you buy an nft hoping it will increase in value at a compounded rate of 8% each year. if you pay $500 for the nft now. how much do you expect it to be worth in 20 years?

The height of the formation is [tex]31meter[/tex], to the nearest meter.

Height, altitude, and elevation all refer to the vertical distance between something's top and bottom or its base but something above it. Height is used to describe everything that may be measured vertically, high or low.

In essence, height relates to elevation, which is the height above a certain level. Furthermore, height refers to any quantifiable distance above a particular level as well as extend upwards (as it is from foot to head). For instance, the elevation of a tower, tree, person, mountain, etc.

We can use the tangent function to calculate the formation's height.

[tex]tan(36^{0} )=\frac{h}{3}[/tex]

Here, we have to multiply both sides by [tex]43[/tex] we get,

[tex]43tan(36^{0} )=h[/tex]

[tex]h=31meter[/tex]

Therefore, the height of the formation to the nearest meter is [tex]31meter[/tex].

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The required mean of the given frequency table is 6.

There are various mean types in mathematics, particularly in statistics.

Each mean helps to summarize a certain set of data, frequently to help determine the overall significance of a specific data set.

For instance, the mean for the collection of values 8, 9, 5, 6, and 7 is 7, since 8 + 9 + 5 + 6 + 7 = 35, and 35/5 = 7.

So, to find the mean:

First, arrange the terms in ascending order as follows:

1, 4, 8, 13

Now, add the middle two numbers and divide by 2 to get the mean:

= 4 + 8/2

= 12/2

= 6

Therefore, the required mean of the given frequency table is 6.

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Answer:

Step-by-step explanation:

665,280 dived by 3 = 221760

221760 = distance

221760 = per 1 hour

I tried my best :}

Zachary will need to earn a score of 23 on his sixth quiz to have a mean score of 22 for all six quizzes.

To find out what score Zachary needs to earn on his sixth quiz to have a mean score of 22,

We need to use the formula for the mean (or average) of a set of numbers:

Mean = (sum of all the scores) / (number of tests)

Zachary's mean score for the first five quizzes is 22.

Using the above as a guide, we have the following:

(20 + 25 + 22 + 18 + 24 + x) / 6 = 22

Evaluate the like terms

(109 + x) / 6 = 22

So, we have

109 + x = 132

Evaluate the like terms

x = 23

Hence, the score must be 23

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Answer:

2. f(t) = 25^(t+1)

Step-by-step explanation:

25 = 25^1

=> 25^(0+1)

15,625 = 25^3

=> 25^(2+1)

9,765,625 = 25^5

=> 25^(4+1)

So f(t) = 25^(t+1)

1 & -3 & 2
2 & 0 & -1

The cross product of two vectors can be calculated using the formula:
\[ \vec{u} \times \vec{v} = \left[ \begin{array}{ccc}
\vec{i} & \vec{j} & \vec{k}\\
u_{1} & u_{2} & u_{3} \\
v_{1} & v_{2} & v_{3}
\end{array} \right] \]

For your given vectors, we can calculate the cross product as follows:
\[ \vec{u} \times \vec{v} = \left[ \begin{array}{ccc}
\vec{i} & \vec{j} & \vec{k}\\
1 & -3 & 2 \\
2 & 0 & -1
\end{array} \right] \]

Which simplifies to:
\[ \vec{u} \times \vec{v} = \left[ \begin{array}{c} 6 \\ 2 \\ -1 \end{array} \right] \]

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Answer:

Scarlett paid $48.51

Step-by-step explanation:

The initial price of the shoes was $49. After using a 10% discount coupon, the price is reduced by $4.90 ($49 x 0.1). Therefore, the price after the discount is $44.10 ($49 - $4.90).

Then, a 10% processing fee is applied to this discounted price. This fee amounts to $4.41 ($44.10 x 0.1).

So the final amount Scarlett paid is the sum of the discounted price and the processing fee:

$44.10 + $4.41 = $48.51

Therefore, Scarlett paid $48.51 for the shoes in the end.

Step-by-step explanation:

If triangle JKL is dilated by a scale factor of 2, this means that all sides of triangle JKL are multiplied by 2 to form triangle J'K'L'.

Therefore, if the measure of side LJ is x, then the measure of side LJ' is 2x. Similarly, if the measure of side LK is y, then the measure of side L'K' is 2y.

To find the measure of side L'J', we can use the fact that the sum of the measures of the angles in any triangle is always 180 degrees.

In triangle JKL, the sum of the measures of angles J, K, and L is 180 degrees. This means that angle JKL is:

angle JKL = 180 - angle J - angle K

Since triangle JKL is dilated by a scale factor of 2, the corresponding angles in triangle J'K'L' are congruent to the angles in triangle JKL. Therefore, angle J'K'L' is also:

angle J'K'L' = 180 - angle J - angle K

Since angle J'K'L' is a straight line (180 degrees), we can find angle J'L'K' by subtracting angles J'K'L' and J'KL':

angle J'L'K' = angle J'KL' - angle J'K'L' = angle JKL - angle J'K'L'

Using the fact that corresponding angles in similar triangles are congruent, we have:

angle J'KL' = angle JKL

angle L'K'J' = angle LKJ

Therefore, angle J'L'K' is:

angle J'L'K' = angle JKL - angle J'K'L' = angle JKL - angle LKJ

Since triangle JKL is a triangle, we know that angle JKL + angle LKJ + angle LJK = 180 degrees. Therefore, we can rewrite the equation for angle J'L'K' as:

angle J'L'K' = (angle JKL + angle LKJ + angle LJK) - 2(angle JKL + angle LKJ)

Simplifying this equation, we get:

angle J'L'K' = angle LJK - angle JKL

Using the fact that corresponding angles in similar triangles are congruent, we have:

angle L'J'K' = angle LJK

angle J'L'K' = angle JKL

Therefore, triangle J'L'K' is similar to triangle JKL, and we can write the ratio of corresponding side lengths as:

L'J' / LJ = K'L' / KL

Since triangle JKL is dilated by a scale factor of 2, we have:

K'L' = 2LK

L'J' / LJ = 2L'K' / LK

Substituting L'K' = 2LK, we get:

L'J' / LJ = 4

Therefore, the measure of side L'J' is four times the measure of side LJ. If the measure of side LJ is x, then the measure of side L'J' is 4x.

The measure of side L'J' in the dilated triangle J'K'L' can be found by multiplying the length of the corresponding side in the original triangle by the scale factor of 2.

In the field of Mathematics, specifically geometry, a dilation is a transformation that alters the size of a figure without changing its shape. If Triangle JKL is dilated by a scale factor of 2 to form Triangle J'K'L', every dimension of Triangle JKL is multiplied by this scale factor to get the corresponding dimension in Triangle J'K'L'.

Therefore, if you want to determine the measure of side L'J' in the new triangle, we need to know the length of the corresponding side in the original triangle (which is LJ). You would then multiply this length by the scale factor. The result would give you the length of side L'J'.

For example, if the length of side LJ in the original triangle is 4 units, the length of the corresponding side L'J' in the dilated triangle would be 4 * 2 = 8 units.

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The monthly payment on a 30-year, $100,000 mortgage with an interest rate of 6 percent is $599.55.

To find the monthly payment on a 30-year, $100,000 mortgage with an interest rate of 6 percent, we need to use the following formula:

Monthly Payment = (Loan Amount * Interest Rate) / (1 - (1 + Interest Rate)^(-Number of Payments))

First, we need to convert the interest rate from a percentage to a decimal by dividing it by 100:

Interest Rate = 6 / 100 = 0.06

Next, we need to calculate the number of payments over the life of the loan. Since there are 12 months in a year, and the loan is for 30 years, the number of payments is:

Number of Payments = 12 * 30 = 360

Now we can plug these values into the formula:

Monthly Payment = ($100,000 * 0.06) / (1 - (1 + 0.06)^(-360)) = $599.55

Therefore, A $100,000 mortgage with a 30 year term and a 6% interest rate has a $599.55 monthly payment.

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Answer:

If 0.25 inch represents 1 foot, then 1 inch represents 4 feet (since 4 x 0.25 = 1).

Therefore, the height of the miniature dog in inches will be:

2.5 feet x 4 = 10 inches

So, the miniature version will be 10 inches tall.

Answer:

Her dog will be 5/8 of an inch tall. You would add 1/4 and 1/4 which equals 2/4. Then, you would split 1/4  = 1 in half so, 1/8 = 0.5

Add 1/8 to 2/4 to get 5/8

Step-by-step explanation:

Answer:

The graph is a parabola that opens downward. Generally, a parabola has the following equation: y=ax2+bx+c If a is positive it opens upward

The correct answer to the listed problems above in their correct scientific notation would be :

1.) 76,450,000 =7.645×10⁷

2.) 0.000327 = 3.27 × 10^-4

3.) 34,000 = 2.4 × 10⁴

4.) 116,000,000,000, = 1.16 × 10¹¹

5.) 0.024= 2.4× 10^-2

6.) 2,800= 2.8× 10³

7.) 100= 1 × 10²

8.) 0.00000qq= q.q × 10^-6

9.) q00,000= q × 10⁵

10.) 0.00633= 6.33 × 10^-3

Scientific notation is defined as a form that can be used to represent cumbersome numbers or very small numbers in a way that is most convenient.

The correct scientific notation of the given values are listed below with reasons:

1.) 76,450,000 =7.645×10⁷ and not =7.645×10⁸ when the decimal point is moved so that there is 1 non-zero value in front of the decimal point, the number of zero is 8 not 7.

2.) 0.000327 = 3.27 × 10^-4 and not 3.27 × 10⁴ as the value is less than 1.

3.) 34,000 = 3.4 × 10⁴. Correctly written.

4.) 116,000,000,000, = 1.16 × 10¹¹. Should be 11 not q.

5.) 0.024= 2.4× 10^-2 and not 24 × 10^-2

6.) 2,800= 2.8× 10³. Correctly written.

7.) 100= 1 × 10² and not 1 × 10³

8.) 0.00000qq= q.q × 10^-6. Correctly written.

9.) q00,000= q × 10⁵ and not q × 10^-5

10.) 0.00633= 6.33 × 10^-3 and not 63.3 × 10^-3. This is because the decimal point should be placed immediate after the first whole number.

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The value of the missing sides using Pythagorean theorem are;

1.  15. 81m

2. 24. 08cm

3. 12. 85 ft

4. 4. 58 in

5. 27. 02 yd

6.  18. 14 mm

7. 16. 58 km

8. 6. 24m

9. 9. 85 miles

10. 14.14 feet

11. 14.14 feet

12. 13. 78  feet

Using the Pythagorean theorem, which states that the square of the longest side of a triangle which is the hypotenuse leg is the sum of the squares of the other two sides of the triangle, that is, the opposite and adjacent.

Then, we have;

x² = y² + z²

Where the parameters are;

From the figures shown, we have that;

1. Adjacent side = 8m

Opposite side = 9m

Using the Pythagorean theorem, we have;

x² = 13² + 9²

Find the squares and substitute

x² = 169 + 81

x² = 250

find the square root

x =√250 = 15. 81m

2. For the triangle, we have;

x² = 16² + 18²

x² = 256 + 324

x = √580 = 24. 08cm

3. For the third triangle:

y² = 19² - 14²

y² = 361 -196

y = √165 = 12. 85 ft

4. For the fourth triangle

z² = 11² - 10²

z² = 121 - 100

z = √21 = 4. 58 in

5. x² = 21² + 17²

x² = 441 + 289

x = 27. 02 yd

6. z² = 27² - 20²

z² = 729 - 400

z = √329 = 18. 14 mm

7. z² = 30² - 25²

Find the squares

z² = 900 - 625

z² = 275

z = √275 = 16. 58 km

8. y² = 8² - 5²

find the squares

y² = 64 - 25

y = √39 = 6. 24m

9. x² = 9² + 4²

Find the squares

x² = 81 + 16

x = √97 = 9. 85 miles

10. Length of the diagonal is the hypotenuse side

x² = 10² + 10²

x² = 100 + 100

x = √200 = 14.14 feet

11. The height of the pole is the opposite side

y² = 15² - 5²

y² = 225 -25

y = √200 = 14. 14 feet

12. the height of the pole is the opposite side

y² = 14² - 4²

y² = 196 - 16

y = √190 = 13. 78  feet

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Answer:

He needs 256 teaspoons of sugar.

Step-by-step explanation:

This one is really simple!
96/3 = 32, meaning that we multiplied the recipe by 32.
This means that your teaspoons of sugar must also be multiplied by 32, which 8x32=256!

The trigonometric equation [tex]\frac{2tanx}{1+tan^2x}[/tex] is equivalent to sin2x

An equation is an expression that shows how numbers and variables are related using mathematical operators like exponent, addition, multiplication, subtraction and division.

Trigonometric identity is used to show relationship between trigonometric functions. Some examples are:

1 + tan²x = 1/cos²x; tanx = cosx/sinx; sin2x = 2sinxcosx

Given:

[tex]\frac{2tanx}{1+tan^2x} \\\\but\ 1+tan^2x=\frac{1}{cos^2x};substituting:\\ \\= \frac{2tanx}{\frac{1}{cos^2x} } \\\\=2tanx*cos^2x\\\\=2*\frac{sinx}{cosx}*cos^2x\\ \\=2sinxcosx[/tex]

= sin2x

The trigonometric equation are equivalent.

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Answer: Last one

Step-by-step explanation:

Therefore, for the first  coordinate point (-x, y), the ultimate reflected points are (-x, -y) and (-x, -y), and for the second point, they are (x, -y) and (-x, -y) .

A coordinate system is a technique used to precisely find points or additional mathematical objects on a distance, such as Euclidean space, by using one or so more values or coordinates. To find a point or item on a double plane, one uses coordinates, which are pairs of numbers. Two numbers called the y and x matrices are used to describe a point's location on a two-dimensional plane. a set of numbers used to identify specific locations.

Here,

The x-coordinate remains constant and the y-coordinate is cancelled when a point on the coordinate plane is reflected over the x-axis (i.e. multiplied by -1). The y-coordinate remains constant and the x-coordinate is cancelled when a point is reflected over the y-axis.

Let's mirror the point (-2, 3) first over the x-axis and then over the y-axis as an illustration:

The x-coordinate remains constant while the y-coordinate is cancelled when reflecting over the x-axis. In light of this, the argument is (-2, -3).

The x-coordinate is cancelled while the y-coordinate remains constant when reflecting over the y-axis. In light of this, the argument is (2, -3).

We can reflect the coordinates (-x, y) and (x, -y) over the x-axis and then over the y-axis:

Therefore, for the first point (-x, y), the ultimate reflected points are

(-x, -y) and (-x, -y), and for the second point, they are (x, -y) and (-x, -y) .

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So, with one acute angle measuring 23.58 degrees and another acute Pythagorean theorem angle measuring 66.42 degrees, we may define this triangle as a right triangle.

The fundamental Euclidean geometry relationship between the three sides of a right triangle is the Pythagorean Theorem, sometimes referred to as the Pythagorean Theorem. This rule states that the areas of squares with the other two sides added together equal the area of the square with the hypotenuse side. According to the Pythagorean Theorem, the square that spans the hypotenuse (the side that is opposite the right angle) of a right triangle equals the sum of the squares that span its sides. It may also be expressed using the standard algebraic notation, a2 + b2 = c2.  

A triangle with the vertices A, B, and C with sides measuring 5, 12, and 13 units is seen in the illustration.

Let's use our triangle to illustrate this theorem:

[tex]a = 5, b = 12, c = 13\\a^2 + b^2 = 25 + 144 = 169\sc \\a^2 = 13^2 = 169\\[/tex]

Our triangle complies with the Pythagorean theorem because a2 + b2 = c2. Triangle ABC is a right triangle as a result, and the side with the length 13 is opposite the right angle.

[tex]A = arcsin(5/13)[/tex] = 23.58 degrees when sin(A)

= opposite/hypotenuse

= 5/13 A.

[tex]arccos(12/13) = 66.42 degrees B = cos(A) = adjacent/hypotenuse = 12/13[/tex]

A, B, and C are all at 90 degrees.

So, with one acute angle measuring 23.58 degrees and another acute angle measuring 66.42 degrees, we may define this triangle as a right triangle.

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Answer:

10:21 AM

Step-by-step explanation:

To get the answer you subtract 4 hours and 47 mins from 3:08 pm

4 hours and 47 minutes before 3:08 pm is 10:21 am.

An expression contains one or more terms with addition, subtraction, multiplication, and division.

We always combine the like terms in an expression when we simplify.

We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.

Example:

1 + 3x + 4y = 7 is an expression.com

3 + 4 is an expression.

2 x 4 + 6 x 7 – 9 is an expression.

33 + 77 – 88 is an expression.

We have,

To find the time 4 hours and 47 minutes before 3:08 pm, we can subtract 4 hours and 47 minutes from 3:08 pm.

First, let's convert 3:08 pm to 24-hour format:

3:08 pm = 15:08

Now we can subtract 4 hours and 47 minutes:

15:08 - 4:47 = 10:21

Therefore,

4 hours and 47 minutes before 3:08 pm is 10:21 am.

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675 cases divided by 135 cases per order equals five orders each year

a) If you order one case at a time, your yearly inventory cost is $6,750. This means that if you order one case of beans each year, it will cost you a total of $6,750.

b) To minimize your annual inventory cost, you should order 135 cases at a time. This is the number of cases that produces the minimum value for the cost equation, C = 200Q + 5,750.

c) By ordering 135 cases at a time instead of one, you will save $3,750 each year in inventory costs.

d) When you are minimizing your annual inventory cost, you should place your orders every five years. This is because 675 cases divided by 135 cases per order equals five orders each year.

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To compute the arithmetic mean of raw CPU time of the benchmark programs, we need to add the raw CPU time of the first 3 benchmark programs (P1 to P3) for each processor and divide the result by 3. Similarly, to compute the geometric mean of raw CPU time of the benchmark programs, we need to multiply the raw CPU time of the first 3 benchmark programs (P1 to P3) for each processor and take the cube root of the result. The same applies to the arithmetic mean and geometric mean of normalized CPU time of the benchmark programs. The total raw CPU time and total normalized CPU time are simply the sum of the raw CPU time and normalized CPU time of the first 3 benchmark programs (P1 to P3) for each processor, respectively.

Using a spreadsheet program, we can compute the following:

(i) Arithmetic mean of raw CPU time of the benchmark programs:
- Proc. A: (300 + 1200 + 600) / 3 = 700
- Proc. B: (600 + 300 + 1200) / 3 = 700
- Proc. C: (1200 + 600 + 300) / 3 = 700

(ii) Geometric mean of raw CPU time of the benchmark programs:
- Proc. A: (300 * 1200 * 600)^(1/3) = 600
- Proc. B: (600 * 300 * 1200)^(1/3) = 600
- Proc. C: (1200 * 600 * 300)^(1/3) = 600

(iii) Arithmetic mean of normalized CPU time of the benchmark programs:
- Proc. A: (1 + 1 + 1) / 3 = 1
- Proc. B: (2 + 0.25 + 2) / 3 = 1.4167
- Proc. C: (4 + 0.5 + 0.5) / 3 = 1.6667

(iv) Geometric mean of normalized CPU time of the benchmark programs:
- Proc. A: (1 * 1 * 1)^(1/3) = 1
- Proc. B: (2 * 0.25 * 2)^(1/3) = 1
- Proc. C: (4 * 0.5 * 0.5)^(1/3) = 1

(v) Total raw CPU time:
- Proc. A: 300 + 1200 + 600 = 2100
- Proc. B: 600 + 300 + 1200 = 2100
- Proc. C: 1200 + 600 + 300 = 2100

(vi) Total normalized CPU time:
- Proc. A: 1 + 1 + 1 = 3
- Proc. B: 2 + 0.25 + 2 = 4.25
- Proc. C: 4 + 0.5 + 0.5 = 5

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Jack is 13 years old now.

To solve this problem, we can use algebra to set up an equation and solve for Jack's age. Let's let J represent Jack's age now, and A represent Anna's age now.

According to the problem, Jack is ten years older than Anna, so we can write:
J = A + 10

Seven years from now, Jack will be twice as old as Anna, so we can write:
J + 7 = 2(A + 7)

Now we can substitute the first equation into the second equation to solve for A:
A + 10 + 7 = 2(A + 7)
A + 17 = 2A + 14
A = 3

Now that we know Anna's age, we can plug it back into the first equation to find Jack's age:
J = 3 + 10
J = 13

So Jack is 13 years old now.

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The value of a₄ is,

⇒ a₄ = 108

To multiply means to add a number to itself a particular number of times. Multiplication can be viewed as a process of repeated addition.

Given that;

The value is,

⇒ a₁ = - 4

And, [tex]a_{n} = - 3a_{n - 1}[/tex]

Now, We can put n = 2;

⇒ a₂ = - 3 a₁

⇒ a₂ = - 3 x - 4

⇒ a₂ = 12

Put n = 3;

⇒ a₃ = - 3 a₂

⇒ a₃ = - 3 x 12

⇒ a₃ = - 36

Put n = 4;

⇒ a₄ = - 3 a₃

⇒ a₄ = - 3 x - 36

⇒ a₄ = 108

Thus, The value of a₄ is,

⇒ a₄ = 108

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Value of a₄ in sequence is -108.

In mathematics, geometric sequence, also known as a a geometric progression, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

Given,

a₁ = 4 and aₙ = -3aₙ₋₁

a₂ = -3a₂₋₁

a₂ = -3a₁

a₂ = -3(4)

a₂ = -12

a₃ =  -3(-12)

a₃ = 36

a₄ = -3a₄₋₁

a₄ = -3a₃

a₄ = -3(36)

a₄ = -108

Hence, -108 is value of a₄ of the sequence.

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Therefore, Roberto needs 26 more inches of lumber. Answer: 26 inches.

an explanation of the relationship between the two phrases on either side of a sign. A single variable and an equal sign are typically present. It is equivalent to this 2. 2x - 4 Equals 2. The variable x is present in the earlier instance.

To solve this problem, we need to convert all the measurements to the same units. Let's convert 6 feet and 8 inches to inches:

6 feet = 6 x 12 inches = 72 inches

6 feet and 8 inches = 72 + 8 = 80 inches

So Roberto needs 80 inches of lumber to repair the deck.

He has 54 inches of lumber, so he needs:

80 - 54 = 26 inches

Therefore, Roberto needs 26 more inches of lumber. Answer: 26 inches.

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The total amount for an  of $5,000 over 5 years at an annual interest rate of 6% if the interest is compounded quarterly is $5386.42.

Interest is the amount a lender or financial institution receives for lending money. Interest can also refer to a shareholder's ownership interest in a company and is usually expressed as a percentage.

Interest rate indicates the cost of borrowing or the reward for saving. Therefore, if you are a borrower, interest rate is the amount charged to borrow money, expressed as a percentage of the total loan amount.  

Solution according to the information given in the question :

Given, Principal(P) = $5,000

           Compounding per year(n) = 4

          Time(in years)(T) = 5

          Interest rate = 6%

Amount = P× (1 + r/n)^t

             = 5000 × (1 + 6/4×100)⁵

            = 5000 × (1.015)⁵

            = 5000 ×1.07

            = $5286.42

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Answer:

277

Step-by-step explanation:

Just add lol

The final polynomial after simplifying 3m^(7)-9m^(5)+2m^(7)-4m^(6)  is   5m⁷-9m⁵-4m⁶.

To simplify the given polynomial, we need to combine like terms. Like terms are terms that have the same variable and exponent.

The given polynomial is: 3m⁷-9m⁵+2m⁷-4m⁶

First, let's identify the like terms:
- 3m⁷ and 2m⁷ are like terms
- -9m⁵ and -4m⁶ are not like terms because they have different exponents

Next, let's combine the like terms:
- 3m⁷ + 2m⁷ = 5m⁷

So, the simplified polynomial is: 5m⁷-9m⁵-4m⁶

This polynomial cannot be further simplified, so it is none of the options listed (binomial, trinomial, or none of these).

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The values of x in the given equation, (x-7)^2=36, are x = 13 and x = 1. The correct options are the first and second options - x = 13 and x = 1

From the question, we are to determine the values of x in the given equation. The given equation is:

(x-7)^2=36

To solve the equation (x-7)^2=36, we can take the square root of both sides and solve for x:

(x-7)^2 = 36

Taking the square root of both sides:

x - 7 = ±6

Adding 7 to both sides:

x = 7 ± 6

So we get two possible values of x:

x = 7 + 6 or x = 7 - 6

x = 13 or x = 1

Hence, the correct values of x are:

x = 13

x = 1

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To find a plane containing a point P not on a given line l and parallel to l, we can follow these steps:

Choose any point Q on the line l.

Find the vector v that connects point Q to point P, i.e., v = P - Q.

Find the direction vector d of the line l.

Take the cross product of the direction vector d and the vector v, denoted by d x v.

The plane that contains point P and is parallel to line l can be represented by the equation ax + by + cz = d, where (a, b, c) is the vector d x v, and x, y, and z are the coordinates of any point on the plane.

Here's an example to illustrate the process:

Let's say we have a line l that passes through points A(1, 2, 3) and B(4, 5, 6), and a point P(7, 8, 9) not on the line. We want to find a plane containing point P and parallel to line l.

Choose any point Q on the line l. Let's choose Q(1, 2, 3) for convenience.

Find the vector v that connects point Q to point P: v = P - Q = (7-1, 8-2, 9-3) = (6, 6, 6).

Find the direction vector d of the line l: d = (4-1, 5-2, 6-3) = (3, 3, 3).

Take the cross product of the direction vector d and the vector v: d x v = <3, 3, 3> x <6, 6, 6> = <0, 18, -18>.

Write the equation of the plane: 0x + 18y - 18z = d. Since the plane is parallel to line l, we know that the coefficients of x, y, and z are all 0, except for the coefficients of y and z, which are the components of the vector d x v. Therefore, the equation simplifies to 18y - 18z = 0, or y - z = 0.

So the plane containing point P(7, 8, 9) and parallel to the line passing through points A(1, 2, 3) and B(4, 5, 6) can be represented by the equation y - z = 0.

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