.Sergei is training to be a weightlifter. Each day he trains at the local gym by lifting a metal bar that has heavy weights attached. He carries out successive lifts. After each lift, the same amount of weight is added to the bar to increase the weight to be lifted. The weights of each of Sergei's lifts form an arithmetic sequence. Sergei's friend, Yuri, records the weight of each lift. Unfortunately, last Monday, Yuri misplaced all but two of the recordings of Sergei's lifts. On that day, Sergei lifted 21 kg on the third lift and 46 kg on the eighth lift. (a) For that day (i) find how much weight was added after each lift; (ii) find the weight of Sergei's first lift. [4] On that day, Sergei made 12 successive lifts. (b) Find the total combined weight of these lifts.

Answer:

a) It would have taken 1 person 14 days to do this amount of work.

b) It would take 1 person 56 days to finish building this house extension.

We know that 1 person can build a house extension in 70 days. This means that in 1 day, 1 person can do 1/70 of the work. In 7 days, 2 people can do 2 * 7 * 1/70 = 2/10 of the work. This means that 1 person can do 1/10 of the work in 7 days. To finish the remaining 8/10 of the work, 1 person would need 7 * 8 = 56 days.

Step-by-step explanation:

The calculated value of t_obs = -4.823.

We can use the formula for the t-value in an independent samples t-test, which is as follows:

\(t = (M_1 - M_2) / \sqrt{ [ (SD_1^2/n1) + (SD_2^2/n_2) ]}\)

Where:

M1, M2 are the means of the two groups,

SD1, SD2 are the standard deviations of the two groups,

n1, n2 are the sizes of the two groups.

Plugging in your values, we get:

t = (34.4 - 38.4) / √[ (2²) + (0.9²) ]

t = -4 / √[ (4/7) + (0.81/7) ]

t = -4 / √[0.5714 + 0.1157]

t = -4 /√[0.6871]

t = -4 / 0.8294

t = -4.823

So, t_obs = -4.823.

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The main answer is: The 2-tailed independent t-test, comparing sample 1 (M=34.4, SD=2) and sample 2 (M=38.4, SD=0.9) with equal sample sizes of 7, at α=0.01, determines if there is a significant difference between their means.

1. To conduct a 2-tailed independent t-test, we can follow these steps:

State the null hypothesis (H0) and alternative hypothesis (H1):

2. Set the significance level (α) to 0.01.

3. Calculate the degrees of freedom (df) using the formula:

df = (n1 + n2) - 2

In this case, both samples have a sample size of 7, so the degrees of freedom would be 12.

4. Calculate the pooled standard deviation (sp) using the formula:

sp = sqrt(((n1 - 1) * s1^2 + (n2 - 1) * s2^2) / df)

where s1 and s2 are the standard deviations of the two samples.

In this case, s1 = 2 and s2 = 0.9.

5. Calculate the t-value using the formula:

t = (M1 - M2) / (sp * sqrt(1/n1 + 1/n2))

where M1 and M2 are the means of the two samples.

In this case, M1 = 34.4 and M2 = 38.4.

6. Determine the critical t-value using the t-distribution table or a statistical software package with α = 0.01 and df = 12.

7. Compare the calculated t-value with the critical t-value. If the calculated t-value falls within the critical region (rejecting region), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Please note that the exact critical t-value and the outcome of the test depend on the specific values calculated in steps 4 and 5.

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Answer:

First rationalize the surd

(√3 +√2)(√3 - √2)/(√3 - √2)(√3 -√2)

The final answer will be 5 + 2√6

So x = 5 + 2√6

I) x + 1/x

5 + 2√6 + 1/5 + 2√6

Find the LCM

The final answer will be 10

ii) x² +1/x²

(5 + 2√6)² + 1/(5 +2√6)²

= 49 + 20√6 + 1/49 + 20√6

Find the LCM

The final answer will be 98

For i) x = 10

ii) x = 98

Hope this helps.

Answer:

i) 10

ii) 98

Step-by-step explanation:

x= (√3+√2)/(√3-√2)

i)

ii)

Answer:

Line FJ is -3/1, while Line GH is 3/-2

Step-by-step explanation:

By completing the square, the equation in standard form is (x - 2)² + (y + 4)² = 4².

In Mathematics and Geometry, the standard form of the equation of a circle is modeled by this mathematical equation;

(x - h)² + (y - k)² = r²

Where:

From the information provided below, we have the following equation of a circle:

x² - 4x + y² + 8y = -4

x² - 4x + (-4/2)² + y² + 8y + (8/2)² = -4 + (-4/2)² + (8/2)²

x² - 4x + 4 + y² + 8y + 16 = -4 + 4 + 16

(x - 2)² + (y + 4)² = 16

(x - 2)² + (y + 4)² = 4²

Therefore, the center (h, k) is (2, -4) and the radius is equal to 4 units.

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Complete Question:

Complete the square to rewrite the following equation in standard form. x² - 4x + y² + 8y = -4.

Horizontal translation: 2 units right.

Vertical translation: 5 units up

Stretch/compression: stretched by a factor of 3.

Reflection: not reflected.

13x + 3x + 2x = 180°

=> 18x = 180°

=> x = 180°/18

=> x = 10°

Ec is a diameter solve for x well i really dont know

In this conversion, we assume that θ is not equal to 0 or any multiple of π, as csc(θ) is undefined for those values.

In rectangular coordinates, the equation r = 2csc(θ) can be expressed as:

x = 2cos(θ)

y = 2sin(θ)

To convert the polar equation r = 2csc(θ) to rectangular coordinates, we need to express the equation in terms of x and y.

In polar coordinates, r represents the distance from the origin (0,0) to a point (x, y), and θ represents the angle between the positive x-axis and the line segment connecting the origin to the point.

To convert r = 2csc(θ) to rectangular coordinates, we can use the following relationships:

x = r * cos(θ)

y = r * sin(θ)

First, let's express csc(θ) in terms of sin(θ):

csc(θ) = 1 / sin(θ)

Now, substitute r = 2csc(θ) into the equations for x and y:

x = (2csc(θ)) * cos(θ)

y = (2csc(θ)) * sin(θ)

Using the relationship between csc(θ) and sin(θ), we can rewrite the equations as:

x = (2/sin(θ)) * cos(θ)

y = (2/sin(θ)) * sin(θ)

Simplifying further:

x = 2cos(θ)

y = 2sin(θ)

Therefore, in rectangular coordinates, the equation r = 2csc(θ) can be expressed as:

x = 2cos(θ)

y = 2sin(θ)

Note: In this conversion, we assume that θ is not equal to 0 or any multiple of π, as csc(θ) is undefined for those values.

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Correct question- How do you convert the polar equation  r = 8cscθ into rectangular form?

x ≈ -1.97 x ≈ -0.26  x ≈ 4.11  x ≈ 5.11  These are the approximate values of x where P(x) equals zero.

To find the zeros of the function P(x) = x⁴ - 4x³ - x² + 20x - 20, we need to solve the equation P(x) = 0.

There is no simple algebraic method to find the exact solutions for quartic equations in general. However, we can use numerical methods or factorization techniques to find the approximate solutions.

Using a numerical method or a graphing calculator, we find that the approximate zeros of the function P(x) are:

x ≈ -1.97

x ≈ -0.26

x ≈ 4.11

x ≈ 5.11

These are the approximate values of x where P(x) equals zero.

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Step-by-step explanation:

remember, a square has 4 equally long sides.

(a)

the area of a square is

side length × side length = (side length)²

hence the name "square".

therefore, the side length is the square root of the area.

sqrt(81) = 9

so, the length of each side is 9.

(b)

the perimeter of a square is

4 × side length

therefore, side length = perimeter/4

49/4 = 12.25

so, the length of each side is 12.25.

The area of the regular polygons with 12 sides(dodecagon) and 5 sides (pentagon) are 389.06 in² and 19.87 in² respectively.

Area of regular polygon = 1/2 × apothem × perimeter

perimeter = (s)side length of octagon × (n)number of side.

apothem = s/[2tan(180/n)].

11 = s/[2tan(180/12)]

s = 11 × 2tan15

s = 5.8949

perimeter = 5.8949 × 12 = 70.7388

Area of dodecagon = 1/2 × 11 × 70.7388

Area of dodecagon = 389.0634 in²

Area of pentagon = 1/2 × 5.23 × 7.6

Area of pentagon = 19.874 in²

Therefore, the area of the regular polygons with 12 sides(dodecagon) and 5 sides (pentagon) are 389.06 in² and 19.87 in² respectively.

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First, we need to set up our two equations. For the picture of this scenario, there is one length (L) and two widths (W) because the beach removes one of the lengths. We will have a perimeter equation and an area equation.

P = L + 2W

A = L * W

Now that we have our equations, we need to plug in what we know, which is the 40m of rope.

40 = L + 2W

A = L * W

Then, we need to solve for one of the variables in the perimeter equation. I will solve for L.

L = 40 - 2W

Now, we can substitute the value for L into L in the area equation and get a quadratic equation.

A = W(40 - 2W)

A = -2W^2 - 40W

The maximum area will occur where the derivative equals 0, or at the absolute value of the x-value of the vertex of the parabola.

V = -b/2a

V = 40/2(2) = 40/4 = 10

Derivative:

-4w - 40 = 0

-4w = 40

w = |-10| = 10

To find the other dimension, use the perimeter equation.

40 = L + 2(10)

40 = L + 20

L = 20m

Therefore, the dimensions of the area are 10m by 20m.

Hope this helps!

Answer:

Width: 10 m

Length: 20 m

Step-by-step explanation:

Hi there!

Let w be equal to the width of the enclosure.

Let l be equal to the length of the enclosure.

1) Construct equations

\(A=lw\) ⇒ A represents the area of the enclosure.

\(40=2w+l\) ⇒ This represents the perimeter of the enclosure. Normally, P=2w+2l, but because one side isn't going to use any rope (sandy beach), we remove one side from this equation.

2) Isolate one of the variables in the second equation

\(40=2w+l\)

Let's isolate l. Subtract 2w from both sides.

\(40-2w=2w+l-2w\\40-2w=l\)

3) Plug the second equation into the first

\(A=lw\\A=(40-2w)w\\A=40w-2w^2\\A=-2w^2+40w\)

Great! Now that we have a quadratic equation, we can do the following:

4) Solve for w

\(A=-2w^2+40w\)

Factor out -2w

\(A=-2w(w-20)\)

For A to equal 0, w=0 or w=20.

The average of 0 and 20 is 10, so the width that will max the area is 10 m.

5) Solve for l

\(40=2w+l\)

Plug in 10 as w

\(40=2(10)+l\\40=20+l\\l=20\)

Therefore, the length of 20 m will max the area.

I hope this helps!

Answer:

Mode is 1, Mean is 3.1 if the '41" is 1 and 4

Step-by-step explanation:

Answer:

sixty-seven thousand ninety-two

The integer that makes the inequality 3(n − 6) < 2(n + 12) false is

The integer that makes the inequality false is solved by substituting the values and solving the inequality

3(n − 6) < 2(n + 12)

for n = -5

substituting the value of n

3(-5 − 6) < 2(-5 + 12)

= -33 < 14

3(n − 6) < 2(n + 12)

for n = 3

substituting the value of n

3(3 − 6) < 2(3 + 12)

= -9 < 30

3(n − 6) < 2(n + 12)

for n = 12

substituting the value of n

3(12 − 6) < 2(12 + 12)

= 18 < 48

3(n − 6) < 2(n + 12)

for n = 42

substituting the value of n

3(42 − 6) < 2(42 + 12)

= 108 < 108 wrong

This is the only wrong solution since 108 = 108

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9514 1404 393

Answer:

  (a) Store A: (2, 40¢), (1, 20¢); Store B: (2, 30¢), (1, 15¢)

  (b) Store A: 20¢; Store B: 15¢; A's cost more

  (c) A: $1.00; B: $0.75

Step-by-step explanation:

If you have done any personal shopping, you are probably aware that the cost of 4 items is 4 times the cost of 1 item. If you are at all familiar with your multiplication tables, you probably know that ...

  2 = 2×1

  4 = 2×2

  6 = 2×3

__

(a) This knowledge makes it fairly easy to fill in the tables in part (a) of this question.

The cost of 4 pens is 2 times the cost of 2 pens. The cost of 2 pens will be half the cost of 4 pens, so will be 80¢/2 = 40¢. Similarly, the cost of 1 pen will be half the cost of 2 pens, so will be 40¢/2 = 20¢. Now the table looks like ...

  Pens, Cost

  4, 80¢

  2, 40¢

  1, 20¢

__

The cost of 6 pens is 3 times the cost of 2 pens, so the cost of 2 pens will be 1/3 of 90¢, or 30¢. The cost of 1 pen is half that, 15¢. So the table for Store B looks like ...

  Pens, Cost

  6, 90¢

  2, 30¢

  1, 15¢

__

(b) As we have seen above, the charge for 1 pen is ...

  Store A: 20¢

  Store B: 15¢

  Store A's pens cost more.

__

(c) The cost of 5 pens can be figured a number of ways. One way is to simply multiply the cost of 1 pen by 5. You can also add the cost of 1 pen to the cost of 4 pens, or subtract the cost of 1 pen from the cost of 6 pens.

  5 pens from Store A = 5 × 20¢ = 100¢ = $1.00

  5 pens from Store B = (cost of 6) - (cost of 1) = 90¢ -15¢ = 75¢ = $0.75

The x-axis denotes blocks and the y-axis denotes the blocks per game .

Scatter - Plot : Scatter plots are an essential type of data visualization that shows relationships between variables . The variables are plotted on the x - axis and the y - axis .

Each of the points on a scatter plot is represented by a dot . It shows how strongly two variables are related .

In the given question , we have to label the axes ,

Labelling axes of a Scatter-Plot :

x-axis denotes blocks

y-axis denotes blocks per game

Hence , the x-axis denotes blocks and the y-axis denotes the blocks per game .

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By using the concept of minima, it can be calculated that

The box which minimizes the cost of materials has a square base of side length 4.93 cm and a height of 1.64 cm.

What is minima of a function?

Minima of a function gives the minimum value of the function in  a certain interval of in the whole domain.

Let the length of the box be l cm, breadth = l cm, Height = h cm

Volume = l \(\times\) l \(\times\) h = \(l^2h\) \(cm^3\)

Now cost (C) = \(l^2+l^2 + 12lh + 12lh = 2l^2 + 24lh\)

                     = \(2l^2 + 24\times \frac{40}{l}\)

                     = \(2l^2 + \frac{480}{l}\)

\(\frac{dC}{dl} = 4l - \frac{480}{l^2}\)

Now,

\(4l - \frac{480}{l^2} = 0\\l^3 = \frac{480}{4}\\l^3 = 120\\l = \sqrt[3]{120} \\l = 4.93 cm\)

h = \(\frac{40}{4.93^2} = 1.64cm\)

\(\frac{d^2C}{dl^2} = 4 + \frac{960}{l^3} > 0\)

Hence cost is minimum

So The box which minimizes the cost of materials has a square base of side length 4.93 cm and a height of 1.64 cm.

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To find x in R² whose coordinate vector relative to the basis B is given, we can express x as a linear combination of the basis vectors in B and solve for the coefficients.

In order to find the coordinate vector of x relative to the basis B, we need to express x as a linear combination of the basis vectors in B. Let's assume that B = {v₁, v₂} is a basis for R², where v₁ and v₂ are vectors in R².

We can express x as a linear combination of v₁ and v₂:

x = c₁v₁ + c₂v₂

Here, c₁ and c₂ are the coefficients or coordinates of x relative to the basis B. These coefficients determine the unique representation of x in terms of the basis vectors.

To find the values of c₁ and c₂, we can solve the system of equations formed by equating the corresponding components of x and the linear combination:

x₁ = c₁v₁₁ + c₂v₂₁

x₂ = c₁v₁₂ + c₂v₂₂

Here, x₁ and x₂ are the components of x, v₁₁ and v₁₂ are the components of v₁, and v₂₁ and v₂₂ are the components of v₂.

By solving this system of equations, we can determine the values of c₁ and c₂, which give us the coordinate vector of x relative to the basis B in R².

Once we find the values of c₁ and c₂, the  coordinate vector of x relative to the basis B can be written as [c₁, c₂]. This vector represents the coefficients or weights that determine the linear combination of the basis vectors to form x.

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The similarity between <2 and <B is that both arcs have a 65° measure.

When two angles add up to 180 degrees, they are said to be each other's auxiliary angles.

For instance, since x + y = 180°, x and y are complementary angles to one another.

The triangle's angles are 50° and 65°.

50° and 65° are the angles on the path.

We are aware that a triangle's total number of angles is 180°. Consequently, the B measure is

50° + 65° + ∠B = 180°

∠B = 180° - (50° + 65°)

∠B = 65°

So, 65 degrees is the gauge of the B.

The total number of angles on the specified line will be due to the fact that all angles are extra angles, and will be 180 degrees.

∠2 + 50° + 65° = 180°

∠2 = = 65°

As a result, 65° is the measure of the 2.

As we can see, both arcs have a measure of 65 degrees.

Therefore, the relationship between 2 and B is that both angles have a value of 65°.

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The probability that any two children will both like lima beans is 0.0016 or 0.16%.

Assuming that every child's choice for lima beans is independent of the other, we will use the binomial distribution to calculate the probability that two kids out of a group of will both like lima beans.

Let p be the chance that a child likes lima beans, which we realize is 0.04 primarily based on the survey. The probability that anyone infant does no longer like lima beans is then 1 - p = 0.96.

The possibility that each children will like lima beans is given through the binomial chance mass feature:

P(X = 2) = (2 choose 2) * \(p^2 * (1 - p)^{(2-2)} = p^2 = (0.04)^{2 }= 0.0016\)

So the chance that any two children will both like lima beans is 0.0016, or 0.16%. that is a very low probability, this means that that it is not likely that any two children selected at random will both like lima beans.

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Answer:

each notebook costs $2.70

each pack of pencils costs $1.50

Step-by-step explanation:

system of equations:

let p = pack of pencils

let n = notebook

3p + 5n = 18

4p + 4n = 16.8

I used the elimination method by multiplying the first equation by 4 and the second equation by -3

4(3p + 5n = 18)  =  12p + 20n = 72

-3(4p + 4n = 16.8)  =  -12p -12n = -50.4

adding the new equations together you get:  8n = 21.6

n = 21.6/8

n = $2.70

solve for 'p':

3p + 5(2.7) = 18

3p + 13.5 = 18

3p = 4.5

p = $1.50

Answer:

I would go with letter B. -7

Step-by-step explanation:

The two factors of a quadratic equation can be multiplied to form the original equation. Let the missing term be a:

y² + 15y + 56 = (y + 7)(y + a)

y² + 15y + 56 = y² + (7 + a)y + 7a

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

Step by Step Solution

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Reformatting the input :

Changes made to your input should not affect the solution:

Step by step solution :

STEP  1 :

Trying to factor by splitting the middle term

1.1     Factoring  y2+14y+49  

The first term is,  y2  its coefficient is  1 .

The middle term is,  +14y  its coefficient is  14 .

The last term, "the constant", is  +49  

Step-1 : Multiply the coefficient of the first term by the constant   1 • 49 = 49  

Step-2 : Find two factors of  49  whose sum equals the coefficient of the middle term, which is   14 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  7  and  7  

                    y2 + 7y + 7y + 49

Step-4 : Add up the first 2 terms, pulling out like factors :

                   y • (y+7)

             Add up the last 2 terms, pulling out common factors :

                   7 • (y+7)

Step-5 : Add up the four terms of step 4 :

                   (y+7)  •  (y+7)

The solution is:

  y  =  -7  

 The solution is:

 y = -14 / 2 = -7

Answer:

Y =-2.5X +3.5

Step-by-step explanation:

x1 y1  x2 y2

-1 6  3 -4

(Y2-Y1) (-4)-(6)=   -10  ΔY -10

(X2-X1) (3)-(-1)=    4  ΔX 4

slope= -2  1/2    

B= 3  1/2    

Y =-2.5X +3.5    

Answer:

About 18.84 square centimetres

Step-by-step explanation:

The surface area of a cone is A=πr²+πrs

Since d=2cm, r=1cm because d=2r

Now we have everything we need to solve:

A=π×1²+π×1×5

A=π+5π=6π

A=6×3.14=18.84cm²

Hope this helps!

Answer:

53 mm

Step-by-step explanation:

Perimeter is the sum of the lengths of all its sides.

Let the length of the unknown side be x mm.

123= 25 +45 +x

x= 123 -25 -45

x= 53

Thus, the unknown side is 53 mm.

Answer:

156 carries

Step-by-step explanation:

Given that:

5 games = 498 yards

Number of yards Gained per carry = 4.5 yards

Target number of yards = 1200

Number of yards left to reach target :

1200 - 498 = 702 yards

Number of carries required = number of yards / yards per carry

Number of carries required = 702 / 4.5

= 156 carries

The indefinite integral of x(5x^2 - 5)^9 dx is:

(5x^2 - 5)^8 / 40 + c

We can find the indefinite integral using the following steps:

1. We can write the integral as (5x^2 - 5)^9 * x^1 dx.

2. We can use the power rule of integration, which states that the integral of x^n dx is x^(n + 1) / (n + 1) + c, where c is the constant of integration.

3. We can simplify the result and add the constant of integration.

The following is the step-by-step solution:

```

∫ x(5x^2 - 5)^9 dx = ∫ (5x^2 - 5)^9 * x^1 dx

= (5x^2 - 5)^9 / 9 + c

```

To check the result, we can differentiate the result and see if we get the original integral.

```

d/dx [(5x^2 - 5)^8 / 40 + c] = (5x^2 - 5)^8 * (10x) / 40 + 0 = x(5x^2 - 5)^8 = ∫ x(5x^2 - 5)^9 dx

```

As we can see, we get the original integral back. Therefore, the answer is correct.

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