y is directly proportional to the square of x. find the percentage increase when x is increased by 5%

The area of the surface generated when the given curve is revolved about the given axis 5x^1/3 is [tex]\frac{\pi }{675} [34\sqrt{34} - 125][/tex].

The curve x = f(y)

The area of the surface around the y-axis from y = a → y = b is:

[tex]\int\limits^a_b {2\pi x\sqrt{1+(\frac{dx}{dy})^2 } } \, dx[/tex]

From the given curve:

[tex]y = 5x^{1/3}[/tex] ; assuming the region bounded by the curve is 0 ≤ y ≤ 1

So;

    [tex]y = 5x^{1/3}[/tex]

⇒ 5x = y³

⇒ x = 1/5 y³

The differential of the above equation Is:

   dx/dy = 1/5 (3y²)

⇒ dx/dy = 3/5 (y²)

Now, we have the area of the surface produced around the curve  [tex]y = 5x^{1/3}[/tex] through the y axis from the region y = 0 to y = 1

∴ [tex]\int\limits^1_0 {2\pi \frac{1}{5}y^3\sqrt{1+(\frac{3}{5} y^2)^2} } \, dx[/tex]

= [tex]\frac{2\pi }{5} \int\limits^1_0 {y^3\sqrt{1+(\frac{9}{25} y^4)} } \, dy[/tex]

= [tex]\frac{2\pi }{5} \int\limits^1_0 {y^3\sqrt{\frac{25+9y^4}{25})} } \, dy[/tex]

= [tex]\frac{2\pi }{5} \int\limits^1_0 {y^3\sqrt{25+9} y^{4} )} } \, dy[/tex]

Let make u = [tex]\sqrt{25+9y^{4} }[/tex]

It implies that:

[tex]u^3 = (25+9y^4) \sqrt{25+9y^4}[/tex]

and, [tex]u = \sqrt{25+9y^{4} }[/tex]

⇒ [tex]du = \frac{1}{2\sqrt{25+9y^4} } (36y^3) dy[/tex]

⇒ [tex]y^3dy = \frac{1}{18} \sqrt{25+9y^{4} }du[/tex]

when y = 0 ;

[tex]u = \sqrt{25+9y^{4} }[/tex]

[tex]u = \sqrt{25+9{(0)^4} }[/tex]

[tex]u = \sqrt{25 }[/tex]

u = 5

when y = 1;

[tex]u = \sqrt{25+9{(1)^4} }[/tex]

[tex]u = \sqrt{34} }[/tex]

∴The equation [tex]\frac{2\pi }{5} \int\limits^1_0 {y^3\sqrt{25+9} y^{4} )} } \, dy[/tex]  can be written as:

= [tex]\frac{\pi }{225} [\frac{u^3}{3} ]\left \{ {{y=\sqrt{34} } \atop {x=5}} \right.[/tex]

= [tex]\frac{\pi }{675} [34\sqrt{34} - 125][/tex]

Therefore,

The area of the surface generated when the given curve is revolved about the given axis 5x^1/3 is [tex]\frac{\pi }{675} [34\sqrt{34} - 125][/tex].

Complete Question:

Find The area of the surface generated when the given curve is revolved about the given axis 5x^1/3.

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The value of 85g increased by 120% is equal to 187g.

In mathematics, a value or ratio that can be expressed as a fraction of 100 is known as a percentage. When determining a percentage of a number, we should first divide it by 100 before multiplying the result by 100. As a result, the proportion refers to a component per 100. The word "percent" denotes a percentage. The letter "%" stands for it.

Now in the given question ,

To increase 85g by 120%, we need to find 120% of 85g and add it to 85g.

To find 120% of 85g, we can first convert 120% to a decimal by dividing by 100:

120% = 120/100 = 1.2

Then we can multiply 1.2 by 85g:

1.2 x 85g = 102g

So, increasing 85g by 120% gives us:

85g + 102g = 187g

Therefore, 85g increased by 120% is equal to 187g.

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Answer:

X=7

Y=6

Step-by-step explanation:

Divide the 1st equation by 2.

2x+3y=32

2x-y=8

When you subtract those 2 numbers, you get that 4y=24.

Y=6

32-6*3=14=2x

X=7

Answer:

y = x + 8  (second option)

Step-by-step explanation:

If Chris's age is x , then Stephanie who is 8 years older should be x + 8

y = x + 8

a) The square root of complex number, 2i is equals to the ±( 1 + i).

b) The square root of complex number, (1 - √3i) is equals to the ±1/√2(√3 - 1). The rectangular coordinates plane of both present in above figure.

Square root of a number is defined as a value which on multiplication by itself, results the original number. If p is the square root of q then it is denoted as p=√q. We have to determine the square root of complex numbers and express them in rectangular coordinates. For a nonzero complex number z, its square roots are √z =√r exp( i(2πk + θ)/2), k = 0,1

(a) The magnitude of 2i is r =√(2² + 0) = 2, and the principal argument is θ = π/2. So, (2i)½ = r½ exp(i (2πk + π/2)/2) , k = 0, 1

For first root, substuting, k = 0

=> (2i)½ = 2½ exp(i (2π×0 + π/2)/2)

=> (2i)½ = √2 exp(i (π/2)/2)

=> (2i)½ = √2 exp(i π/4) = √2(cos(π/4) + i sin(π/4)

=> (2i)½ = √2( 1/√2 + i(1 /√2)) = 1 + i

For second root substuting, k = 1

=> (2i)½ = 2½ exp(i (2π×1 + π/2)/2)

=> (2i)½ = √2 exp(i (5π/2)/2)

=> (2i)½ = √2 exp(i (5π/4) = √2( cos(5π/4) + i sin(5π/4)

=> (2i)½ = √2(- 1/√2 + i(-1 /√2)) = - (1 + i)

Hence, square root of 2i is ±( 1 + i).

b) 1 -√3i

The magnitude and principal argument of 1 −√3i are respectively, r = √(1² + (√3)²) = 2 and θ = tan⁻¹ ( -√3/1) = - π/3. Similar to part(a), square root of (1 - √3i) is written as (1 - √3i)½ = r½ exp(i (2πk - π/3)/2), k= 0, 1

For first root , k = 0

(1 - √3i)½ = √2 exp(i (2π×0 - π/3)/2)

=> (1 - √3i)½ = √2 exp(i (- π/6) )

=> (1 - √3i)½ = √2( cos(-π/6) - i sin(π/6))

=> (1 - √3i)½ = √2( cos(π/6) - i sin(π/6))

=> (1 - √3i)½ = √2( √3/2 - i (1/2)) = 1/√2(√3 - 1)

For second root, k = 1

(1 - √3i)½ = √2 exp(i (2π×1 - π/3)/2)

=> (1 - √3i)½ = √2 exp(i (5π/3)/2)

=> (1 - √3i)½ = √2 exp(i (5π/6))

=> (1 - √3i)½ = √2 (cos(5π/6) + i sin(5π/6))

=> (1 - √3i)½ = √2 (- √3/2 + i(1/2) ) = - 1/√2(√3 - i)

so, (√1- √3i ) is ± 1/√2(√3 - 1). Hence, we got all required square roots.

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Complete question:

1. Find the square roots of (a) 2i; (b) 1 - √3i and express them in rectangular coordinates.

The manager's claim that the new stadium can seat exactly 45% more people than previously is not correct. Instead, the new stadium can accommodate 145% of the previous number of seats, which is an increase of 45% over the original capacity

If a football club increases the number of seats in its stadium by 45%, it means that the new stadium can accommodate 145% of the previous number of seats.

Suppose the previous capacity of the stadium was x seats. Then, the new capacity of the stadium can be calculated by multiplying the previous capacity by 1.45 (which represents the 45% increase in capacity):

New capacity = x × 1.45 = 1.45x

However, the manager of the club claims that the new capacity of the stadium is exactly 45% more than the previous capacity, which means that the new capacity should be:

New capacity = x + (0.45x) = 1.45x

This equation suggests that the increase in capacity is not 45%, but rather an increase of the original capacity by 45%. This is not the same as the increase of 45% over the original capacity.

Therefore, the manager's claim that the new stadium can seat exactly 45% more people than previously is not correct. Instead, the new stadium can accommodate 145% of the previous number of seats, which is an increase of 45% over the original capacity.

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Answer: 5, 6, 7, 8, 9, or the numbers 5-9 inclusive.

Step-by-step explanation: The Triangle Inequality Conjecture states that any two sides of a triangle added up will be larger than the third. To easily prove this, you can add the two smaller sides to see if it is bigger than the larger side. Using this conjecture, we can calculate that 5 works, 6 works, 7 works, 8 works (8 is larger than 7 so now we do 3+7), and 9 works.

The mean of the data is 16.5 and the standard deviation is 7.28

The given data set is a small sample of 12 observations.

To determine the type of distribution, we can first create a histogram or a boxplot of the data.

A histogram of the data shows that the distribution is unimodal and slightly right-skewed.

Alternatively, we can calculate the skewness of the data. If the skewness is close to zero, then the data is approximately symmetric. If the skewness is positive, then the data is right-skewed. If the skewness is negative, then the data is left-skewed.

Calculating the skewness of the data set, we get:

skewness = (n / ((n - 1) * (n - 2))) * Sum[(xi - x-bar)^3 / s^3]

where n is the sample size, x-bar is the sample mean, s is the sample standard deviation, and Sum is the sum of the values in the data set.

Using this formula, we get a skewness of approximately 0.456, which indicates that the distribution is slightly right-skewed.

Based on these findings, we can conclude that the distribution of the data set is approximately normal, but slightly right-skewed.

To find the best measure of center and spread, we can calculate the sample mean and sample standard deviation, respectively.

Sample mean:

mean = (1 + 7 + 11 + 14 + 17 + 17 + 17 + 21 + 21 + 23 + 23 + 26) / 12 = 16.5

Sample standard deviation:

s = sqrt((1/11) * [(1 - 16.5)^2 + (7 - 16.5)^2 + ... + (26 - 16.5)^2]) = 7.28

Therefore, the best measure of center for this data set is the sample mean of 16.5, and the best measure of spread is the sample standard deviation of 7.28.

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The formula for calculating the z-score is z = (x-μ)/σ.

where x is the raw score, μ is the population mean, and σ is the population standard deviation. As the formula suggests, the Z-score is simply the raw score minus the population mean divided by the population standard deviation.

A positive Z-score indicates that the raw score is higher than the average. For example, a z-score of +1 is one standard deviation above the mean.

A negative z-score indicates that the raw score is below average. For example, a Z-score of -2 is two standard deviations below the mean.

Another way to interpret z-scores is to create a standard normal distribution (also known as the z-score distribution or probability distribution).

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Answer:

Step-by-step explanation:

Answer: 11

Step-by-step explanation: Just took the test

The length of the apothem of the pentagon is approximately 11 meters.

Polygon is defined as a geometric shape that is composed of 3 or more sides these sides are equal in length, and an equal measure of angle at the vertex,

Examples of polygons, equilateral triangles, squares, pentagons etc.

Here,
To solve this problem, we can use the formula for the area of a regular pentagon:

A = (5/2) × s × a

Where A is the area of the pentagon, s is the length of one side, and a is the apothem (the distance from the center of the pentagon to the midpoint of a side).

We know that the area of the pentagon is 118.25 square meters and the length of one side is 4.3 meters. We can substitute these values into the formula and solve for the apothem:

118.25 = (5/2) × 4.3 × a

Divide both sides by (5/2) x 4.3:

a = 118.25 / ((5/2) × 4.3)
  = 118.25 / 10.75 ≈ 11 meters

Therefore, the length of the apothem of the pentagon is approximately 11 meters.

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Answer:

3) Likely

Step-by-step explanation:

75% chance

After all these transformations, point A' will lie on the left side of the y-axis with the same y-coordinate as A, and point A''' will lie on the right side of the y-axis with the same x-coordinate as A.

What is parallelogram ?

A parallelogram is a quadrilateral with two pairs of parallel sides.

If parallelogram ABCD is reflected over the y-axis, all the x-coordinates of the vertices will change sign while the y-coordinates remain the same. This means that A will be reflected to a point A' on the left side of the y-axis with the same y-coordinate as A.

If the reflected parallelogram is then reflected over the x-axis, all the y-coordinates of the vertices will change sign while the x-coordinates remain the same. This means that A' will be reflected to a point A'' below the x-axis with the same x-coordinate as A'.

Finally, if the reflected and flipped parallelogram is rotated 180 degrees, all points will be reflected over the origin. This means that A'' will be reflected to a point A''' on the right side of the y-axis with the same x-coordinate as A'' (which is the same as A).

Therefore, after all these transformations, point A' will lie on the left side of the y-axis with the same y-coordinate as A, and point A''' will lie on the right side of the y-axis with the same x-coordinate as A.

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Therefore , the solution of the given problem of area comes out to be

a new bedroom door would cost $142.52 in total.

Its surface area serves as a fair guide of its total size. The environment is considered when calculating the right triangle squared. Everything's overall size is determined by its surface area. The volume of water inside a cuboid is equal to the sum of the contents from each of its twelve vertical sides. To determine the box's dimensions, use the formula below: 2lh, 2lw, and 2hW symbolise the area (SA). The full surface area of the flexible shape serves as the area's representation.

Here,

The area of the door is:

Area = height x width

Area = 7 ft x 4 ft

Area = 28 sq ft

The cost per square foot is $5.09, so the total cost of the new door would be:

Cost = Area x cost per square foot

Cost = 28 sq ft x $5.09/sq ft

Cost = $142.52

Therefore, a new bedroom door would cost $142.52 in total.

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Alternate interior angles - Congruent

Corresponding angles - Neither

Same side interior angles - Supplementary

Alternate angles are a type of angle formed when a straight line intersects two parallel lines. When this happens, alternate angles are formed on opposite sides of the transversal (the line that intersects the parallel lines).

Alternate angles are also known as "alternate interior angles" because they are located inside the parallel lines, but on opposite sides of the transversal. They are called "alternate" because they are located on opposite sides of the transversal and they are congruent, meaning they have the same angle measure.

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The conversion value of pounds to kilogram as per requirement is given by 141 pounds =  ( 63.9576 ) kilograms.

One kilograms is equivalent to 2.20462 pounds.

This implies ,

⇒ 2.20462 pounds = 1 kilograms

Divide both the side by 2.20462 we get,

⇒( 2.20462 / 2.20462 ) pounds = ( 1 / 2.20462 ) kilograms

⇒  1 pounds = ( 0.4536 ) kilograms

Now, Multiply both the side by 141  we get,

⇒ 141 pounds = ( 141 × 0.4536 ) kilograms

⇒ 141 pounds =  ( 63.9576 ) kilograms

Therefore, the conversion factor of kilogram to pounds is 2.20462 this implies 141 pounds =  ( 63.9576 ) kilograms .

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The probability of getting at least one head in two flips of a coin is 3/4 or 0.75, which means that there is a 75% chance of getting at least one head in two flips of a coin.

To find the probability of at least one head in two flips of a coin, we can use the complement rule. The complement of the event "at least one head" is "no heads."

The probability of getting no heads in two flips of a coin is (1/2) x (1/2) = 1/4.

Therefore, the probability of getting at least one head in two flips of a coin is:

1 - (probability of no heads) = 1 - 1/4 = 3/4

The probability of getting at least one head in two flips of a coin is 3/4 or 0.75, which means that there is a 75% chance of getting at least one head in two flips of a coin.

In other words, if you flip a coin twice, the probability of getting at least one head is relatively high, and you are more likely to get at least one head than to get no heads at all.

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Step-by-step explanation:

They can be found by considering where the second derivative changes signs. In similar to critical points in the first derivative, inflection points will occur when the second derivative is either zero or undefined.

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It take 20 years for Eva's initial deposit to double?.

A quick and easy method of calculating the interest charge on a loan is called a Simple interest.

Given that;

Eva deposits $2,000 in a savings account with an interest rate of 5% compounded continuously.

Here, Principal amount = $2000

Amount = $4000

Interest rate = 5%

Now, We know that;

⇒ Interest = PRT

Substitute all the values, we get;

⇒ 2000 = 2000 × 0.05 × T

⇒ T = 1/0.05

⇒ T = 100/5

⇒ T = 20 years

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Ans x/2 = 22/4

A word problem is a few sentences describing a "real-life" scenario where a problem needs to be solved using math.

To answer the question above, we equate the amount of cake mix initially to the amount of water needed to make a mug cake

4  tablespoons of cake mix = 2 tablespoons of water

we then equate the amount of cake mix in the final moment to the amount of water needed to make a mug cake

22  tablespoons of cake mix = X tablespoons of water

Now we divide both and these will give us the answer.

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The correct answer is option B and it is equal to 30% of 81 because it shows 0.30 x 81.

We have to find that how much is equal to the 30% of 81.

We can use the percentage formula here and that is,

%age = Obtained value/total value x 100

Here in this case, we have to find the obtained value and the total value is 81 and the percentage that we are going to get  is equal to 30.

So, putting all the values, we get,

obtained value = 30 x 81/100

obtained value = 0.30 x 81

Hence, the value 0.30 x 81 is equal to 30 % of 81. hence option B is the correct answer.

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Complete question - Which of the following options have the same value as 30% of 81?

A. 0.03 x 81

B. 0.3 x 81

C. 0.03 x 81

D. 3 x 81

The prepositional phrase would be B. of the bees.

The prepositional phrase would be C. he took shelter.

A prepositional phrase is a group of words that includes a preposition (such as "in," "on," "at," "by," "with," etc.) and a noun or pronoun object, along with any associated modifiers.

Prepositional phrases can be used in a variety of ways, including to describe location, time, possession, and other relationships between words in a sentence. In the given statements, the prepositional phrases would be "of the bees" and " he took shelter. "

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The antiderivative of sin is -cos(x) + C.

The antiderivative of a function is the inverse operation of differentiation which is is also known as its indefinite integral.

So, the integral of sin

                 ∫sin(x) = -cos(x) + C

where C = constant of integration.

This can be verified by differentiating -cos(x) + C with respect to x, which gives sin(x).

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The statement is false, the perimeter of the trapezoid shown is 23 units

To find the perimeter of a trapezoid, you need to add the lengths of all four sides of the trapezoid.

A trapezoid is a four-sided figure with two parallel sides (the bases) of different lengths. The other two sides are called the legs.

The formula for the perimeter of a trapezoid is:

Perimeter = sum of all sides = base1 + base2 + leg1 + leg2

where base1 and base2 are the lengths of the parallel sides and leg1 and leg2 are the lengths of the non-parallel sides.

So, to find the perimeter of a trapezoid, you need to measure or know the length of all four sides and then add them up using the formula above.

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Use the trapezoid shown to mark each statement below as true or false. if false, please rewrite the statement correctly in the space below the statement!

Q: The perimeter of the trapezoid shown is 22 units.

True false?

In a recent year, Saudi Arabia produced 8.2 million barrels of crude oil per day, compared to 6.4 million barrels per day for the United States. This was 28% more than what the United States produced.

Saudi Arabia now consumes 1.8 million more barrels of crude oil per day than the United States does. Crude Oil from Saudi Arabia: In January 2023, production was reported at 10,319,000 barrels per day, down from 10,475,000 barrels per day in the previous year.

In November 2022, crude oil production in Saudi Arabia was 10,468 thousand barrels per day, down from 10,957 thousand barrels per day the year before.

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Slope of the given graph is 2.

The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.

The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.

The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is

m=y₂-y₁/x₂-x₁

Let us take two points to find the slope of line.

The two points are (4, 1) and (6,5)

m=5-1/6-4

m=4/2

m=2

Hence, slope of the given graph is 2.

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Online calculators can solve for sides, angles, area, and perimeter of special right triangles, such as 45-45-90 and 30-60-90.

There are numerous web-based number crunchers that can assist you with addressing exceptional right triangles, like the 45-45-90 and 30-60-90 triangles. A few instances of extraordinary right triangle mini-computers are:

Calculator.net: This site has a unique right triangle number cruncher that can settle for the sides, points, and area of 45-45-90 and 30-60-90 triangles.

Omni Number cruncher: This site has a unique right triangle mini-computer that can settle for the sides, points, region, and border of 45-45-90 and 30-60-90 triangles.

Mathsisfun.com: This site has an intelligent apparatus that can help you comprehend and address exceptional right triangles, including 45-45-90 and 30-60-90 triangles.

These adding machines can save you time and assist you with rapidly tackling extraordinary right triangles, particularly assuming that you are chipping away at calculation issues or developments.

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The complete question is:

Explain about special right triangles calculator.

In 1 billion, the probability of division being incorrect would be : {1/(11 x 10⁹)²}.

In science, the probability of an event is a number that indicates how likely the event is to occur.

Given is that the manufacturer initially claimed that the chance of any particular division being incorrect was only 1 in 11 billion.

The probability of division being incorrect would be -

P{incorrect} = (1/11 x 10⁹)

In 11 billion, the probability of division being incorrect is (1/11 x 10⁹).

Then, in 1 billion, the probability of division being incorrect would be -

{1/(11 x 10⁹)²}

Therefore, in 1 billion, the probability of division being incorrect would be : {1/(11 x 10⁹)²}.

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Answer:

Step-by-step explanation:

0.0869

Answer:

Step-by-step explanation:

Forming linear equations, the cost of a t-shirt and a notebook will be $1 and $6 respectively.

What is a linear equation, exactly?

When the greatest power of a variable is 1, the equation is said to be linear. Another term for it is a one-degree equation. The standard form of a linear equation with one variable is Ax + B = 0. There are three variables in this equation: x, A, and B. A denotes a coefficient. A linear equation with only two variables is commonly written as Ax + By = C. There are three more constants in addition to the constant C: the variables x and y, the coefficients A and B, and the variables.

Now,

Given that

club A sold 2 t-shirts and 3 notebooks, and made $20.

Club B sold 2 t-shirts and 1 notebook, for a total of $8.

Let x and y be the cost of t-shirt and notebook respectively

then 2x+3y=20--->1

and 2x+y=8 --->2  will be the equations formed.

Now subtracting 2 from 1

we get 2x-2x+3y-y=20-8

2y=12

y=6

then  2x+6=8

x=1

Hence,

          the cost of a t-shirt and a notebook will be $1 and $6 respectively.

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