Describe and correct the error in finding the circumference of ⊙C

The radius of gyration of the plate about the x-axis is [tex]6 \sqrt{6} / 5[/tex] units.

To find the radius of gyration of a plate covering the region bounded by [tex]y = x^2[/tex], x = 6, and the x-axis with respect to the x-axis, we need to use the formula:

[tex]k_x = \sqrt{(I_x / A)}[/tex]

where [tex]k_x[/tex] is the radius of gyration, [tex]I_x[/tex] is the moment of inertia of the plate about the x-axis, and A is the area of the plate.

We can calculate the area A of the plate as follows:

[tex]A = \int\limits^6_0 { x^2}\, dx\\= [x^3/3]\ from\ 0\ to\ 6\\= 72[/tex]

To find the moment of inertia [tex]I_x[/tex], we can use the formula:

[tex]I_x = \int\ {y^2} \, dA[/tex]

where y is the perpendicular distance of an element of area [tex]dA[/tex] from the x-axis. We can express y in terms of x as y = x². Therefore, we have:

[tex]dA = y dx = x^2 dx\\I_x = \int\limits^6_0 { x^2 (x^2)} dx\\= \int\limits^6_0 {x^4}\, dx\\= [x^5/5]\ from\ 0\ to\ 6\\= 6^5/5[/tex]

Substituting these values into the formula for [tex]k_x[/tex], we get:

[tex]k_x = \sqrt{(I_x / A)}\\= \sqrt{((6^5/5) / 72)}\\= \sqrt{(6^3 / 5)}\\= 6 \sqrt{6} / 5[/tex]

Therefore, the radius of gyration of the plate about the x-axis is [tex]6 \sqrt{6}/ 5[/tex] units.

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Answer: C

Step-by-step explanation:

If both belts are​ used, it will take 2.25 minutes to move the cans to the storage area

To solve this problem, we need to use the concept of rate of work. We know that the larger belt can move 200 pounds of aluminum in 3 minutes, which means its rate of work is 200/3 = 66.67 pounds per minute. Similarly, the smaller belt can move the same quantity of cans in 9 minutes, which means its rate of work is 200/9 = 22.22 pounds per minute.

When both belts are used together, their rates of work add up, so the total rate of work is 66.67 + 22.22 = 88.89 pounds per minute. We can use this rate of work to find how long it will take to move the cans to the storage area.

Let's assume that it takes x minutes to move the cans using both belts. Then, we can set up the following equation:

88.89x = 200

Solving for x, we get:

x = 2.25 minutes

Therefore, it will take 2.25 minutes to move the cans to the storage area using both belts.

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Answer:

2

Step-by-step explanation:

0.50/0.25 = 50/25

= 2

Dividing 0.50/0.25 no. is same as 50/25 when we multultiply by 100/ 100 so it is 2

ans. = 2

0.50 / 0.25

5/10  x  100/5

=2

The region bounded by the given curves is a triangle with vertices at (0,0), (1,1), and (1,0). When this region is revolved around the line y=3, we obtain a solid with a hole in the middle.
To find the volume of this solid, we can use the method of cylindrical shells. Imagine slicing the solid into thin cylindrical shells with radius r and height Δy. The volume of each shell is approximately 2πrΔy times the thickness of the shell.

The distance between the axis of rotation (y=3) and the line y=1 is 2 units. Therefore, the radius of each cylindrical shell is r = 3 - y. The height of each shell is Δy = dx, where x is the distance from the y-axis.

To set up the integral, we need to express x in terms of y. Since the region is bounded by y=x and y=1, we have x=y for 0<=y<=1. Therefore, the integral for the volume of the solid is:
V = ∫[0,1] 2π(3-y)x dx
 = 2π ∫[0,1] (3-y)y dx

Evaluating this integral, we get:
V = 2π [3y^2/2 - y^3/3] from 0 to 1
 = 2π (3/2 - 1/3)
 = 2π/3

Therefore, the volume of the solid obtained by rotating the region bounded by y=x, y=1 about y=3 is (2/3)π cubic units.

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Answer: (670 -110)/8

Step-by-step explanation:

Months: 110 the first month

*Now since she paid 670 in total she subtracts the 110 from the first month which is 560, and now there is an event amound paid in the other months so you divide 560 by 8 since 8 months are left over resulting in $70 each month.

January,

February,

March,

April,

May,

June,

July,

August,

Based on the above, Keenan paid $9,420 in federal income tax in 2009.

Let X be the amount Keenan paid in taxes in 2009.

According to the problem, Keenan paid 70% less in 2010 than he did in 2009. This means that he paid only 30% of what he paid in 2009, since 100% - 70% = 30%.

We can express this mathematically as:

0.30X = 2,826

To solve for X, we can divide both sides of the equation by 0.30:

X = 2,826 ÷  0.30

X = 9,420

Therefore, Keenan paid $9,420 in federal income tax in 2009.

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The coordinates of point P along the directed line segment AB with a ratio of 1:4 are (-5, -2).

Since the ratio of AP to PB is 1:4, we can use the midpoint formula to find the coordinates of point A. The midpoint formula is

((x₁ + x₂)/2, (y₁ + y₂)/2)

Plugging in the coordinates of points P and B, we get:

((4(-7) - 2)/5, (4(-5) + 0)/5) = (-30/5, -20/5) = (-6, -4)

we can use the point-slope formula to find the equation of the line segment AB:

(y - (-4)) = (1/5)(x - (-6))

Simplifying this equation, we get:

y = (1/5)x + 2

Finally, we can use the given ratio of 1:4 to find the coordinates of point P. Since the ratio of AP to PB is 1:4, we can use the ratio formula to find the coordinates of point P:

(x, y) = (4(-5) + (-2))/5, (4(-2) - (-4))/5) = (-30/5, 12/5) = (-6, 2.4)

Rounding off to one decimal place, we get the coordinates of point P as (-5, -2).

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Answer:

-17 and -5

Step-by-step explanation:

-5 x -17 = 85

-5 + -17 = -22

The customer purchased 2 small baskets and 6 large baskets.

Let x be the number of small baskets and y be the number of large baskets that the customer purchases.

We can set up a system of two equations based on the information given:

Equation 1: x + y = 8 (The total number of baskets purchased is 8)

Equation 2 3x + 5y = 36:  (The total amount paid for the baskets is $36)

To solve this system, we can use either substitution or elimination method.

Using substitution method:

From Equation 1, we have x = 8 - y.

Substitute this into Equation 2:

3(8 - y) + 5y = 36

24 - 3y + 5y = 36

2y = 12

y = 6

Now, we can substitute y = 6 back into Equation 1 to find x:

x + 6 = 8

x = 2

Therefore, the customer purchased 2 small baskets and 6 large baskets.

Using elimination method:

We can multiply Equation 1 by 3 and subtract it from Equation 2 to eliminate x:

 3x + 5y = 36

- (3x + 3y = 24)

   2y = 12

    y = 6

Now, we can substitute y = 6 back into either Equation 1 or Equation 2 to find x. Let's use Equation 1:

x + 6 = 8

x = 2

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The angle of the tops of two vertical poles of heights 20 m and 15 m joined by a taut wire 12 m long slope of the wire = 24.6 °

Height of the 1st vertical pole = 20m

Height of the second vertical pole = 15m

Difference of their height = 5 m

Length of the taut wire = 12m

Using trigonometry ratio of sin we get

Perpendicular = 5 m

Hypotenuse = 12 m

Sin A = Perpendicular/ hypotenuse

Sin A = 5/12

A = [tex]sin^{-1} (5/12)[/tex]

A = 24.6 °

The angle of slope of the wire = 24.6°

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The volume of the given cone is 402.1 cubic feet if the slant height is 10ft and the length is 6ft.

To calculate the volume of a cone, the formula used is :

V = (1/3) * π * [tex]r^2[/tex] * h

Here, the radius is the unknown term. we need to calculate the radius of the cone. We can use the Pythagorean theorem to find the radius of the cone.

[tex]l^2 = r^2 + h^2[/tex]

[tex]10^2 = r^2 + 6^2[/tex]

[tex]r = \sqrt{(10^2 - 6^2)}[/tex]

radius = 8 ft

V = (1/3) *  π * [tex]r^2[/tex] * h

V = (1/3) * π *[tex]8^2[/tex] * 6

V = (1/3) * π * 384

V =  402.1 cubic feet

Therefore we can infer that the volume of the given cone is  402.1 cubic feet.

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The complete question is:

'Find the volume of this cone. Round to the nearest tenth.

slant height = 10ft

length = 6ft

Barak should buy the nails from the Hammer Company as it is cheaper than buying from the Nail Company.

Let's first calculate the cost of buying 550 nails from each company:

Nail Company:

Cost of 1 nail = £4.15 + (20% of £4.15) = £4.15 + £0.83 = £4.98 (rounded to 2 decimal places)

Cost of 50 nails = £4.98 x 50 = £249

Cost of 550 nails = £249 x 11 = £2739

Hammer Company:

Cost of 1 nail = £2.95/25 = £0.118 (rounded to 3 decimal places)

Cost of 75 nails (buy 100 get 25 free) = 100 x £0.118 x 3 = £35.40

Cost of 550 nails = 550 x £0.118 = £64.90

Therefore, Barak should buy the nails from the Hammer Company as it is cheaper than buying from the Nail Company.

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If the cost after the coupon is $21.25 and she decides to add a 20% tip, she will be adding $4.25 for the tip.

Find out how much a 20% tip would be on a cost of $21.25 after applying a coupon.
Identify the total cost after the coupon.
In this case, the cost is $21.25.
Determine the percentage for the tip.
The tip percentage is given as 20%.
Convert the percentage to a decimal.
To do this, divide the percentage by 100. So, 20% divided by 100 is equal to 0.2.
Multiply the total cost by the tip percentage in decimal form.
Now, multiply $21.25 (total cost) by 0.2 (tip percentage as a decimal).
$21.25 x 0.2 = $4.25
Calculate the tip amount.
The tip amount is $4.25.

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The mean of a sample is computed by summing all the data values and dividing the sum by the number of items in the sample. Thus, the correct answer is d.

Option a is incorrect because the mean of a sample is not always equal to the mean of the population, unless the sample is a complete representation of the population (which is often not the case).

Option b is incorrect because the mean of a sample can be greater than, equal to, or smaller than the mean of the population, depending on the sampling method and the characteristics of the population.

Option c is incorrect because the sample mean is computed by summing the data values and dividing the sum by the number of items in the sample minus one only if the sample is taken from a normally distributed population and the standard deviation of the population is unknown. Otherwise, the sample mean is computed by dividing the sum of the data values by the number of items in the sample.

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The measurements from greatest to least would be ordered as follows:

First, we need to convert all the units to the same unit. Let's convert everything to inches, since that is the smallest unit.

6 yards = 6 x 3 = 18 feet

18 feet = 18 x 12 = 216 inches

2 1/2 feet = 2 x 12 + 6 = 30 inches

So now we have:

6 yards = 216 inches

2 1/2 feet = 30 inches

45 inches = 45 inches

Putting these in order from greatest to least, we have:

216 inches, 45 inches, 30 inches

Therefore, the measurements from greatest to least would be ordered as follows:

6 yards, 45 inches, 2 1/2 feet

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The full question is:
Put the measurements from greatest to least. 45 inches, 6 yards, and 2 1/2 feet

Step-by-step explanation:

just convert the 1/3 and times 3 to both it's numerator and denominator.

once you have the same denominator as the other mixed number, you can start to minus.

Answer:

2 1/9 hrs

Step-by-step explanation:

4 1/3 = 13/3 = 39/9

2 2/9 = 20/9

39/9 - 20/9 = 19/9 = 2 1/9 hrs

or,

(4 - 2) + (3/9 - 2/9) = 2 1/9 hrs

The upper and lower bound of the cross-sectional area of the insulation, would be A = [ 3129, 3137 ] cm².

The upper and lower bound of A would be found by the formula :

A = π x ( R ² - r ² )

The upper bound is therefore:

= π x  (( 48. 75 / 2) ² - ( 19.2 45 / 2) ²)

= π x ( 1183. 0625 - 184. 857025 )

= π x 998. 205475

= 3, 137 cm²

The lower bound will then be:

= π x ( ( 48. 65 / 2 ) ²- (19. 255 / 2) ²)

= π x ( 1180. 9225 - 184. 963025)

= π x 995. 959475

= 3, 129 cm²

The interval form is therefore A = [ 3129, 3137 ] cm²

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Answer:

2(2(3) + 2(5) + 3(5)) = 2(6 + 10 + 15) = 2(31)

= 62

D is the correct answer.

Where the function f(x) = x² + 2x - 3 is given, note that the x-intercepts of the function f(x) are -3 and 1, and the minimum value of the function is -4. See the attached graph.

To find the minimum and maximum points of the function f(x), we can complete the square:

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

= (x + 1)^2 - 4

We can see that the function is in the vertex form f(x) = a(x - h)^2 + k, where the vertex is (-1, -4).

Since the coefficient of the x^2 term is positive, the parabola opens upwards, and the vertex is the minimum point. Therefore, the minimum value of the function f(x) is -4.

To find the x-intercepts, we can set f(x) = 0:

(x + 1)^2 - 4 = 0

(x + 1)^2 = 4

Taking the square root of both sides, we get:

x + 1 = ±2

x = -1 ± 2

Therefore, the x-intercepts of the function f(x) are x = -3 and x = 1.

In summary, the x-intercepts of the function f(x) are -3 and 1, and the minimum value of the function is -4, which occurs at the vertex (-1, -4).

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The probability of a randomly selected tire from this sample having a lifespan greater than 45,000 miles is approximately 0.0639 or 6.39%.

We can use the central limit theorem to approximate the distribution of the sample mean.

In this case, the population mean is 44,000 miles and the population standard deviation is 7,650 miles. We are taking a sample of 120 tires, so the standard deviation of the sample mean is:

σ/√n = 7,650/√120 = 698.68

To find the probability that the sample mean will be longer than 45,000 miles, we need to standardize the sample mean using the formula:

z = (x - μ) / (σ / √n)

where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

Plugging in the values, we get:

z = (45,000 - 44,000) / (7,650 / √120) = 1.527

We can then look up the probability corresponding to a z-score of 1.527 in a standard normal distribution table or using a calculator. The probability of a randomly selected tire from this sample having a lifespan greater than 45,000 miles is approximately 0.0639 or 6.39%.

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The amount of time Carl represent is 25-m on the problems.

The statement "Davis spent 25 minutes working on math problems. Carl worked on math problems for m fewer minutes" means that Carl spent some amount of time working on math problems, but that amount is m minutes less than what Davis spent.

To represent the amount of time Carl worked on math problems, we can use the variable X. We know that X is equal to the amount of time Carl worked on math problems, and that X is equal to 25 minus m.

This is because Davis spent 25 minutes on math problems, and Carl worked on them for m fewer minutes. So if we subtract m from 25, we get the amount of time Carl worked on math problems.

Therefore, the equation X = 25 - m represents the amount of time Carl worked on math problems, where X is the amount of time in minutes and m is the number of minutes Carl worked less than Davis.

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.....................

Statement 2 is false because while the median score of Group A is higher than Group B, it is not 10 points greater as claimed.

Statement 1 is false because the box plots provide limited information, making it impossible to determine whether at least 50% of students in each group scored more than 165 push-ups.

Statement 3 is false because Group A has more variability in push-up scores than Group B, as indicated by the larger interquartile range (IQR) of Group A.

Looking at the box plots, we can see that the median score of Group A is higher than Group B, but it is not 10 points greater. Therefore, statement 2 is False.

We cannot determine whether at least 50% of students in each group scored more than 165 push-ups. The box plots only show us the quartiles and the minimum and maximum values, so we do not know the exact number of students who scored above 165 push-ups. Therefore, statement 1 is False.

The interquartile range (IQR) of Group A is greater than the IQR of Group B, indicating that Group A has more variability in push-up scores than Group B. Therefore, statement 3 is False.

Hence, All the statement are False.

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Answer: Therefore, the length of a blue bead is 2.5 cm, and the length of a green bead is 1.2 cm. And the length of the bracelet is:

10 × (2.5 + 1.2) = 37 cm.

Step-by-step explanation:

To represent the length of the bracelet, we need to determine the length of each repetition of the basic pattern and then multiply it by the number of times the pattern is repeated.

The length of each repetition of the basic pattern is the sum of the length of one blue bead and one green bead, which is:

B + 1.2 cm

Since the basic pattern is repeated 10 times, the total length of the bracelet is:

10 × (B + 1.2) cm

And we know that the total length of the bracelet is 37 cm, so we can set up an equation:

10 × (B + 1.2) = 37

Simplifying the equation, we can divide both sides by 10:

B + 1.2 = 3.7

Subtracting 1.2 from both sides, we get:

B = 2.5

Therefore, the length of a blue bead is 2.5 cm, and the length of a green bead is 1.2 cm. And the length of the bracelet is:

10 × (2.5 + 1.2) = 37 cm.

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The answer of addition is 94.

The phrase "the addition" refers to combining two or more numbers. Adding two numbers is indicated by the plus sign (+), therefore adding three is written as three plus three.

To add 8 and 78, we can start by adding the ones place digits, which are 8 and 7.

8 + 7 = 15

Since 15 is greater than 10, we need to regroup 10 ones as 1 ten and carry it over to the tens place. We can represent this with place value blocks by moving a rod of 10 ones from the ones place to the tens place.

So we have:

 8

+78

---

```

 8

+78

---

 16 (write 6 in the ones place and carry 1 to the tens place)

```

Now we can add the tens place digits, which are 1 (carried over) and 8:

1 + 8 = 9

So the final result is:

 8

+78

-----

86

Therefore, 8 plus 78, plus 7 is:

```

 8

+78

+ 7

---

 94

```

So the answer of addition is 94.

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The volume of the rectangular prism is 105 cubic meters.

To find the volume of the rectangular prism, we can use the formula V = lwh, where V is the volume, l is the length, w is the width, and h is the height.

Since there are three pairs of congruent faces, we can deduce that the areas of the three pairs of faces represent the three dimensions of the rectangular prism. The areas are 9, 25, and 49 square meters, which are the squares of the sides' lengths.

Take the square root of each area to find the corresponding side lengths:

√9 = 3 meters
√25 = 5 meters
√49 = 7 meters

Now, apply the formula to find the volume:

V = lwh = 3 × 5 × 7 = 105 cubic meters.

The volume of the rectangular prism is 105 cubic meters.

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Answer:

Carlos=1.5041

Mykala=2.6991

William=4.1350

Emily=4.1773

Caleb has 30 quarters.

What is arithmetic?

Mathematical arithmetic is the study of the properties of the standard operations on numbers, such as addition, subtraction, multiplication, division, exponentiation, and root extraction.

Here, we have

Given: Caleb has 51 coins (nickels, dimes, and quarters) in a jar, totaling $9. He has three more nickels than dimes.

We have to find out how many quarters Caleb has.

Let x be nickel,

y be dimes and

z be quarters

x + y + z = 51.....(1)

1 quartes = 25 cents

1 dimes = 10 cents

1 nickel = 5 cents

Now, the total dollar is $9,

5x/100 + 10y/100 + 25z/100 = 9

5x + 10y + 25z = 900

x + 2y + 5z = 180....(2)

and

y + 3 = x...(3)

Solving equation(1) and (2), we get

From (1)

x + x-3 + z = 51

2x + z = 54....(4)

From (2)

x + 2(x -3) + 5z = 180

3x + 5z = 186...(5)

Now, by solving equations (4) and (5), we get

x = 12

z = 30

Now,

y + 3 = x

y + 3 = 12

y = 9

Hence, Caleb has 30 quarters.

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The statement "the series converges to 31.25" is true about the given series.

Given series is 25 + 5 + 1 + ....

We can clearly see that given series is infinite geometric series.

First term is a=25

common ratio is r = 5/25

= 1/5

We know that the formula of sum of an infinite geometric series is

S = a / (1 - r)

S = 25 / (1 - 1/5)

S = 25/(4/5)

S = (25*5)/4

S = 125/4

S = 31.5

Therefore, the sum of the infinite geometric series is 31.25.

Since, the sum of the series is a finite number, we can say that the series converges.

Therefore, the statement "the series converges to 31.25" is true about the given series.

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