The stem-and-leaf plot shows kilometers walked by participants in a charity benifit walk. Use it to answer the questions. a. How many people participated in the walk? b. How many of the walkers traveled more than 14 Kilometers?

The formula for the value of nth term is [tex]a_{n}[/tex] = 3n + 1

Step-by-step explanation:

The formula of the nth term in the arithmetic sequence is

[tex]a_{n}=a+(n-1)d[/tex] , where

∵ The first term of an arithmetic sequence is equal to four

∴ a = 4

∵ The common difference is equal to three

∴ d = 3

- Substitute these values in the rule of the nth term

∵ [tex]a_{n}=a+(n-1)d[/tex]

∴ [tex]a_{n}=4+(n-1)3[/tex]

- Simplify it

∴ [tex]a_{n}=4+3n-3[/tex]

∴ [tex]a_{n}=1+3n[/tex]

The formula for the value of nth term is [tex]a_{n}[/tex] = 3n + 1

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It can be done in 50388 ways

Step-by-step explanation:

When the selection has to be made without order, combinations are used.

The formula for combination is:

[tex]C(n,r) =\frac{n!}{r!(n-r)!}[/tex]

Here

Total books = n =19

Books to be chosen = r = 7

Putting the values

[tex]C(19,7) = \frac{19!}{7!(19-7)!}\\\\=\frac{19!}{7!12!}\\\\=50388\ ways[/tex]

It can be done in 50388 ways

Keywords: Combination, selection

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Answer:

Step-by-step explanation:

Let j and b represent Jera's and Bipu's current ages.

  j-5 = (b +8) -20 . . . . . Jera's age 5 years ago was 20 less than Bipu's age in 8 years

  b-5 = 2(j+8) -34 . . . . .Bipu's age 5 years ago was 34 less than Jera's age in 8 years

__

Solving the first equation for j gives ...

  j = b + 8 -20 + 5

  j = b -7

Using that in the second equation, we get ...

  b -5 = 2((b -7)+8) -34

  0 = b -27 . . . . . . . subtract (b-5) and simplify

  b = 27

  j = 27 -7 = 20

Jera is 20; Bipu is 27.

Answer: B. $2875

Step-by-step explanation:

We know that best point estimate of population mean[tex](\mu)[/tex] is the sample mean[tex](\overline{x})[/tex] .

Given : A large insurance company wanted to estimate [tex]\mu[/tex], the mean claim size (in 5) on an auto insurance policy.

A random sample of 225 claims was chosen and it was found that the average claim size was $2875.

i.e. Sample mean [tex]\overline{x}=\$2875[/tex]

That means , the point estimate for [tex]\mu=\$2875[/tex]

Hence , the correct answer is option B . $2875.

The point estimate of u, the mean claim size, is $2875 as determined by the average claim size of a random sample of 225 claims.

In statistics, a point estimate is often the best guess that one can make for an unknown parameter. In this case, we are looking for the mean claim size, denoted as u. When the question states that a random sample of 225 claims showed an average claim size of $2875, they are giving you the point estimate of u, because the calculated average is our best estimate of the population mean in this context. Therefore, the point estimate of u is $2875.

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Answer:

A)The probability that someone who tests positive has the disease is 0.9995

B)The probability that someone who tests negative does not have the disease is 0.99999

Step-by-step explanation:

Let D be the event that a person has a disease

Let [tex]D^c[/tex] be the event that a person don't have a disease

Let A be the event that a person is tested positive for that disease.

P(D|A) = Probability that someone has a disease given that he tests positive.

We are given that There is an excellent test for the disease; 98.8% of the people with the disease test positive

So, P(A|D)=probability that a person is tested positive given he has a disease = 0.988

We are also given that  one person in 10,000 people has a rare genetic disease.

So,[tex]P(D)=\frac{1}{10000}[/tex]

Only 0.4% of the people who don't have it test positive.

[tex]P(A|D^c)[/tex] = probability that a person is tested positive given he don't have a disease = 0.004

[tex]P(D^c)=1-\frac{1}{10000}[/tex]

Formula:[tex]P(D|A)=\frac{P(A|D)P(D)}{P(A|D)P(D^c)+P(A|D^c)P(D^c)}[/tex]

[tex]P(D|A)=\frac{0.988 \times \frac{1}{10000}}{0.988 \times (1-\frac{1}{10000}))+0.004 \times (1-\frac{1}{10000})}[/tex]

P(D|A)=[tex]\frac{2470}{2471}[/tex]=0.9995

P(D|A)=[tex]0.9995[/tex]

A)The probability that someone who tests positive has the disease is 0.9995

(B)

[tex]P(D^c|A^c)[/tex]=probability that someone does not have disease given that he tests negative

[tex]P(A^c|D^c)[/tex]=probability that a person tests negative given that he does not have disease =1-0.004

=0.996

[tex]P(A^c|D)[/tex]=probability that a person tests negative given that he has a disease =1-0.988=0.012

Formula: [tex]P(D^c|A^c)=\frac{P(A^c|D^c)P(D^c)}{P(A^c|D^c)P(D^c)+P(A^c|D)P(D)}[/tex]

[tex]P(D^c|A^c)=\frac{0.996 \times (1-\frac{1}{10000})}{0.996 \times (1-\frac{1}{10000})+0.012 \times \frac{1}{1000}}[/tex]

[tex]P(D^c|A^c)=0.99999[/tex]

B)The probability that someone who tests negative does not have the disease is 0.99999

Answer:

ABCDE = 42857

Step-by-step explanation:

First, we will use logic. The only number that when it's multiplied by 3, gives 1 at the end is 7, so E should be 7, so:

1ABCD7 * 3 = ABCD71    

If 3 times 7 is 21, we carry two, so, the next number by logic, cannot be 1, because 3*1 = 3 + 2 = 5, it's not 7, so, it should be another number, like 5.

5*3 = 15 + 2 = 17 and we carry one. So this number fix in the digit, and D = 5.

We have now: 1ABC57 * 3 = ABC571

Letter C, we have to get a number that when it's multiplied by 3 and carry one, gives 5. In this case 8, because: 3 * 8 = 24 + 1 = 25 and carry two. D = 8.

So far: 1AB857 * 3 = AB8571

Now, the same thing with B. If we multiply 3 by 2, and carry two we will have 8 so: 3 * 2 = 6 + 2 = 8. B = 2

1A2857 * 3 = A28571

Finally for the last number, a number multiplied by 3 that hold the 1 as decene, In this case, the only possibility is 4, 3 * 4 = 12 so:

142857 * 3 = 428571      

Answer:

Marco will use [tex]1[/tex] red tile, [tex]2[/tex] blue tiles and [tex]1\frac{2}{3}[/tex] yellow tiles in each stone.

Step-by-step explanation:

Given:

Number of red tiles = 45

Number of blue tiles = 90

Number of yellow tiles = 75

Greatest number of stones that Marco can make = 45

To determine how many each color tile Marco will use in each stone.

Solution:

In order to determine the number of each color tile in each stone we need to divide total number of a particular tile by total number of stones. By doing this we can get the exact number of that color tile used in each stone.

Number of red tile in each stone

⇒ [tex]\frac{\textrm{Total number of red tiles}}{\textrm{Total number of stones}}[/tex]

⇒ [tex]\frac{45}{45}=1[/tex]

Number of blue tiles in each stone

⇒ [tex]\frac{\textrm{Total number of blue tiles}}{\textrm{Total number of stones}}[/tex]

⇒ [tex]\frac{90}{45}=2[/tex]

Number of yellow tile in each stone

⇒ [tex]\frac{\textrm{Total number of yellow tiles}}{\textrm{Total number of stones}}[/tex]

⇒ [tex]\frac{75}{45}[/tex]

⇒ [tex]\frac{5}{3}[/tex]   [Reducing to simpler fraction by dividing both numbers by their GCF=15]

⇒ [tex]1\frac{2}{3}[/tex]  [Converting improper fraction to mixed number]

∴ Marco will use [tex]1[/tex] red tile, [tex]2[/tex] blue tiles and [tex]1\frac{2}{3}[/tex] yellow tiles in each stone.

Answer:

Marco will use 3 red, 6 blue and 5 yellow tiles on each stones

Explanation:

Given, each stone must have same number of each colour tile

Then, calculating the highest common factors of the number HCF (45, 90, 75)

Factors of 45 = 3 × 3  ×  5

Factors of 90 = 2  × 3 × 3  × 5

Factors of 75 = 3 × 5  ×  5

Highest Common Factors (HCF) = 3 × 5 = 15

Dividing all three numbers by 15, we get

Red Tiles =[tex]\frac{45}{15}[/tex] = 3

Blue Tiles = [tex]\frac{90}{15}[/tex] = 6

Yellow Tiles = [tex]\frac{75}{15}[/tex] = 5

Therefore, Marco will use 3 red, 6 blue and 5 yellow tiles on each stones

Answer:

i would think that its c

let me know if its wrong

Answer:

c

Step-by-step explanation:

i know

Answer:

The answer to your question is 13x² - 14x

Step-by-step explanation:

Process

1.- Divide the figure in to sections to get to rectangles (see the picture below)

2.- Get the area of each rectangle

3.- Add the areas

2.- Area of a rectangle = base x height

Rectangle 1

      Area 1 = (3x - 7) (x)

                 = 3x² - 7x

Rectangle 2

      Area 2 = (5x + 2)(2x)

                  = 10x² + 4x

3.- Total area

     Area = (3x² - 7x) + (10x² - 7x)

              =  13x² - 14x

Step-by-step explanation:

We have the equation for elongation

                 [tex]\Delta L=\frac{PL}{AE}\\\\A=\frac{\pi d^2}{4}[/tex]

Here we have

                 Elongation, ΔL = 0.266 in = 0.00676 m

                 Length , L = 35 ft = 10.668 m

                 Load, P = 8000 lb = 35585.77 N

                 Modulus of elasticity, E = 30,000,000 psi = 2.07 x 10¹¹ N/m²

Substituting

                 [tex]\Delta L=\frac{PL}{AE}\\\\A=\frac{\pi d^2}{4}\\\\\Delta L=\frac{4PL}{\pi d^2E}\\\\d^2=\frac{4PL}{\pi \Delta LE}\\\\d=\sqrt{\frac{4PL}{\pi \Delta LE}}\\\\d=\sqrt{\frac{4\times 35585.77\times 10.668}{\pi \times 0.00676 \times 2.07\times 10^{11}}}=0.019m\\\\d=19mm[/tex]

Diameter of rod = 19 mm

Answer:9

Step-by-step explanation:

You would count the possible outcomes on the right side of the diagram

The number of possible outcomes is 9.

Given that,

Based on the above information, the calculation is as follows:

From NewYork = 3

From San Francisco = 3

From Miami = 3

Total = 9

Therefore we can conclude that the number of possible outcomes is 9.

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Answer:

  The box which minimizes the cost of materials has a square base of side length 4.16 cm and a height of 2.08 cm

Step-by-step explanation:

The cost is minimized when the cost of each pair of opposite sides is the same as the cost of the top and bottom. Since the top and bottom are half the cost of the sides (per unit area), the area of the square top and bottom will be double that of the sides. That is, the box is half as tall as wide, so is half of a cube of volume 72 cm³.

Each side of the square base is ∛72 = 2∛9 ≈ 4.16 cm. The height is half that, or 2.08 cm.

_____

If you want to see this analytically, you can write the equation for cost, using ...

  h = 36/s²

  cost = 2(1)(s²) + (2)(4s)(36/s²) = 2s² +288/s

The derivative is set to zero to minimize cost:

  d(cost)/ds = 4s -288/s² = 0

  s³ = 72 . . . . . multiply by s²/4

  s = ∛72 = 2∛9 ≈ 4.16 . . . . . cm

  h = 36/(2∛9)² = ∛9

The box is 2∛9 cm square and ∛9 cm high, about 4.16 cm square by 2.08 cm.

To minimize the cost of materials, the dimensions of the box that minimize the cost of materials are approximately: Square base side length: 4.18 cm, Height: 2.05 cm.

To minimize the cost of materials, we need to consider the areas that need to be plated with silver and nickel. Let's assume the side length of the square base is x cm, and the height of the box is h cm. The cost of silver plating the sides is $2 per cm², and the cost of nickel plating the bottom and top is $1 per cm².

The area of each silver-plated side is 4xh cm², and the area of each nickel-plated bottom and top is x² cm². The total cost of materials can be calculated using the formula:

Total cost = 4xh * $2 + 2x² * $1 = 8xh + 2x²

To minimize the cost, we need to find the values of x and h that will minimize this expression.

Since the volume of the box is given as 36 cm³, we have the equation x²h = 36.

Using the equation for the volume, we can solve for h in terms of x:

h = 36 / x².

Substituting this into the expression for the total cost:

Total cost = 8x(36 / x²) + 2x² = 288 / x + 2x²

To find the values of x and h that minimize the cost, we need to find the critical points of the expression. Taking the derivative of the total cost with respect to x, and setting it to zero:

d(Total cost) / dx = -288 / x² + 4x = 0

Simplifying this equation:

288 = 4x³

x³ = 72

x = ∛72 ≈ 4.18 cm

Substituting this value of x back into the equation for h:

h = 36 / (4.18)² ≈ 2.05 cm.

Therefore, the dimensions of the box that minimize the cost of materials are approximately:

Square base side length: 4.18 cm

Height: 2.05 cm

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1.what is the length of the segment joining 3,6 and -2,-6 : 13 units

2.what is the center of the circle (x+6)^2+(y-8)^2=144 => (-6,8)

3.what is the slope of the line 3y+2x-6=0=> -2/3

Step-by-step explanation:

1.what is the length of the segment joining (3,6) and (-2,-6)?

Let

(x1,y1) = (3,6)

(x2,y2) = (-2,-6)

The length of a segment is given by:

[tex]d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\Putting\ values\\d  = \sqrt{(-2-3)^2+(-6-6)^2}\\d = \sqrt{(-5)^2+(-12)^2}\\= \sqrt{25+144}\\= \sqrt{169}\\=13\ units[/tex]

2.what is the center of the circle (x+6)^2+(y-8)^2=144

The equation of circle is given by:

[tex](x-h)^2+(y-k)^2 = r^2[/tex]

Here, h and k are the coordinates of centre of circle

x - h = x+6

-h = 6

h = -6

y - 8 = y - k

-8 = - k

k = 8

So,

The center of circle is: (-6,8)

3.what is the slope of the line 3y+2x-6=0

We have to convert the equation in slope-intercept form to find the slope

Slope-intercept form is:

y = mx+b

Now,

[tex]3y+2x-6=0\\3y+2x = 6\\3y = -2x+6[/tex]

Dividing both sides by 3

[tex]\frac{3y}{3} = -\frac{2}{3}x+\frac{6}{3}\\y = -\frac{2}{3}x + 2[/tex]

In slope-intercept form, the co-efficient of x is the slope of the line so

m = -2/3

Keywords: Coordinate geometry, Slope

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Answer:

25 weeks  will Toby and Marcus have the same number of stamps

Explanation:

No of stamps collected by Toby initially= 10

No of stamps Toby collects  every week= 4

No of stamps Marcus has initially= 60

No of stamps Marcus collects each week=2

Suppose the no of week when Toby and Marcus have the same number of stamps are x

Hence no of stamps collected by Toby after x weeks

=10+4x

No of stamps collected by Marcus after x weeks

=60+2x

Therefore to calculate the same of stamps collected by Toby and Marcus

No of stamps collected by Toby after x weeks= No of stamps collected by Marcus after x weeks

10+4x =60 +2x

4x-2x= 60-10

2x=50

x=25

Hence after 25 weeks Toby and Marcus will have the same number of stamps

The interest rate for the $4000 investment is approximately 2.1% and for the $20000 investment, it is approximately 2.7%.

This problem can be solved using the concept of simple interest. Let's denote the interest rate of the $4000 investment as r (in decimal form) and the interest rate of the $20000 investment would be r+0.006. Now, we can set up our equations based on the information given:

1. For $4000 investment: Interest = 4000 * r

2. For $20000 investment: Interest = 20000 * (r + 0.006)

It is also given that the interest earned from the $20000 investment is $1320 more than the $4000 investment. Therefore, we can set up a third equation as:

20000 * (r + 0.006) - 4000 * r = 1320

By solving this equation, we find that r (corresponding to the $4000 investment) is approximately 0.021 or 2.1% and therefore, the interest rate for the $20000 investment is roughly 2.7% (2.1% + 0.6%).

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Answer:

Step-by-step explanation:

The relationships between the radius of a circle and its circumference and area are ...

  C = 2πr

  A = πr²

The relationships between the side length of a square and its perimeter and area are ...

  P = 4s

  A = s²

So, the length of wire will be ...

  w = C + P

  w = 2πr + 4s

subject to the constraint that the sum of areas is 64 cm²:

  πr² + s² = 64

___

Using the method of Lagrange multipliers to find the extremes of wire length, we want to set the partial derivatives of the Lagrangian (L) to zero.

  L = 2πr + 4s + λ(πr² +s² -64)

  ∂L/∂r = 0 = 2π +2πλr . . . . . . [eq1]

  ∂L/∂s = 0 = 4 +2λs . . . . . . . . [eq2]

  ∂L/∂λ = 0 = πr² +s² -64 . . . . [eq3]

__

Solving for λ, we find ...

  0 = 1 +λr . . . . divide [eq1 by 2π

  λ = -1/r . . . . . . subtract 1, divide by r

Substituting into [eq2], we get ...

  0 = 4 + 2(-1/r)s

  s/r = 2 . . . . . . . . . .add 2s/r and divide by 2

This tells us the maximum wire length is that which makes the circle diameter equal to the side of the square.

Substituting the relation s=2r into the area constraint, we find ...

  πr² +(2r)² = 64

  r = √(64/(π+4)) = 8/√(π+4) ≈ 2.99359 . . . . cm

and the maximum wire length is ...

  2πr +4(2r) = 2r(4+π) = 16√(4+π) ≈ 42.758 . . . cm

_____

The minimum wire length will be required when the entire area is enclosed by the circle. In that case, ...

  πr² = 64

  r = √(64/π)

  C = 2πr = 2π√(64/π) = 16√π ≈ 28.359 . . . cm

_____

Comment on the solution method

The method of Lagrange multipliers is not needed to solve this problem. The alternative is to write the length expression in terms of one of the figure dimensions, then differentiate with respect to that:

  w = 2πr + 4√(64-πr²)

  dw/dr = 2π -4πr/√(64-πr²) = 0

  64 -πr² = 4r²

  r = √(64/(π+4)) . . . . same as above

_____

Comment on the graph

The attached graph shows the relationship between perimeter and circumference for a constant area. The green curve shows the sum of perimeter and circumference, the wire length. The points marked are the ones at the minimum and maximum wire length.

The minimum length of the wire can be found by setting up a function to represent the total length of the wire and using calculus to minimize it. The maximum length of the wire is undefined because the length of the wire can increase indefinitely as the radius of the circle decreases.

To solve this problem, we use the formulas for the perimeters of a square and a circle, and the fact that the sum of their areas should equal 64 cm2. The perimeter of a square is 4s and the circumference of a circle is [tex]2\pi r[/tex], where s and r represent the side length of the square and the radius of the circle, respectively. The area of a square is s2 and the area of a circle is πr2.

The total length of the wire is the sum of the perimeter of the square and the circumference of the circle. The total area enclosed by the wire, according to the problem, should be 64 cm2.

To find the minimum length of wire needed, we can use calculus to minimize the function representing the length of the wire. The maximum length of the wire is undefined because as the radius of the circle approaches zero, the side length of the square and therefore the length of the wire can increase indefinitely.

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Answer:

3. [tex]\displaystyle 1\frac{1}{3} = x[/tex]

2C. [tex]\displaystyle III.[/tex]

2B. [tex]\displaystyle I.[/tex]

2A. [tex]\displaystyle II.[/tex]

1. [tex]\displaystyle Set-Builder\:Notation: {x|7, 0 ≠ x} \\ Interval\:Notation: (-∞, 0) ∪ (0, 7) ∪ (7, ∞)[/tex]

Step-by-step explanation:

3. See above.

2C. The keyword is ratio, which signifies division, so you would choose "III.".

2B. The keyword is percent, which signifies multiplication of a ratio by 100, so you would choose "I.".

2A. The keyword is total, which signifies addition, so you would choose "II.".

1. Base this off of the denominator. Knowing that the denominator CANNOT be zero, you will get this:

[tex]\displaystyle x^2 - 7x \\ x[x - 7] = 0; 7, 0 = x \\ \\ Set-Builder\:Notation: {x|7, 0 ≠ x} \\ Interval\:Notation: (-∞, 0) ∪ (0, 7) ∪ (7, ∞)[/tex]

I am joyous to assist you anytime.

Answer:

\[(-\infty ,0)\cup (0,7)\cup (7,\infty )\]

Step-by-step explanation:

Given expression is \[14 - 2x / x^{2} - 7x\]

For this rational expression to be valid it must satisfy the constraint that the denominator is not equal to 0.

This implies that \[x^{2} - 7x = 0\]  should be false.

In order words \[x*(x-7) = 0\] should be false.

Or, x=0, x=7 must be false.

Hence the domain restriction that applies is as follows :

\[(-\infty ,0)\cup (0,7)\cup (7,\infty )\]

For this case we have the following equation:

[tex]I = \frac {nE} {nr + R}[/tex]

We must clear the variable "n", for them we follow the steps below:

We multiply by [tex]nr + R[/tex] on both sides of the equation:

[tex]I (nr + R) = nE[/tex]

We apply distributive property on the left side of the equation:

[tex]Inr + IR = nE[/tex]

Subtracting [tex]nE[/tex] from both sides of the equation:

[tex]Inr-nE + IR = 0[/tex]

Subtracting IR from both sides of the equation:

[tex]Inr-nE = -IR[/tex]

We take common factor n from the left side of the equation:

[tex]n (Ir-E) = - IR[/tex]

We divide between Ir-E on both sides of the equation:

[tex]n = - \frac {IR} {Ir-E}[/tex]

Answer:

[tex]n = - \frac {IR} {Ir-E}[/tex]

Answer:

2

Step-by-step explanation:

y=a sin x

amplitude=|a|

a=-2

amplitude=2

Answer:

y(t) = a sin(ωt).

Step-by-step explanation:

The graph of the motion starts at y-0 t = 0 so we use sine in the equation

y(t) = A sin (2π t / T) where A = the amplitude and T = the period so here we  can write:

Displacement at t = y(t) = a sin(2π/ 2π/ω)t

y(t) = a sin(ωt)

This is about graph of simple harmonic motion.

y(t) = a sin (ωt)

y(t) = A sin ωt

Where;

A is amplitude

ω is angular frequency

y(t) is the displacement at time(t)

ω can also be expressed as;

ω = 2π/T

Where T is period.

Thus;

y(t) = A sin (2π/T)t

Period; T = 2π/ω

Thus

y(t) = A sin (2π/(2π/ω))t

2π will cancel out to give;

y(t) = A sin (ωt)

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Answer:

The dimensions of the pen that minimize the cost of fencing are:

[tex]x \approx 12.25 \:ft[/tex]

[tex]y \approx 8.17 \:ft[/tex]

Step-by-step explanation:

Let [tex]x[/tex] be the width and [tex]y[/tex] the length of the rectangular pen.

We know that the area of this rectangle is going to be [tex]x\cdot y[/tex].The problem tells us that the area is 100 feet, so we get the constraint equation:

[tex]x\cdot y=100[/tex]

The quantity we want to optimize is going to be the cost to make our fence. If we have chain-link on three sides of the pen, say one side of length [tex]y[/tex] and both sides of length [tex]x[/tex], the cost for these sides will be

[tex]10(y+2x)[/tex]

and the remaining side will be fence and hence have cost

[tex]20y[/tex]

Thus we have the objective equation:

[tex]C=10(y+2x)+20y\\C=10y+20x+20y\\C=30y+20x[/tex]

We can solve the constraint equation for one of the variables to get:

[tex]x\cdot y=100\\y=\frac{100}{x}[/tex]

Thus, we get the cost equation in terms of one variable:

[tex]C=30(\frac{100}{x})+20x\\C=\frac{3000}{x}+20x[/tex]

We want to find the dimensions that minimize the cost of the pen, for this reason, we take the derivative of the cost equation and set it equal to zero.

[tex]\frac{d}{dx} C=\frac{d}{dx} (\frac{3000}{x}+20x)\\\\\mathrm{Apply\:the\:Sum/Difference\:Rule}:\quad \left(f\pm g\right)'=f\:'\pm g'\\\\C'(x)=\frac{d}{dx}\left(\frac{3000}{x}\right)+\frac{d}{dx}\left(20x\right)\\\\C'(x)=-\frac{3000}{x^2}+20[/tex]

[tex]C'(x)=-\frac{3000}{x^2}+20=0\\\\-\frac{3000}{x^2}x^2+20x^2=0\cdot \:x^2\\-3000+20x^2=0\\-3000+20x^2+3000=0+3000\\20x^2=3000\\\frac{20x^2}{20}=\frac{3000}{20}\\x^2=150\\\\\mathrm{For\:}x^2=f\left(a\right)\mathrm{\:the\:solutions\:are\:}x=\sqrt{f\left(a\right)},\:\:-\sqrt{f\left(a\right)}\\\\x=\sqrt{150},\:x=-\sqrt{150}[/tex]

Because length must always be zero or positive we take [tex]x=\sqrt{150}[/tex] as only value for the width.

To check that this is indeed a value of [tex]x[/tex] that gives us a minimum, we need to take the second derivative of our cost function.

[tex]\frac{d}{dx} C'(x)=\frac{d}{dx} (-\frac{3000}{x^2}+20)\\\\C''(x)=-\frac{d}{dx}\left(\frac{3000}{x^2}\right)+\frac{d}{dx}\left(20\right)\\\\C''(x)=\frac{6000}{x^3}[/tex]

Because [tex]C''(\sqrt{150})=\frac{6000}{\left(\sqrt{150}\right)^3}=\frac{4\sqrt{6}}{3}[/tex] is greater than zero, [tex]x=\sqrt{150}[/tex] is a minimum.

Now, we need values of both x and y, thus as [tex]y=\frac{100}{x}[/tex], we get

[tex]x=\sqrt{150}=5\sqrt{6}=12.25[/tex]

[tex]y=\frac{100}{\sqrt{150}}=\frac{10\sqrt{6}}{3}\approx 8.17[/tex]

The dimensions of the pen that minimize the cost of fencing are:

[tex]x \approx 12.25 \:ft[/tex]

[tex]y \approx 8.17 \:ft[/tex]

Answer:

Y - intercept of equation of-line A is y = 1.

Y - intercept of equation of-line B is y = -2.

Step-by-step explanation:

Given:

For Iine A:

[tex]y+ 1 =\frac{1}{5}\times (x+10)[/tex]

For line B:

x = -2  then y = 2

x = -1  then y = 0

x = 0  then y = -2

x = 1  then y = -4

To Find:

Y- intercepts of Line A and Line B.

Solution:

Intercepts: Where the line cut X axis called X- intercept and where cut Y axis is called Y- intercept.

Y-intercept mean x coordinate will be 0

Therefore Put x = 0 in Line A we get

[tex]y+ 1 =\frac{1}{5}\times (0+10)\\y+1=\frac{10}{5}\\ y+1=2\\y= 2-1\\y=1[/tex]

Y - intercept of equation of-line A is y = 1.

For line B

See where x coordinate is 0 ,Therefore we have,

y = -2

Y - intercept of equation of-line B is y = -2.

Answer:

  14 feet

Step-by-step explanation:

There are 3 feet in 1 yard, so 36 feet in 12 yards. The remaining ribbon will be the original amount less the amount used.

  36 - 22 = 14 . . . . feet remaining

After decorating the blankets, Olga will have 14 feet of ribbon remaining. The conversion from yards to feet and subtraction calculates this remaining amount accurately.

Calculating Remaining Ribbon

To determine how much ribbon Olga has left after decorating the blankets, we need to perform a couple of conversions and a subtraction.

First, let's convert the total ribbon from yards to feet:

→ 1 yard = 3 feet

→ 12 yards = 12 * 3

                 = 36 feet

Next, Olga uses 22 feet of ribbon to decorate the blankets:

→ Total ribbon in feet: 36 feet

→ Ribbon used: 22 feet

Now, subtract the amount used from the total:

→ Remaining ribbon = 36 feet - 22 feet

                                 = 14 feet

Olga will have 14 feet of ribbon remaining.

Answer:

a. $104.17 monthly interest.

b. 120 monthly payments.

c. Total interest of $12,500.

Step-by-step explanation:

a.

I = Prt

I = (20000 x 0.0625 x 1) = 1250 annually

for monthly Interest payment divide the answer by 12;

1250/12 = $104.17 monthly

b.

12 x 10 = 120 monthly payments

c.

I = Prt

I = $20,000 x 0.0625 x 10

I = $12,500

Answer:

Does not exceed 2% in both cases.

Step-by-step explanation:

Given that a  manufacturer of interocular lenses will qualify a new grinding machine if there is evidence that the percentage of polished lenses that contain surface defects does not exceed 2%.

Sample proportion = [tex]\frac{6}{250} \\=0.024[/tex]

Create hypothesses as

[tex]H_0: p = 0.02\\H_a : p >0.02[/tex]

(Right tailed test at 5% significance level)

P difference = 0.004

Std error = 0.0089

test statistic Z = p diff/std error = 0.4518

p value = 0.326

Since p >alpha, we accept nullhypothesis

b) For confidence interval 97% we have

Margin of error = 2.17* std error = 0.0192

Confidence interval

= [tex](0.024-0.0191, 0.024+0.0191)\\= (0.0049, 0.0431)\\[/tex]

Since 2% = 0.02 lies within this interval we accept null hypothesis.

Does not exceed 2%

Since

Final answer:

To determine whether the new grinding machine can be qualified, we need to test the hypothesis that the percentage of polished lenses with surface defects does not exceed 2%.

Explanation:

To determine whether the new grinding machine can be qualified, we need to test the hypothesis that the percentage of polished lenses with surface defects does not exceed 2%. The null hypothesis (H₀) is that the proportion of defective lenses is equal to or less than 2%, while the alternative hypothesis (Ha) is that the proportion of defective lenses is greater than 2%. We can use a one-sample proportion test to analyze the data.

(a) The hypotheses are:

H₀: p ≤ 0.02 (proportion of defective lenses)

Ha: p > 0.02 (proportion of defective lenses)

With a significance level of α = 0.05, we can calculate the p-value from the sample data. Using a normal approximation to the binomial distribution, we find that the p-value is 0.0165.

(b) The question in part (a) can also be answered using a confidence interval. We can calculate a confidence interval for the proportion of defective lenses and see if it includes or excludes the value of 0.02. If the confidence interval includes 0.02, it suggests that the machine can be qualified. If the confidence interval does not include 0.02, it suggests that the machine cannot be qualified.

Answer:

You didn't write the expression

Step-by-step explanation:

Answer:Glass ceiling

Step-by-step explanation:

Samantha is feeling the effect of glass ceiling .A glass ceiling is a term used to describe an unseen barrier that prevents a particular demographic (usually applied to minorities) from increasing in a hierarchy beyond a certain level.

Here The phrase “glass ceiling” is used to describe the difficulties faced by women when trying to move to higher roles in a male-dominated hierarchy.

Answer:

3.3 in other words between 2-5

Step-by-step explanation:

33% of 100 is 33 so

33% of 10 is 3.3

Answer:

œ=50

Step-by-step explanation:

Solve the equation: 120+œ+60+15+20 = 130+35+2œ

Simplifies as follows: œ+215 = 2œ+165

œ = 215 - 165 = 50

Answer:

The speed in the first part is 16 mph.

Step-by-step explanation:

The total distance is 48 miles. The total time is 4 hours.

In the second part of the trip, he was traveling at speed s for distance d for 2 hours.

In the first part of the trip he was traveling at twice the speed, or 2s, for distance 48 - d, for 2 hours.

speed = distance/time

distance = speed * time

First part of trip:

48 - d = 2s * 2

d + 4s = 48   Equation 1

Second part of the trip:

d = s * 2

d = 2s    Equation 2

Equations 1 and 2 form a system of equations in two unknowns, d and s.

d + 4s = 48

d = 2s

Substitute 2s for d in equation 1.

2s + 4s = 48

6s = 48

s = 8

The speed in the second part is 8 mph.

The speed in the first part is 2s = 2(8) = 16.

The speed in the first part is 16 mph.

Check:

d = 2s = 2(8) = 16

48 - d = 48 - 16 = 32

The second part is 16 miles. The first part is 32 miles.

16 miles at 8 mph takes 2 hours.

32 miles at 16 mph takes 2 hours.

2 hours + 2 hours = 4 hours.

Our answer is correct.

Answer: The speed in the first part is 16 mph.

Answer:  

The speed in the first part is 16 mph.  

Step-by-step explanation:

The total distance is 48 miles. The total time is 4 hours.  

In the second part of the trip, he was traveling at speed s for distance d for 2 hours.  

In the first part of the trip he was traveling at twice the speed, or 2s, for distance 48 - d, for 2 hours.  

speed = distance/time  

distance = speed * time  

First part of trip:  

48 - d = 2s * 2  

d + 4s = 48   Equation 1  

Second part of the trip:  

d = s * 2  

d = 2s    Equation 2  

Equations 1 and 2 form a system of equations in two unknowns, d and s.  

d + 4s = 48  

d = 2s  

Substitute 2s for d in equation 1.  

2s + 4s = 48  

6s = 48  

s = 8  

The speed in the second part is 8 mph.  

The speed in the first part is 2s = 2(8) = 16.  

The speed in the first part is 16 mph.  

Check:  

d = 2s = 2(8) = 16  

48 - d = 48 - 16 = 32  

The second part is 16 miles. The first part is 32 miles.  

16 miles at 8 mph takes 2 hours.  

32 miles at 16 mph takes 2 hours.  

2 hours + 2 hours = 4 hours.  

Our answer is correct.

Answer: The speed in the first part is 16 mph