"

4 types of "fruit-baskets"

7 flower arrangements

So, total number of different gifts that can be created is: 10 + 4 + 7 = 21.

⇒ If a customer wants to order 2 items of different kinds, they can choose from:

10 varieties of greeting-cards and 4 types of fruit baskets :

(10 × 4 = 40 choices)

10 varieties of greeting cards and 7 flower arrangements :

(10 × 7 = 70 choices)

4 types of fruit-baskets and 7 flower arrangements :

(4 × 7 = 28 choices)

So, number of different gifts that can be created is: 40 + 70 + 28 = 138.

⇒ If a customer wants to order 3 items of different kinds, they can choose from:

10 varieties of greeting cards, 4 types of fruit baskets, and 7 flower arrangements (10 × 4 × 7 = 280 choices),

So, the total number of different gifts that can be created is : 280.

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The probability of TJ pulling 2 white socks is 0.125 or 12.5%.

There are 16 socks in total, so the probability of the first sock being white is 6/16.

Since we are not replacing the first sock, there are now 15 socks left, including 5 white socks.

So the probability of the second sock being white, given that the first sock was white, is 5/15.

To find the probability of both events occurring, we multiply the probabilities:

P(white and white) = P(white on first draw) × P(white on second draw, given that the first was white)

P(white and white) = (6/16) × (5/15)

P(white and white) = 1/8

So the probability of pulling 2 white socks is 1/8 or approximately 0.125.

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Answer:

3 cm.

Step-by-step explanation:

Let's use the formula for the perimeter of a rectangle, which is P = 2l + 2w, where P is the perimeter, l is the length, and w is the width.In this case, we have:P = 16 cm (given)

l = k cm (given)

w = 5 cm (given)Substituting these values into the formula, we get:

16 cm = 2(k cm) + 2(5 cm)

Simplifying, we get:

16 cm = 2k cm + 10 cm

Subtracting 10 cm from both sides, we get:6 cm = 2k cm

Dividing both sides by 2, we get:

3 cm = k

Therefore, the value of k is 3 cm.

[tex]a = \sqrt{ {8}^{2} - {6}^{2} } \\ \\ = \sqrt{64 - 36 }\\ \\ =\sqrt{ 28} \\ \\ = \sqrt{4 \times 7} \\ \\ = 2 \sqrt{7} [/tex]

Layla must pay the agency a fee of $1,426.67 for their services in helping her secure her new job.

One month's pay = Annual salary / 12 months

One month's pay = $53,500 / 12

One month's pay = $4,458.33

Next, we need to determine the agency fee, which is equal to 32% of one month's pay:

Agency fee = 32% x One month's pay

Agency fee = 0.32 x $4,458.33

Agency fee = $1,426.67

Therefore, Layla must pay the agency a fee of $1,426.67 for their services in helping her secure her new job. Layla's agency fee is determined by taking 32% of her monthly pay, which is approximately $4,458.33. This results in a fee of approximately $1,426.67 that Layla must pay to the agency.

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A Pentagon on a coordinate plane would be neither a relation nor a function.

This is because it does not have a unique output for every input, which is a requirement for a function. This is because a function must have a unique output for each input, while a pentagon may have multiple points (outputs) for a given x-coordinate (input) due to its shape. Additionally, it does not satisfy the vertical line test, which is a requirement for a relation. Therefore, it is not a relation or a function.

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The volume of the unit cube is 1 cubic unit, and since it is divided into identical rectangular prisms, each of those prisms has the same volume. Therefore, the volume of one of the identical prisms is also 1 cubic unit.

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The system of inequalities that has a solution set that is a line is [x + y ≥ 3; x + y ≤ 3]; option A

A system of inequalities is a set of two or more inequalities that are solved simultaneously to find the values of variables that satisfy all the inequalities in the system.

Considering the given system of inequalities:

The system of inequalities that has a solution set that is a line is:

[x + y = 3;]

This is because the solution set of this equation is a line with slope -1 passing through the point (3, 0) and (0, 3).

Therefore, the system of inequalities [x + y ≥ 3; x + y ≤ 3] has a solution set that is a line.

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The characteristics of the hyperbola with equation (x+12)²/36-(y-15)²/100=1 are:

Comparing the given equation with the standard form, we have:

(h, k) = (-12, 15)

a² = 36

b² = 100

Taking the square root of a² and b², we get:

a = 6

b = 10

c² = 36 + 100

c² = 136

c ≈ 11.66

The foci of the hyperbola are located at (-12, 15 + 11.66) and (-12, 15 - 11.66), which are approximately (-12, 26.66) and (-12, 3.34).

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The area of the cross-section is 16 sq. cm. Thus, option C is the correct answer.

Vertices sides =  2, 3, 6, and 7

Divide part face = parallel to the face of vertices

It is given that a square face is present in the middle of the cube. The area of the cross-section of the cube results from the plane and cube intersection.

To find the distance between the square face of the cube and the length of the side:

distance = [tex]\sqrt{[(x^{2} - x1)^2 + (y^{2} - y1)^2 + (z^{2} - z1)^2]}[/tex]

we can use the coordinates of any two adjacent sides to find the distance.

distance = [tex]\sqrt{[(3-2)^2 + (3-2)^2 + (3-1)^2] }[/tex]

distance = [tex]\sqrt{11}[/tex]

To calculate the area of the face of the cube:

area = [tex]side^{2}[/tex]

area = [tex]\sqrt{(11)^2}[/tex]

area = 11

The area of the cross-section can be estimated as:

area = (1/2) x 11 x 4) + 5 vertices of plane

area = 16 sq. cm

Therefore we can infer that the area of the cross-section is 16 sq. cm

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The divergence of F is 2y cos(z) + eˣ cos(z) + 3xe³, and the curl of F is -eˣcos(z)i - 3xe³j + (eˣsin(z) + 2cos(z))k.

To find the divergence and curl of the vector field F(x, y, z) = 2y cos(z)i + eˣ sin(z)j + xe³k, we need to apply the appropriate operators.

The divergence of F is given by:

div F = ∇ · F = (∂/∂x)i + (∂/∂y)j + (∂/∂z)k · (2y cos(z)i + eˣ sin(z)j + xe³k)

where ∇ is the del operator.

Calculating the dot product, we get:

div F = 2y cos(z) + eˣ cos(z) + 3xe³

Therefore, the divergence of F is:

div F = 2y cos(z) + eˣ cos(z) + 3xe³

Now, let's find the curl of F. The curl of F is given by:

curl F = ∇ × F = ( (∂/∂y)(xe³) - (∂/∂z)(eˣsin(z)) )i - ( (∂/∂x)(2ycos(z)) - (∂/∂z)(xe³) )j + ( (∂/∂x)(eˣsin(z)) - (∂/∂y)(2ycos(z)) )k

Calculating the partial derivatives, we get:

(∂/∂y)(xe³) = 0

(∂/∂z)(eˣsin(z)) = eˣcos(z)

(∂/∂x)(2ycos(z)) = 0

(∂/∂z)(xe³) = 3xe³

(∂/∂x)(eˣsin(z)) = eˣsin(z)

(∂/∂y)(2ycos(z)) = -2cos(z)

Substituting these values, we get:

curl F = (0 - eˣcos(z))i - (0 - 3xe³)j + (eˣsin(z) - (-2cos(z)))k

Simplifying, we get:

curl F = -eˣcos(z)i - 3xe³j + (eˣsin(z) + 2cos(z))k

Therefore, the divergence of F is 2y cos(z) + eˣ cos(z) + 3xe³, and the curl of F is -eˣcos(z)i - 3xe³j + (eˣsin(z) + 2cos(z))k.

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The total number of students who were surveyed is 575

The graph likely displays a horizontal axis and a vertical axis.

In this graph, the horizontal axis may indicate two categories, such as "juniors" and "seniors," and the vertical axis shows the number of students in each category who responded "yes," "no," or "not sure."

To find the total number of students surveyed, we need to look for the bar that represents the entire group of juniors and seniors. Typically, this bar will be labeled as "total," "all," or something similar. Once we locate this bar, we can add up the number of students represented by the bar. The value will be displayed on the vertical axis, and it should correspond to the sum of the bars for juniors and seniors separately.

=> 100 + 200 + 125 + 150 = 575

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At the end of the weekend, Elisa had the most biscuits (168). So, the answer is (E) Elisa baked the most biscuits on Saturday.

To determine who baked the most biscuits on Saturday, we need to calculate how many biscuits each child had at the end of the weekend.

Anna had 24 biscuits, Berta had 25, Charlie had 26, David had 27, and Elisa had 28.

Let's start with the child who had twice as many biscuits as on Saturday. We can divide their total number of biscuits by 2 to get the number they had on Saturday.

If we try this calculation for each child, we find that only Elisa's total number of biscuits (28) is evenly divisible by 2. Therefore, Elisa must be the child who had twice as many biscuits as on Saturday, meaning she had 14 biscuits on Saturday.

We can use a similar process to determine how many biscuits each child had on Saturday:

- The child who had three times as many biscuits as on Saturday must have had a total of 42 biscuits, which means they had 14 biscuits on Saturday.
- The child who had four times as many biscuits as on Saturday must have had a total of 56 biscuits, which means they had 14 biscuits on Saturday.
- The child who had five times as many biscuits as on Saturday must have had a total of 70 biscuits, which means they had 14 biscuits on Saturday.
- The child who had six times as many biscuits as on Saturday must have had a total of 84 biscuits, which means they had 14 biscuits on Saturday.

Now we can add up the number of biscuits each child had on Saturday:

- Anna had 24 biscuits.
- Berta had 25 biscuits.
- Charlie had 26 biscuits.
- David had 27 biscuits.
- Elisa had 14 biscuits.

Therefore, David baked the most biscuits on Saturday with 27.
To determine who baked the most biscuits on Saturday, we need to consider the information given about the multiplication factors (twice, 3 times, 4 times, 5 times, and 6 times) and the initial number of biscuits baked by each child.

1. Anna baked 24 biscuits.
2. Berta baked 25 biscuits.
3. Charlie baked 26 biscuits.
4. David baked 27 biscuits.
5. Elisa baked 28 biscuits.

Now, let's apply the multiplication factors and see which child had the most biscuits at the end of the weekend:

1. Anna: 24 x 2 = 48
2. Berta: 25 x 3 = 75
3. Charlie: 26 x 4 = 104
4. David: 27 x 5 = 135
5. Elisa: 28 x 6 = 168

At the end of the weekend, Elisa had the most biscuits (168). So, the answer is (E) Elisa baked the most biscuits on Saturday.

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A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

The dice has eight sides, hence the theoretical probability of rolling a six is given as follows:

1/8 = 0.125 = 12.5%.

The experimental probabilities are obtained considering the trials, hence:

The more trials, the closer the experimental probability should be to the theoretical probability.

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The lower and upper quartiles are calculated by finding the average of two values when there is an even number of data points in the dataset. Specifically, the lower quartile (Q1) is the average of the middle two values when the dataset is sorted in ascending order, and the upper quartile (Q3) is the average of the middle two values when the dataset is sorted in descending order.

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