Find the average rate of change of f(x)=x²+2x+1 from x=4 to x=6.

The value of x, considering the Pythagorean Theorem, is given as follows:

x = 9.

The Pythagorean Theorem states that in the case of a right triangle, the square of the length of the hypotenuse, which is the longest side,  is equals to the sum of the squares of the lengths of the other two sides.

Hence the equation for the theorem is given as follows:

c² = a² + b².

In which:

The sides for the problem are given as follows:

15 and 8 (segment of 8 is the radius, which is the distance from the center to a point on the circumference of the circle).

Hence the hypotenuse is obtained as follows:

h² = 15² + 8²

h² = 289

h = 17.

Hence the value of x is obtained as follows:

x = 17 - 8

x = 9.

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The first four terms of the sequence t/n = n(6n - 2) are: 4, 20, 48, 88.

Let's substitute n = 1:

t/1 = 1(6(1) - 2)

t = 1(6 - 2)

t = 4

The first term of the sequence (when n = 1) is 4.

Now, let's substitute n = 2:

t/2 = 2(6(2) - 2)

t = 2(12 - 2)

t = 2(10)

t = 20

The second term of the sequence (when n = 2) is 20.

Next, let's substitute n = 3:

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

t = 3(18 - 2)

t = 3(16)

t = 48

The third term of the sequence (when n = 3) is 48.

Lastly, let's substitute n = 4:

t/4 = 4(6(4) - 2)

t = 4(24 - 2)

t = 4(22)

t = 88

The fourth term of the sequence (when n = 4) is 88.

Therefore, the first four terms of the sequence t/n = n(6n - 2) are: 4, 20, 48, 88.

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The volume of the solid is given as follows:

55 ft³.

The volume of a rectangular prism, with dimensions length, width and height, is given by the multiplication of these dimensions, according to the equation presented as follows:

Volume = length x width x height.

The total volume would be given as follows:

V = 9 x 6 x 1.25

V = 67.5 ft³.

The volume of the removed part is given as follows:

V = 1.25 x 2.5 x (9 - 3 - 2)

V = 12.5 ft³.

Hence the volume of the solid is given as follows:

67.5 - 12.5 = 55 ft³.

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Answer:

57 kilograms. .........................................................................kg

Solving a simple equation, we can see that x = 35.

The 3 lines with an arrow that go to the left are parallel lines, thus, the relation between the sets of values in each of the sides must be the same ones.

Then we can take the quotients between the respective values, and these quotients must be equal, so we can write:

30/12 = x/14

And now we can solve that equation for x, we will get:

x = 14*(30/12)

x = 35

That is the value of x.

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The amount of money that will produce Rs. 15000 at the end of 6 month for 10years at 16% is Rs. 3750

Compound interest is the interest you earn on interest.

For example, if you have $100 and it earns 5% interest each year, you'll have $105 at the end of the first year. At the end of the second year, you'll have $110.25.

Compound interest is expressed as;

A = P( 1+r/n) ^{nt}

where A is the amount

n is the number of time its compounded

t is time

A = 15000

r = 16/100 = 0.16

n = 6/12 = 0.5

t = 10 years

15000 = P( 1 + 0.16/0.5) ^{ 0.5× 10}

15000 = P( 1 + 0.32)⁵

15000 = P( 1.32)⁵

15000 = 4P

P = 15000/4

P = Rs. 3750

Therefore the amount that will be deposited to give Rs. 15000 is Rs. 3750.

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Answer:

4,12,36,108,324.....

Step-by-step explanation:

The sequence represented by the equation [tex]a=4(3)^{n-1}[/tex]is a geometric sequence with first term 5 and common ratio 3. The first 4 terms of the sequence are:

[tex]a_1 = 4(3)^{1-1} =4(3)^0 = 4[/tex]

[tex]a_2 =4(3)^{2-1} = 4(3)^1 = 12[/tex]

[tex]a_3 =4(3)^{3-1} = 4(3)^2 = 36[/tex]

[tex]a_4 = 4(3)^{4-1} =4(3)^3 = 108[/tex]

[tex]a_5=4(3)^{5-1} =4(3)^4=324[/tex]

The sequence can be written as:

[tex]a_n = 4(3)^{n-1}[/tex]

where n is the term number.

Therefore, the sequence is 4,12,36,108,324.....

Answer:

The answer is 4,12,36,108,324

Step-by-step Explanation:

a1=4(3)¹‐¹=4(3)⁰=4

a2=4(3)²‐¹=4×3=12

a3=4(3)³‐¹=4(3)²=4×9=36

a4=4(3)⁴‐¹=4(3)³=4×27=108

a5=4(3)⁵‐¹=4(3)⁴=4×81=324

Answer:

B, C

Step-by-step explanation:

x^2 = 5x - 6

x = 2 or x = 3

The measure of angle ABC is given as follows:

m < ABC = 55º.

First we look to the exterior angle of angle C in the bottom parallel line, as a ray is formed, hence the missing angle measure is given as follows:

50 + 75 + x = 180

125 + x = 180

x = 55º.

The exterior angle relative to angle B is supplementary with the angle of 55º, as they are consecutive angles, hence it’s measure is obtained as follows:

y + 55º = 180

y = 125º.

By the exterior angle theorem, an interior angle is supplementary with it’s respective exterior angle, hence the measure of angle ABC is given as follows:

m < ABC + 125 = 180

m < ABC = 55º.

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Answer:  The jar contains a total of 18 coins (8 pennies + 3 nickels + 5 dimes + 2 quarters). To get more than 4 cents, we need to choose a nickel, dime, or quarter. There are 3 nickels, 5 dimes, and 2 quarters, for a total of 10 coins that satisfy this condition. Therefore, there are 10 ways to choose a coin from the jar that is more than 4 cents.

The exponential value equation is A = 1/√x = 1/x^(-1/2)

Given data ,

Let the exponential equation be represented as A

Now , the value of A is

A = x ^ ( 1/6 ) / x ^ ( 2/3 )

On simplifying , we get

The different Laws of exponents are:

mᵃ×mᵇ = mᵃ⁺ᵇ

mᵃ / mᵇ = mᵃ⁻ᵇ

( mᵃ )ᵇ = mᵃᵇ

mᵃ / nᵃ = ( m / n )ᵃ

m⁰ = 1

m⁻ᵃ = ( 1 / mᵃ )

So , A = x ^ ( 1/6 - 2/3 )

On further simplification , we get

A = x ^ ( -3/6 )

A = x ^ ( -1/2 )

So , the value of A is

A = 1/√x

Hence , the exponential equation is solved

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The calculated value of x in the right triangle is 12.5

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

The right triangle

The value of x can be calculated using the following cosine rule

So, we have

cos(28) = 11/x

make x the subject of the formula

So, we have

x = 11/cos(28)

Evaluate the quotient

This gives

x = 12.5

Hence, the value of x is 12.5

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Darlene's net proceeds from the investment into shares would be $ 8, 279 .

First, find the initial broker fee:

=  Initial investment × Broker fee rate

= 20, 000 x 1. 5 %

= 20, 000 x 0. 15

= $ 300

The total fees incurred to buy and sell are:

= 300 initial broker + 21 later broker

= $ 321

The net proceeds would then be:

= Final value of stock - Initial investment - Total fees

= 28, 600 - 20, 000 - 321

= 8, 600 - 321

= $ 8, 279

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Full question is:

Darlene purchases $20,000 worth of stock on her brokers advice and pays her broker a 1.5% broker fee. She sells her stock when it increases to $ 28,600  two years later, and uses a discount broker who charges $21 per trade. Compute Darlene's net proceeds after the broker fees are taken out.

Answer:

x = 24.5

Step-by-step explanation:

27/(x + 7) = (27 - 6)/x

27x = (x + 7)(21)

27x = 21x + 147

6x = 147

x = 24.5

Answer:

x = 31.5

Step-by-step explanation:

given a line parallel to a side of a triangle and it intersects the other two sides then it divides those sides proportionally , that is

[tex]\frac{6}{27-6}[/tex] = [tex]\frac{7}{x}[/tex]

[tex]\frac{6}{21}[/tex] = [tex]\frac{7}{x}[/tex] ( cross- multiply )

6x = 21 × 7 = 147 ( divide both sides by 6 )

x = 24.5

Step-by-step explanation:

limit for n going to what ?

I think you mean n going to ±infinity.

for very large n (when n goes to infinity) such a fraction is reduced to the ratio of the terms with the highest exponents of n. the terms with lower exponents simply get less and less impact with growing n, and fade away for really, really large n going to infinity.

so, this limit evaluation is reduced to

lim n²/(3n³ + n²) = 1/(3n + 1)

which is again for very large n

lim 1/(3n), which goes to 1/infinity = 0

the same for going to 1/-infinity = 0

if you meant e.g. limit for n going to 0, then for the limit we set n to 0 and see, that all terms are turning 0, except for the constants -5 in the numerator and +7 in the denominator.

so, that limit is then

-5/7 = -0.714285714...

The total cost of materials needed to build the house is $4374.75.

Calculate the area of the front and back of the house.

Area = Length x Width

Area = 3.5 m x 2.5 m = 8.75 m²

Calculate the area of the sides of the house.

Area = 2 x Length x Height

Area = 2 x 3.5 m x 5 m = 35 m²

Calculate the total cost of glass.

Cost of glass = Area x Cost per square meter

Cost of glass = 8.75 m² x $37 / m² = $323.75

Calculate the total cost of roof tiles.

Cost of roof tiles = Area x Cost per square meter

Cost of roof tiles = 35 m² x $95 / m² = $3325

Calculate the total cost of reinforcement metal.

Cost of reinforcement metal = Perimeter x Cost per meter

Cost of reinforcement metal = 2 x (Length + Width) x $54 / m = $726

Add the costs of glass, roof tiles, and reinforcement metal to get the total cost of materials.

Total cost = $323.75 + $3325 + $726 = $4374.75

Therefore, the total cost of materials needed to build the house is $4374.75.

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The unknown length of the similar figure is:

x = 15 units

Two figures are similar if they have the same shape but different sizes. The corresponding angles are equal and the ratios of their corresponding sides are also equal.

Using the above concept, we can equate the ratio of the corresponding sides and solve for the unknown lengths. That is:

TV/AC = TU/AB

x/10 = 6/4

x/10 = 1.5

x = 10 * 1.5

x = 15 units

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Based on the analysis, the correct transformation that maps triangle △UVW to triangle △EFG is option C: (x, y) ⟶ (2x, 2y), which represents an enlargement or dilation of the figure.

To determine which transformation maps triangle △UVW to triangle △EFG, we need to analyze the given options and understand the effects of each transformation.

A. (x, y) ⟶ (0.5x, 0.5y):

This transformation scales the coordinates of a point by a factor of 0.5 in both the x and y directions.

It represents a reduction or contraction of the figure.

Triangle △UVW would be scaled down to a smaller size, so option A is not the correct transformation.

B. (x, y) ⟶ (–0.5x, 0.5y):  

This transformation reflects the coordinates of a point across the y-axis and scales the x-coordinate by a factor of -0.5 and the y-coordinate by a factor of 0.5.

It represents a reflection and scaling.

Triangle △UVW would be reflected across the y-axis and scaled, which does not match the transformation to △EFG.

Therefore, option B is not the correct transformation.

C. (x, y) ⟶ (2x, 2y): This transformation scales the coordinates of a point by a factor of 2 in both the x and y directions.

It represents an enlargement or dilation of the figure.

Triangle △UVW would be scaled up to a larger size, which aligns with the transformation to △EFG.

Therefore, option C could be the correct transformation.

D. (x, y) ⟶ (–2x, 2y):

This transformation reflects the coordinates of a point across the x-axis and scales the x-coordinate by a factor of -2 and the y-coordinate by a factor of 2.

It represents a reflection and scaling.

Triangle △UVW would be reflected across the x-axis and scaled, which does not match the transformation to △EFG.

Thus, option D is not the correct transformation.

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The surface area of the rectangular prism is 1468 square inches

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

17 in by 16 in by 14 in

The surface area of the rectangular prism is calculated as

Area = 2 * (17 * 16 + 17 * 14 + 16 * 14)

Evaluate

Area = 1468

Hence, the area is 1468 square inches

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Question

What is the surface area of this rectangular prism?

17 in. by 16 in. by 14 in.​

The calculated number of plants the flower bed can contain is 120

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

Dimensions = 16 m by 9 m

So, the area of the flower bed is

Area = 16 * 9

Evaluate

Area =  144

Also, we have

Each plant requires a space of 1.2m x 1m?

This means that

Plant area = 1.2 * 1

Plant area = 1.2

So, we have

Plants = 144/1.2

Evaluate

Plants = 120

Hence, the number of plants is 120

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Answer:

Step-by-step explanation:

To calculate the number of plants that can be planted in the rectangular flower bed, we need to calculate the area of the flower bed and divide it by the space required for each plant.

The area of the flower bed is calculated by multiplying its length and breadth.

So, the area of the flower bed is 16m x 9m = 144 sq.m.

Each plant requires a space of 1.2m x 1m = 1.2 sq.m.

Therefore, the number of plants that can be planted in the flower bed is:

144 sq.m. ÷ 1.2 sq.m./plant = 120 plants.

So, you can plant 120 plants in the rectangular flower bed.

The graph of the given function is attached.

Given is a function for the rainfall in 5 months in inches.

We need to display the data,

So, as we can see that the data is not showing any proportion or pattern,

So, it can be displayed as a line chart.

Hence the chart is attached for the function.

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The properties of equality that shows justification of how the equation is solved is explained below.

Given the equation, 17/3 - 3/4x = 1/2x + 5, we have the following steps and justification which explains the property of equality that was used:

17/3 - 3/4x = 1/2x + 5 [given]

17/3 - 3/4x - 17/3 = 1/2x + 5 - 17/3 [subtraction property of equality]

3/4x = 1/2x - 2/3 [simplification]

3/4x - 1/2x = 1/2x - 2/3 - 1/2x [subtraction property of equality]

-5/4x = -2/3 [simplification]

-5/4x * -4/5 = -2/3 * -4/5 [multiplication property of equality]

x = 8/15 [simplification]

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The length of the hypotenuse of the given right triangle is 117 millimeters because it is the longest side and it satisfies the conditions of the Pythagorean theorem.

The length of the hypotenuse of a right triangle with side lengths 45 millimeters, 108 millimeters, and 117 millimeters is 117 millimeters.

The hypotenuse of a right triangle is always the side opposite the right angle, and it is the longest side in a right triangle.

In this case, the side lengths are given as 45 millimeters, 108 millimeters, and 117 millimeters.

By comparing the lengths, we can see that 117 millimeters is the longest side, which makes it the hypotenuse.

To further understand why the hypotenuse is the longest side, we can refer to the Pythagorean theorem.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.

Mathematically, it can be written as c² = a² + b²,

where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.

In this case, if we square the lengths of the sides, we get 45² = 2025, 108² = 11664, and 117² = 13689.

By comparing these values, we can see that 13689 is the largest, which corresponds to the length of the hypotenuse, 117 millimeters.

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Answer:

1) 5m

2) 0.25tan

Step-by-step explanation:

Answer:

25

Step-by-step explanation:

There are only additions, so do each addition, in the order it appears, from left to right.

-11 + 3 + 10 + (-25) + 48 =

= -8 + 10 + (-25) + 48

= 2 + (-25) + 48

= -23 + 48

= 25

For this problem, you need an on-hand calculator.

Problem:

sin (pi/8)

Step 1: divide pi by 8.

[tex]\pi[/tex]/8

=0.39269908169

Step 2: Find Cosine of (0.39269908169)

This is the part where you need a calculator.

Locate the SIN button on your graphing calculator.

Press the SIN button, then continue to type 0.39269908169 after the parenthesis.

The calculator gives a value of .0068538383

Final answer:

0.068538383

The values of the function are 5 and 1.5 and the domain is,

⇒ - ∞ < x < ∞

Here, The equation of the function is given as

f(x) = 5 (1.3)ˣ

The above function is an exponential function.

An exponential function is represented as

f(x) = abˣ

Where

a represents the initial value, b represents the rate of growth or decay

While x and f(x) are the input and output values

Hence, By comparison, we have

a = 5 and b = 1.3

This means that the function values are 5 and 1.3.

Since, the function is an exponential function.

The domain of an exponential function is the set of all real values

Hence, the domain of f(x) is the set of all real values.

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The solution of the inequality equation is determined as y > 3.

A solution for an inequality in y is a number such that when we substitute that number for y we have a true statement.

For the given inequality equation, which is;

y + 5 > 8

The solution of the inequality equation is calculated as follows;

y + 5 > 8

Subtract 5 from both sides of the equation as shown below;

y + 5 - 5 > 8 - 5

y > 3

The solution of the inequality equation is determined as 3, so we can conclude that any value less than  3 is not true and the proposed solution of 1 is not true or correct for the inequality equation.

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