Help me with this question

Since [tex]a_1,a_2,a_3,\cdots[/tex] are in arithmetic progression,

[tex]a_2 = a_1 + 2[/tex]

[tex]a_3 = a_2 + 2 = a_1 + 2\cdot2[/tex]

[tex]a_4 = a_3+2 = a_1+3\cdot2[/tex]

[tex]\cdots \implies a_n = a_1 + 2(n-1)[/tex]

and since [tex]b_1,b_2,b_3,\cdots[/tex] are in geometric progression,

[tex]b_2 = 2b_1[/tex]

[tex]b_3=2b_2 = 2^2 b_1[/tex]

[tex]b_4=2b_3=2^3b_1[/tex]

[tex]\cdots\implies b_n=2^{n-1}b_1[/tex]

Recall that

[tex]\displaystyle \sum_{k=1}^n 1 = \underbrace{1+1+1+\cdots+1}_{n\,\rm times} = n[/tex]

[tex]\displaystyle \sum_{k=1}^n k = 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}2[/tex]

It follows that

[tex]a_1 + a_2 + \cdots + a_n = \displaystyle \sum_{k=1}^n (a_1 + 2(k-1)) \\\\ ~~~~~~~~ = a_1 \sum_{k=1}^n 1 + 2 \sum_{k=1}^n (k-1) \\\\ ~~~~~~~~ = a_1 n +  n(n-1)[/tex]

so the left side is

[tex]2(a_1+a_2+\cdots+a_n) = 2c n + 2n(n-1) = 2n^2 + 2(c-1)n[/tex]

Also recall that

[tex]\displaystyle \sum_{k=1}^n ar^{k-1} = \frac{a(1-r^n)}{1-r}[/tex]

so that the right side is

[tex]b_1 + b_2 + \cdots + b_n = \displaystyle \sum_{k=1}^n 2^{k-1}b_1 = c(2^n-1)[/tex]

Solve for [tex]c[/tex].

[tex]2n^2 + 2(c-1)n = c(2^n-1) \implies c = \dfrac{2n^2 - 2n}{2^n - 2n - 1} = \dfrac{2n(n-1)}{2^n - 2n - 1}[/tex]

Now, the numerator increases more slowly than the denominator, since

[tex]\dfrac{d}{dn}(2n(n-1)) = 4n - 2[/tex]

[tex]\dfrac{d}{dn} (2^n-2n-1) = \ln(2)\cdot2^n - 2[/tex]

and for [tex]n\ge5[/tex],

[tex]2^n > \dfrac4{\ln(2)} n \implies \ln(2)\cdot2^n - 2 > 4n - 2[/tex]

This means we only need to check if the claim is true for any [tex]n\in\{1,2,3,4\}[/tex].

[tex]n=1[/tex] doesn't work, since that makes [tex]c=0[/tex].

If [tex]n=2[/tex], then

[tex]c = \dfrac{4}{2^2 - 4 - 1} = \dfrac4{-1} = -4 < 0[/tex]

If [tex]n=3[/tex], then

[tex]c = \dfrac{12}{2^3 - 6 - 1} = 12[/tex]

If [tex]n=4[/tex], then

[tex]c = \dfrac{24}{2^4 - 8 - 1} = \dfrac{24}7 \not\in\Bbb N[/tex]

There is only one value for which the claim is true, [tex]c=12[/tex].

Answer:

B

Step-by-step explanation:

The beginning temperature is -12 and then it rises 5 degrees each hour at the end of the game it is 32 degrees.

Using the normal distribution, it is found that:

a. A male sailor whose waist is 34.1 inches is at the 87.5th percentile.

b. 5.7% of male sailors requires custom uniform pants.

The z-score of a measure X of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The mean and the standard deviation are given, respectively, by:

[tex]\mu = 32.6, \sigma = 1.3[/tex]

For item a, the percentile is the p-value of Z when X = 34.1, hence:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

Z = (34.1 - 32.6)/1.3

Z = 1.15

Z = 1.15 has a p-value of 0.875.

Hence 87.5th percentile.

For item b, the proportion who does not require an special order is the p-value of Z when X = 36 subtracted by the p-value of Z when X = 30, hence:

X = 36:

Z = (36 - 32.6)/1.3

Z = 2.62

Z = 2.62 has a p-value of 0.996.

X = 30:

Z = (30 - 32.6)/1.3

Z = -2

Z = -2 has a p-value of 0.023.

0.996 - 0.023 = 0.943.

Hence the proportion who requires an special order is:

1 - 0.943 = 0.057.

5.7% of male sailors requires custom uniform pants.

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The perimeter of a given figure is a measure of the addition of each individual length of the sides of the figure. Thus the dimensions that would produce the largest area of the field are; length = 300 m, and width = 200 meters.

The perimeter of a given figure is a measure of the addition of each length of the sides of the figure. It always has the unit as that of the given sides of the figure.

So in the given question, the perimeter of the fence required = 1200 meters.

Thus, let the length of the enclosed rectangle be represented by l and its width by w. Thus,

Perimeter = 2l + 3w

1200 = 2l + 3w

Thus, let l be equal to 300, we have;

1200 = 2(300) + 3w

1200 - 600 = 3w

w = 200

Thus the dimensions of the field that would produce the largest area are; length = 300 m and width = 200 m.

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Answer:

length = 24 meters
breadth = 16 meters

Step-by-step explanation:

Let L be length and B be breadth

From the first fact we get the equation
L = B + 8         (1)

We know that perimeter = 2(L+B) and this is given as 80

So 2(L+B) = 80

L + B = 80/2 = 40
or
L = 40 - B              (2)

If we add equations (1) and (2) we can eliminate B

We get 2L = B + 8 + 40 - B =48

L = 48/2 = 24

Substituting for L in equation (1) we get

24 = B + 8  ==>

B = 24-8 =16

Cross-check

Perimeter = 2 (L+B) = 2(24 + 16) = 2(40) = 80

Hence check OK

The given set of functions are not linearly independent.

Given,

[tex]f_{1} (x) = x\\f_{2} (x) = x^{2} \\f_{3} (x) = 6x-2x^{2}[/tex]

We need,

[tex]c_{1} f_{1} (x)+c_{2} f_{2} (x)+c_{3} f_{3}(x)=0[/tex]

Substituting the values in equation we get,

[tex]c_{1} x+c_{2} x^{2} +c_{3} (6x-2x^{2} )=0\\[/tex]

Computing the equation we get,

[tex]c_{1} x+c_{2} x^{2} +c_{3} 6x-c_{3} 2x^{2}=0[/tex]

[tex](c_{1} +6c_{3} )x+(c_{2} -2c_{3} x^{2} =0[/tex]

This resolves to two equations

[tex](c_{1} +6c_{3})x =0\\(c_{2} -2c_{3} )x^{2} =0[/tex]

These will have an infinite set of solutions:

[tex]c_{1} =-6c_{3} \\c_{2} =2c_{3}[/tex]

Two functions are said to be linearly independent if neither function is a constant multiple of the other.

Here, it is clear that the given functions are not linearly independent.

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Step-by-step explanation:

Answer:

The all common Factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24

Shaded area = area of the hexagon – area of the pentagon + area of the square – area of the equilateral triangle. This can be obtained by finding each shaded area and then adding them.

From the question we can say that the Hexagon has three shapes inside it,

Also it is given that,

An equilateral triangle is shown inside a square inside a regular pentagon inside a regular hexagon.

From this we know that equilateral triangle is the smallest, then square, then regular pentagon and then a regular hexagon.

A pentagon is shown inside a regular hexagon.

An equilateral triangle is shown inside a square.

The expression for total shaded region would be written as,

Shaded area = Area of first shaded region + Area of second shaded region

Hence,        

⇒ Shaded area  = area of the hexagon – area of the pentagon + area of the  square – area of the equilateral triangle.

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Answer:

Regular Hexagon

Regular Pentagon

Square

Equilateral Triangle

Step-by-step explanation:

E2020 Geometry B!! :3

Answer:

[tex]X1 = 6-\sqrt{2} , X2 = 6+\sqrt{2}[/tex]

Step-by-step explanation:

Answer: x= √2 + 6, - √2 + 6

Step-by-step explanation:

The probability of getting a reading between 1.50 and 2.25 is; 0.00546

We are given the following information in the question:

Mean; μ = 0 °C

Standard Deviation; σ = 1 °C

We are given that the distribution of thermometer readings is a bell shaped distribution that is a normal distribution.

Formula for z-score is;

z = (x' - μ)/σ

P(Between 1.50 degrees and 2.25 degrees) is expressed as;

P(1.5 ≤ x ≤ 2.25)

= P((1.5 - 0)/1 ≤ z ≤ (2.25 - 0)/1))

= P(z ≤ 2.25) -  P(z < 1.5)

= 0.0546 = 5.46%

The graph that correctly describes this is the first graph

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Answer: 2 and 3, or 3 and 2

Step-by-step explanation:

a + b = 5 so a = 5 - b

Substitute a = 5 - b into ab = 6; (5 - b)b =  6

5b - [tex]b^{2}[/tex] = 6

[tex]b^{2}[/tex] - 5b + 6 = 0

(b - 2)(b - 3) = 0

So b is either 2 or 3

So a is either 3 or 2 depending on what b is

[tex]d = \sqrt{ \frac{4h}{5} } \\ square \: both \: sides \\ d {}^{2} = \frac{4h}{5} \\ multiply \: both \: sides \: by \: 5 \\ 5d {}^{2} = 4h \\ divide \: both \: sides \: by \: 4[/tex]

[tex]h = \frac{5d {}^{2} }{4} [/tex]

The local minimum value of the function f(x) = 6 + 9x² - 6x³ is 6 and the local maximum value of the function f(x) = 6 + 9x² - 6x³ is 9.

For given question,

We have been given a function f(x) = 6 + 9x² - 6x³

We need to find the local maximum and local minimum of the function  f(x)

First we find the first derivative of the function.

⇒ f'(x) = 0 + 18x - 18x²

⇒ f'(x) = - 18x² + 18x

Putting the first derivative of the function equal to zero, we get

⇒ f'(x) = 0

⇒ - 18x² + 18x = 0

⇒ 18(-x² + x) = 0

⇒ x (-x + 1) = 0

⇒ x = 0    or    -x + 1 = 0

⇒ x = 0     or    x = 1

Now we find the second derivative of the function.

⇒ f"(x) = - 36x + 18

At x = 0 the value of second derivative of function f(x),

⇒ f"(0) = - 36(0) + 18

⇒ f"(0) = 0 + 18

⇒ f"(0) = 18

Here, at x=0, f"(x) > 0

This means, the function f(x) has the local minimum value at x = 0,  which is given by

⇒ f(0) = 6 + 9(0)² - 6(0)³

⇒ f(0) = 6 + 0 - 0

⇒ f(0) = 6

At x = 1 the value of second derivative of function f(x),

⇒ f"(1) = - 36(1) + 18

⇒ f"(1) = - 18

Here, at x = 1, f"(x) < 0

This means, the function f(x) has the local maximum value at x = 1,  which is given by

⇒ f(1) = 6 + 9(1)² - 6(1)³

⇒ f(1) = 6 + 9 - 6

⇒ f(1) = 9

So, the function f(x) = 6 + 9x² - 6x³ has local minimum at x = 0 and local maximum at x = 1.

Therefore, the local minimum value of the function f(x) = 6 + 9x² - 6x³ is 6 and the local maximum value of the function f(x) = 6 + 9x² - 6x³ is 9.

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Answer:

Step-by-step explanation:

Answer: 797.4 m²

Step-by-step explanation:

The surface area is just the total of the areas of each face of of a solid. In this solid, we have 2 triangles and 3 rectangles.

We know that the two triangles of this solid are congruent, so they will have the same area. Since the area of a triangle is [tex]\frac{1}{2}bh[/tex], two triangles would have an area of [tex]bh[/tex]. Hence, the total area is

[tex]A=9 * 15\\A=135[/tex]

The area of a rectangle is lw, where l is the length and w is the width. Let's find the total area of all of them.

[tex]A=9*16+15*16+17.4*16[/tex]

All of the areas are a product of some number and 16. This makes sense as the length of this prism is 16. We can un-distribute this 16 to make the calculation easier.

[tex]A=16(9+15+17.4)\\A=16(41.4)\\A=662.4[/tex]

We can add both totals to get the total surface area of the solid.

[tex]135+662.4\\=797.4[/tex]

The surface area of this right triangular prism is 797.4 m².

By critically observing the graph, we can infer and logically deduce the following points:

A graph can be defined as a type of chart that's commonly used to graphically represent data on both the horizontal and vertical lines of a cartesian coordinate, which are the x-axis and y-axis.

A linear function can be defined as a type of function whose equation is graphically represented by a straight line on the cartesian coordinate.

This ultimately implies that, the data of a linear graph are directly proportional and as such, as the value on the x-axis increases or decreases, the values on the y-axis also increases or decreases.

By critically observing the graph which models the changes in temperature over a specific period of time (in years), we can infer and logically deduce the following points:

In conclusion, there are four (4) points of intersection on this graph.

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If T is the midpoint of PQ and PT = 3x+3, TQ = 7x-9, then PT = 12 units.

Determining the Value of PT

It is given that,

T is the midpoint of PQ ........ (1)

PT=3x+3 ......... (2)

TQ=7x-9 .......... (3)

From (1), the distance from P to T and the distance from T to Q will be equal.

⇒ PT = TQ [Since, a midpoint divides a line into two equal segments]

Hence, equating the equations of PT and TQ given in (2) and (3) respectively, equal, we get the following,

3x + 3 = 7x - 9

or 7x - 9 = 3x + 3

or 7x - 3x = 9 + 3

or 4x = 12

or x = 12/4

⇒ x = 3

Substitute this obtained value of x in equation (2)

PT = 3(3) + 3

PT = 9 + 3

PT = 12 units

Thus, if T is the midpoint of PQ, then the measure of PT and TQ is equal to 12 units.

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The values that exist two standard deviations above and below the mean are 681 and 265.

The given parameters exist:

Mean = 473

Standard deviation =104

The values above and below the mean exist calculated utilizing:

[tex]$x=\bar{x} \pm n \sigma$[/tex]

In this case n = 2.

So, we have, [tex]$x=\bar{x} \pm 2 \sigma$[/tex]

The value above is:

x = 473 + 2 [tex]*[/tex] 104 = 681

The value below is:

x = 473 - 2 [tex]*[/tex] 104 = 265

Therefore, the values that exist two standard deviations above and below the mean are 681 and 265.

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The height of the cone in terms of y is h = 12x⁴y

The cylinder and the cone have the same volume.

Volume of a cylinder = πr²h

where

Therefore,

Volume of a cylinder = πx²y

volume of a cone = 1 / 3 πr²h

where

Therefore,

πx²y = 1 / 3 × π × (1 / 2x)² × h

πx²y = πh / 12x²

πx²y × 12x² / π = h

h = 12x⁴y

Therefore, the height of the cone in terms of y is h = 12x⁴y

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Answer:

16u²

Step-by-step explanation:

It is a regular parallelogram

we have points (3, 4) and (4, 0).

we also have the origin which is (0, 0) and the difference between (0, 0) and (4, 0) on the x axis is 4 and since it is a regular shape, that means the top right corner = (3, 4) + (4, 0), so it is (7, 4). we know the base is 4 now. the vertical height = (7, 4) - (1, 0) which is (6, 4). now we are looking at difference in y. which is between (4, 0) and (6, 4), so the difference is 4.

now we just do 4 x 4 since it is bh and you get 16 units ²

The second store offered the better buy at the price of $92.00.

What is better buy?

Better buy refers to the lowest price out of the prices offered by the three stores.

In order to determine the lowest price, the no discount price of the first store needs to be compared to the prices of  two other stores, bearing that after-discount price is the pre-discount price multiplied by 1 minus the discount rate.

First store price=$95.00

Second store after-discount price=pre-tax discount price*(1-discount rate)

pre-discount price=$115

discount rate=20%

Second store after-discount price=$115*(1-20%)

Second store after-discount price=$92.00

Third store after-discount price=pre-tax discount price*(1-discount rate)

pre-discount price=$105

discount rate=10%

Third store after-discount price=$105*(1-10%)

Third store after-discount price=$94.50

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Answer:

D. Midpoint...

Step-by-step explanation:

I hope it helps You:)

Using proportions, it is found that:

A. The vegetables weigh 66 pounds.

B. 0.5 pints are dispensed after 25 seconds.

A proportion is a fraction of a total amount, and the measures are related using a rule of three. Due to this, relations between variables, either direct or inverse proportional, can be built to find the desired measures in the problem.

4 and 1/8 pounds is equivalent to 4.125 pounds. Since each pound has 16 ounces, the weight of the vegetable is of:

W = 16 x 4.125 = 66 pounds.

Every second, 0.16 gallons are dispensed. Hence the amount dispensed in 25 seconds is:

A = 25 x 0.16 = 4 gallons.

Each gallon has 8 pints, hence the number of pints dispensed is:

4/8 = 0.5 pints.

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The distance between the ships changing at 92.29 Knots

                              [tex]d(t) = \sqrt{(18t)^{2} + (40+17t)^{2} } \\[/tex]

The rate of change of distance is -

                              [tex]\frac{dd}{dt} = \frac{36t + 2(40 + 17t)17}{2\sqrt{18t^{2} + (40 + 17t) } }[/tex]

after putting t = 5 into this rate of change ,

we get, answer = 92.29

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Answer:

Part 1) Option B. y = 10x + 45

Part 2) The score is 95

Step-by-step explanation:

Linear equation that best describes the given model

Let

x ---> number of hours students spent studying

y ---> their score on the test

Looking at the line that was fit to the data to model the relationship

The slope is positive

The y-intercept is the point (0,45)

For x=1, y=55 ----> point (1,55)

Find the slope

The formula to calculate the slope between two points is equal to

substitute the points (0,45) and (1,55)

Find the equation of the line in slope intercept form

we have

substitute

Part 2) Estimate the score for a student that spent 5 hours studying.

For x=5 hours

substitute in the linear equation and solve for y

To conduct a test of hypothesis with a small sample, we make an assumption that the population is normally distributed .

What is normal distribution?

A probability distribution that is symmetric about the mean is the normal distribution, sometimes referred to as the Gaussian distribution. It demonstrates that data that are close to the mean occur more frequently than data that are far from the mean.

The normal distribution appears as a "bell curve" on a graph.

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Hi there,

please see below for solution steps :

‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗

⨠ add 8 and 6

[tex]\sf{2x^2+14}[/tex]

⨠ factor the 2 out

[tex]\sf{2(x^2+7)}[/tex]

Since we cannot simplify this more, we know that we've simplified completely.                                                                                   [tex]\small\pmb{\sf{Frozen \ melody}}[/tex]

‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗

Using the Fundamental Counting Theorem, it is found that there are 1024 ways of delivering the letters.

It is a theorem that states that if there are n things, each with [tex]n_1, n_2, \cdots, n_n[/tex] ways to be done, each thing independent of the other, the number of ways they can be done is:

[tex]N = n_1 \times n_2 \times \cdots \times n_n[/tex]

In this problem, each house has a correct letter, however the letter cannot be used for the house, hence the parameters are given as follows:

n1 = n2 = n3 = n4 = n5 = 5 - 1 = 4.

Thus the number of ways is:

N = 4 x 4 x 4 x 4 x 4 = 4^5 = 1024.

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