How do I evaluate the limit? How do I know which theorems to use

Answer:

7n + 14

Step-by-step explanation:

Multiply the number outside the parentheses by each quantity in the parentheses.

7(n + 2) = 7 × n + 7 × 2 = 7n + 14

Answer:

Step-by-step explanation:

distribute the 7 to the n.

then distribute the 7 to the 2.

=7n+14

The value of x is 4/3

The perimeter of a figure is the total sum of the length of all its external sides or boundary.

Given that the rectangle and triangle have the same perimeter, then;

perimeter of the rectangle = 2 (length + width)

                                        = 2[(x + 3) + (x -2)}

                                       = 2(x + 1)

                                       = 2x + 2

perimeter of the triangle = (2x - 2) + (2x - 3) + (x + 3)

                                        = 5x - 2

So that,

perimeter of rectangle = perimeter of triangle

2x + 2 = 5x - 2

2 + 2 = 5x - 2x

4 = 3x

x = 4/3

The value of x is 4/3.

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The aacrued amount in the account in 15 years to the nearest dollar is $9,393.

The formula accrued amount in a compounded interest is expressed as;

A = P( 1 + r/n )^( n × t )

Where A is accrued amount, P is principal, r is interest rate and t is time.

Given the data in the question;

First,lets convert R as a percent to r as a decimal

r = R/100

r = 3.6/100

r = 0.036

Now, plug in the values

A = P( 1 + r/n )^( n × t )

A = $5,500( 1 + 0.036/2 )^( 2 × 15 )

A = $5,500(1 + 0.018)^(30)

A = $5,500(1.018)^(30)

A = $9,393

Therefore, the accrued amount is $9,393.

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The difference is 7.76 × 10³ in scientific notation.

What is scientific notation?

Scientific notation is a way of expressing a number as a number between 1 and to multiplied by a power of 10.

We can perform the subtraction in the usual way, using the fact that the numbers are already in scientific notation:

(8.4 × 10³) − (6.4 × 10²) = 8.4 × 1000 − 6.4 × 100

= 8400 − 640

= 7760

To write this in scientific notation, we can express the result as a number between 1 and 10 multiplied by an appropriate power of 10. In this case, we can write:

7760 = 7.76 × 10³

Therefore, the difference is 7.76 × 10³ in scientific notation.

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Answer:

5 1/2

Step-by-step explanation:

1 7/8 + 3 5/8

first start by adding the whole numbers

1 + 3 = 4

then add the fractions (only add the numerators)

7/8 + 5/8 = 12/8

12/8 is the same thing as 1 4/8

1 4/8 is the same thing as 1 1/2

Now add the whole number you added above

4 + 1 1/2 = 5 1/2

So, you answer to 1 7/8 + 3 5/8 is 5 1/2

Hope this helps!

There are 21 green apples and 7 red apples

There are 28 apples in a basket

The ratio of red to green is 3:1

The first step is to calculate the total ratio, this can be achieved by adding the number of red to green apple

= 3+1

Total ratio= 4

The next step is to divide the total number of apples which is 28 by 4

= 28/4

= 7

The number of green apples can be calculated as follows

= 7 × 3

= 21

The number of red apples can be calculated as follows

= 7 × 1

= 7

Hence there are 21 green apples and 7 red apples

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Answer:

30

Step-by-step explanation:

[tex]x^{2} +7=71 \iff x^{2} =71-7[/tex]

[tex]x^{2} = 64 \iff x^{2} = 8^{2}[/tex]

[tex]x_{1} = - 8, \ x_{2} = 8[/tex]

Answer:

8/3

Step-by-step explanation:

To find how many miles are in one hour, we need to scale up the distance covered in 1/4 hour to the distance covered in 1 hour.

We can start by finding the distance covered in 1 hour:

2/3 mile in 1/4 hour = (2/3) ÷ (1/4) = 8/3 miles in 1 hour

Therefore, the ball covers 8/3 miles in one hour.

Answer:

8/3 miles an hour

Step-by-step explanation:

Because you can travel 2/3 in 1/4 (or 0.25) of an hour, that means you can travel 4 times that distance in 1 hour, assuming the velocity is constant.

To find this then, we simply multiply 2/3 by 4  (2/3*4/1) to get our final answer of 8/3 miles an hour.

The error occurs in step 2, because according to BODMAS we need to perform multiplication before addition.

The acronym BODMAS rule is used to keep track of the correct order of operations to perform while resolving mathematical statements. BODMAS stands for Brackets, Order of Powers or Roots, Division, Multiplication, Addition, and Subtraction. In some circumstances, the prefix "of" is used as well. This indicates that numerous operator expressions must only be spelled out in this sequence, from left to right. We begin by resolving the brackets, then go on to powers or roots, division or multiplication (depending on which comes first from the left side of the expression), and lastly subtraction or addition (depending on which comes on the left side).

According to the BODMAS multiplication must be performed before addition.

Hence, the error occurs in step 2, because according to BODMAS we need to perform multiplication before addition.

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Answer:

w = 102

m = 78

n = 58

Step-by-step explanation:

∠m is an exterior angle of the triangle on the left

exterior angle = sum of alternate interior angles

m° = 36° + 42°

m  = 78

m and w are supplementary angles

m + w = 180

78 + w = 180

w = 180 - 78

w = 102

w is an exterior angle of the triangle on the right

w = n + 44

102 = n + 44

102-44 = n

n = 58

Option A is correct, 5(3f+2) expression also represents the total cost of frames.

An expression is combination of variables, numbers and operators.

The expression 15f+10 represents the total cost of f frames.

We need to find the equivalent expression which represents the total cost of frames.

Equivalent expressions are expressions that work the same even though they look different.

Let us find the GCF of 15 and 10, which is 5 to factor out the expression.

5(3f+2)

Hence, 5(3f+2) expression also represents the total cost of frames.

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Answer:281.92 cm

Step-by-step explanation:

ok so we have a triangle and the area of a triangle is bh/2 so 12.8x8.81/2= 56.384 and since we have 5 of these 56.384x5=281.92Cm

The total cost for 3 pounds of grapes, 3 pounds of bananas, and 2 pounds of pears is $11.32. The solution has been obtained by using linear equation.

What is a linear equation?

One is the linear equation's degree. In linear equations with exponents greater than 1, it is clear that there are no variables. A straight line is produced by the graph's equation.

We are given that the price of grapes, g, is $2.19 per pound. The price of bananas, b, is $0.59 per pound. The price of pears, p, is $1.49 per pound.

From this, we get an equation as

Total Cost = 2.19g + 0.59b + 1.49p

Now, the total cost for 3 pounds of grapes, 3 pounds of bananas, and 2 pounds of pears will be

⇒Total Cost = 2.19(3) + 0.59(3) + 1.49(2)

⇒Total Cost = 6.57 + 1.77 + 2.98

⇒Total Cost = $11.32

Hence, the total cost for 3 pounds of grapes, 3 pounds of bananas, and 2 pounds of pears is $11.32.

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Question: The price of grapes, g, is $2.19 per pound. The price of bananas, b, is $0.59 per pound. The price of pears, p, is $1.49 per pound.

What is the total cost for 3 pounds of grapes, 3 pounds of bananas, and 2 pounds of pears?

Answer:

q=2.1

Step-by-step explanation:

2q+3.07=7.27

take away 3.07 from both sides

2q=4.20

divide both sides by 2

q=2.10

q=2.1

The value of the functions are,

⇒ f (- 10) = 20201

⇒ f (2) = 17

⇒ f (3) = 143

A relation between a set of inputs having one output each is called a function. and an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).

Given that;

The value of function is,

⇒ f (x) = 2x⁴ + x² - 10x + 1

Now, We can find all the values as;

At x = - 10;

⇒ f (x) = 2x⁴ + x² - 10x + 1

⇒ f (- 10) = 2 (- 10)⁴ + (- 10)² - 10 (- 10) + 1

⇒ f (- 10) = 20000 + 100 + 100 + 1

⇒ f (- 10) = 20201

At x = 2;

⇒ f (x) = 2x⁴ + x² - 10x + 1

⇒ f (2) = 2 (2)⁴ + (2)² - 10 (2) + 1

⇒ f (2) = 32 + 4 - 20 + 1

⇒ f (2) = 17

At x = 3;

⇒ f (x) = 2x⁴ + x² - 10x + 1

⇒ f (3) = 2 (3)⁴ + (3)² - 10 (3) + 1

⇒ f (3) = 162 + 9 - 30 + 1

⇒ f (3) = 143

Thus, The value of the functions are,

⇒ f (- 10) = 20201

⇒ f (2) = 17

⇒ f (3) = 143

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We can see that there are 9 possible outcomes for spinning the spinner two times.

By pairing each potential result for the first spin with each potential result for the second spin, we can list all the outcomes and establish the sample space for spinning the spinner twice. Since there are three possible results for each spin, spinning the spinner twice has a total of 3 x 3 = 9 outcomes.

Making a table that displays every scenario is one technique to list the sample space:

Spin 1 Spin 2

Blue  Blue

Blue Orange

Blue  Red

Red          Blue

Red   Orange

Red          Red

Orange  Blue

Orange  Orange

Orange   Red

Listing all the potential pairs of outcomes is another way to list the sample space:

Blue, Blue, Blue, Orange, Blue, Red, Red, Blue, Orange, Red, and Orange, Red; Blue, Blue, Blue, Orange, Blue, Red; and Orange, Blue, Orange, Orange, Red

In either case, we can see that spinning the spinner twice can result in 9 different results.

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The scatter plot shows negative correlation because as the plotted value of x increases, the values of y generally decreases.

In this scenario, the x-values would be plotted on the x-axis (x-coordinate) of the scatter plot while the y-values would be plotted on the y-axis (y-coordinate) of the scatter plot through the use of Microsoft Excel.

From the scatter plot (see attachment) which models the relationship between the x-values and y-values, a linear equation for the line of best fit is given by

y = -0.441x + 6.085

In conclusion, we can reasonably infer and logically deduce that the scatter plot most likely indicates an inverse relationship between the x-values and y-values.

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The length of AD is 3 units.

We begin from the equation:

| ² + (AD)² = (2x)²

An equation is described as a formula that expresses the equality of two expressions, by connecting them with the equals sign.

Next step we take is simplifying the right-hand side by squaring 2x:

| ² + (AD)² = 4x²

Substituting x=√3, we get:

| ² + (AD)² = 4(√3)²

| ² + (AD)² = 4(3)

| ² + (AD)² = 12

(AD)² = 12 - ²

(AD)² = 12 - 3

(AD)² = 9

AD = √9

AD = 3 units

Therefore, the length of AD is 3 units.

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Answer: 4.8625

Step-by-step explanation:

Based on the graph, it seems that choosing Company A may be the most popular option among residents.

What is a bar graph?

A bar graph, also called a bar chart, is a kind of data visualization that uses rectangular bars to represent data. Each bar's height or length corresponds to the value it stands for. Bar graphs are frequently used to compare and present various data sets or to demonstrate how data has changed over time.

The vertical axis (y-axis) in a conventional bar graph indicates the values or quantities being measured, while the horizontal axis (x-axis) represents the categories or variables being compared. Each bar in the graph represents a separate category or variable, and they are arranged along the x-axis. Each bar's height or length denotes the value or amount related to that category or variable.

Depending on the volume and complexity of the data being presented, bar graphs can be simple or complex. They can be applied to emphasize patterns or trends in the data, demonstrate changes over time, or compare data across several groups or categories. Among the many disciplines that employ bar graphs for data analysis are business, economics, social sciences, and natural sciences.

Based on the given graph, it appears that Patrick is responsible for choosing the company his local community will use for waste disposal. The graph shows the percentage of residents who prefer each of the four companies, labeled A, B, C, and D.

It seems that Company A is the most popular choice, with approximately 40% of residents preferring it. Company B is the second most popular choice, with approximately 30% of residents preferring it. Company C is the least popular choice, with only about 10% of residents preferring it. Company D falls in between Company B and Company C, with approximately 20% of residents preferring it.

Patrick's decision will ultimately depend on various factors, including the cost, reputation, and reliability of each company, as well as the specific needs and preferences of the community. However, based on the graph, it seems that choosing Company A may be the most popular option among residents.

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Step-by-step explanation:

To find the zeros of the function f(x), we need to find the values of x that make f(x) equal to zero.

f(x) = (x^2 - 13x + 40)(x^2 - 2x + 3)

Setting f(x) equal to zero, we can use the zero product property to solve for x:

(x^2 - 13x + 40)(x^2 - 2x + 3) = 0

Either (x^2 - 13x + 40) = 0 or (x^2 - 2x + 3) = 0.

Solving for the first equation, we can use the quadratic formula:

x = [13 ± √(13^2 - 4(1)(40))] / 2(1)

x = [13 ± √(169 - 160)] / 2

x = [13 ± √9] / 2

So, the solutions for the first quadratic equation are x = 8 and x = 5.

Solving for the second equation, we can again use the quadratic formula:

x = [2 ± √(2^2 - 4(1)(3))] / 2(1)

x = [2 ± √4] / 2

So, the solutions for the second quadratic equation are x = 1 and x = 2.

Therefore, the zeros of the function f(x) are x = 8, x = 5, x = 1, and x = 2.

The required frequency distribution table is below in the solution part.

The relative frequency of the data is the ratio of the given frequency to the total frequency in the observation.

To create a frequency distribution with 5 classes, we first need to find the range of the data, which is the difference between the largest and smallest values:

Range = 10

Next, we divide the range by the number of classes we want (5) to get the class width:

Class width = Range / Number of classes

Class width = 10 / 5

Class width = 2

Now, we can create the frequency distribution table:

Class      Frequency

0-1                5

2-3                29

4-5                16

6-7                14

8-9                 2

10-11               2

___________________

Total             73

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The linear equations that represents the problem are c = 40d + 20, P = 6.07 + 0.17(y - 2002), c = fixed fee + 7.75p and d = -93t + 1007

To solve this problem, we need to write linear equations that represent each scenario

1. Total cost (C) to rent the trench digger for d days can be represented as:

C = 40d + 20

2. The amount of renewable power (P) produced in year y between 2002 and 2005 can be represented as:

P = 6.07 + 0.17(y - 2002)

3. The cost (C) of a birthday party for p people can be represented using the two given data points as:

For 10 people: 97.50 = fixed fee + 10(cost per person)

For 20 people: 175 = fixed fee + 20(cost per person)

Subtracting the first equation from the second equation gives:

77.50 = 10(cost per person)

Hence, the cost per person is:

cost per person = 7.75

Therefore, the rule for the cost (C) of a birthday party for p people is:

C = fixed fee + 7.75p

4. Jimmy's distance (d) from home after t hours of driving can be represented using the two given data points as:

After 3 hours: d = 740

After 6 hours: d = 461

Subtracting the first equation from the second equation gives:

d - 740 = -279

Hence, the distance (d) from home after t hours of driving can be represented as:

d = -93t + 1007

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The polynomial function is  [tex]f(x) = x^3+(\sqrt3-3)x^2+(2-3\sqrt3)x+2\sqrt3[/tex].

The standard polynomial is given by:

f(x) = a(x-p)(x-q)(x-r) ...

Where a is the leading coefficient and p and q are the zeros.

Here We are given that the zeros are x = 1,2,and −√3.

Now leading Coefficient a = 1, p=1 , q=2 and r=−√3

Then the function is,

=> f(x) = 1(x-1)(x-2)(x-(−√3))

=> f(x)=(x-1)(x-2)(x+√3)

=> f(x)=[tex](x^2-2x-x+2)(x+\sqrt3)[/tex]

=> [tex]f(x)=x^2-3x+2(x+\sqrt3)[/tex]

=>[tex]f(x)=x^3+\sqrt3 x^2-3x^2-3\sqrt3 x+2x+2\sqrt3[/tex]

=> [tex]f(x) = x^3+(\sqrt3-3)x^2+(2-3\sqrt3)x+2\sqrt3[/tex]

Hence the polynomial function is [tex]f(x) = x^3+(\sqrt3-3)x^2+(2-3\sqrt3)x+2\sqrt3[/tex].

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The scale factor required to dilate triangle ABC so that its image, A'B'C', is congruent to triangle XYZ is 1.2.

The scale factor is the ratio of corresponding sides of two similar figures. To find the scale factor, divide the length of the corresponding sides of the two triangles.

To find the scale factor between triangle ABC and triangle XYZ, divide the length of the corresponding sides of the two triangles. The length of side AC in triangle ABC is 10 cm, and the length of side XZ in triangle XYZ is 24 cm. Thus, the scale factor is 24/10, or 1.2.

Therefore, to dilate triangle ABC so that its image, A'B'C', is congruent to triangle XYZ, a scale factor of 1.2 is required.

In summary, the scale factor required to dilate triangle ABC so that its image, A'B'C', is congruent to triangle XYZ is 1.2. This is determined by dividing the length of the corresponding sides of the two triangles.

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The value of the sides are;

x = 6

HW = 37

AM = 32

WM = 44

It is important to note that one of the properties of an isosceles trapezoid is that the non-parallel sides are equal to each other.

From this deduction, we then have;

HW = AM

Given that;

HW = 6x + 1

AM = 3x + 19

Equate the lengths

6x + 1 = 3x + 19

collect the like terms

6x - 3x = 19 - 1

3x = 18

Make 'x' subject

x = 6

Then, HW = 37 and AM = 37

Substitute the value of x in

HA = 6x - 4 = 32

WM = 4x + 20 = 44

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Answer:

75°

Step-by-step explanation:

The ratio of two adjacent angles is 3: 5

the total ratio = 3+5=8

the sum of the two adjacent angles = 120°

The measure of the larger angle

[tex] \frac{5}{8} \times 120 \\ = \frac{600}{8} \\ =75 [/tex]

Therefore the larger angle is 75°.