use the binomial series to find the maclaurin series for the function. f(x) = 1 (1 + x)4. I know what the binomial series is, and I expanded, using the given information, into a polynomial, but I'm stuck with trying to turn the Maclaurin polynomial into a series with the sigma form.

The inequality for different candles to buy.

10x + 25y > 500

The inequality for different candles to sell

x + y ≤ 25

It shows a relationship between two numbers or two expressions.

There are commonly used four inequalities:

Less than = <

Greater than = >

Less than and equal = ≤

Greater than and equal = ≥

We have,

Small candles = x

Large candles = y

Cost of small candles = $10

Cost of large candles = $25

Now,

The inequality that makes more than $500.

10x + 25y > 500

Now,

The inequality to sell 25 or fewer candles.

x + y ≤ 25

Thus,

The inequalities are:

10x + 25y > 500

x + y ≤ 25

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The minimum area is 126.025 at [tex]$x=\frac{60 \pi}{(4+\pi)}[/tex] is 26.39, maximize the combined area of the circle and the​ square is 60.

Let the length of the wire (l)=60

Suppose the wire is cut into two parts as follows:

The length of the wire used for circle is x and the length of the wire used for square is (l-x)=(60-x).

So, length of the circle (x)= Perimeter of the of the circle

[tex]$$(x)=2 \pi r$$[/tex]

[tex]$r=\frac{x}{2 \pi} \quad[/tex] , r is the radius of the circular part.

Also, length of square (60-x)= Perimeter of the of the square

(60-x)=4 a

[tex]$a=\frac{60-x}{4} \quad$[/tex], a is the side of the square.

Hence, the total combined area of the circular region and the square region say A(x) is;

A(x)=area of circle + area of square

[tex]\\& A(x)=\pi r^2+a^2\end{aligned}$$[/tex]

Put the values of 'r' and 'a' and proceed as follows;

[tex]$$\begin{aligned}& A(x)=\pi\left(\frac{x}{2 \pi}\right)^2+\left(\frac{60-x}{4}\right)^2 \\& =\pi \frac{x^2}{4 \pi^2}+\left(15-\frac{x}{4}\right)^2\end{aligned}$$[/tex]

[tex]$=\frac{x^2}{4 \pi}+\left(15-\frac{x}{4}\right)^2$$[/tex]

So, [tex]$A(x)=\frac{x^2}{4 \pi}+\left(15-\frac{x}{4}\right)^2$[/tex].

For optimization, find the Critical points of [tex]$A^{\prime}(x)$[/tex];

Differentiate [tex]$A(x)=\frac{x^2}{4 \pi}+\left(15-\frac{x}{4}\right)^2$[/tex] with respect to x;

[tex]$$A^{\prime}(x)=\frac{2 x}{4 \pi}+2\left(15-\frac{x}{4}\right)\left(-\frac{1}{4}\right)$$[/tex]

Put [tex]$A^{\prime}(x)=0$[/tex] for critical points;

[tex]$$\begin{aligned}& A^{\prime}(x)=\frac{2 x}{4 \pi}+2\left(15-\frac{x}{4}\right)\left(-\frac{1}{4}\right)=0 \\& \frac{2 x}{4 \pi}=\left(15-\frac{x}{4}\right)\left(\frac{1}{2}\right) \\& \frac{4 x}{4 \pi}=\left(15-\frac{x}{4}\right) \\& \frac{4 x}{4 \pi}+\frac{x}{4}=15 \\& \frac{x(4+\pi)}{4 \pi}=15 \\& x=\frac{60 \pi}{(4+\pi)} \\& x=26.39\end{aligned}$$[/tex]

x=26.39 is the required critical point.

For maximum or minimum, find [tex]$A^{\prime \prime}(x)$[/tex] as follows:

[tex]$$\begin{aligned}& A^{\prime \prime}(x)=\frac{d}{d x}\left(\frac{2 x}{4 \pi}+\left(15-\frac{x}{4}\right)\left(-\frac{1}{2}\right)\right) \\& A^{\prime \prime}(x)=\frac{2}{4 \pi}+\frac{1}{8}\end{aligned}$$[/tex]

As [tex]$A^{\prime \prime}(x)=\frac{2}{4 \pi}+\frac{1}{8} > 0$[/tex] for every value of x. So, [tex]$A^{\prime \prime}(x) > 0$[/tex] for every critical point.

Hence, by Second Derivative test; A(x) is minimum at [tex]\\ $$x=\frac{60 \pi}{(4+\pi)}$.[/tex]

Therefore, minimum area is given as follows:

[tex]$$\begin{aligned}& A\left(\frac{60 \pi}{(4+\pi)}\right)=\left(\frac{60 \pi}{(4+\pi)}\right)^2 \frac{1}{4 \pi}+\left(15-\frac{60 \pi}{4(4+\pi)}\right)^2 \\\\& \text { As } x=\frac{60 \pi}{(4+\pi)}=26.39\end{aligned}$$[/tex]

So,

[tex]$$\begin{aligned}& A\left(\frac{60 \pi}{(4+\pi)}\right)=\left(\frac{60 \pi}{(4+\pi)}\right)^2 \frac{1}{4 \pi}+\left(15-\frac{60 \pi}{4(4+\pi)}\right)^2 \\& A(26.39)=(26.39)^2 \frac{1}{4 \pi}+\left(15-\frac{26.39}{4}\right)^2 \\& A(26.39)=55.42030+70.6020 \\& A(26.39)=126.025\end{aligned}$$[/tex]

Therefore, the minimum area is 126.025 at [tex]$x=\frac{60 \pi}{(4+\pi)}[/tex] is 26.39.

b. Note that, maximum area cannot be determined by the critical points as there is only one critical point that too corresponds to the minimum area.

For maximum area; consider the following;

Put x=0 and x=60 in A(x);

[tex]$$\begin{aligned}& A(0)=(0)^2 \frac{1}{4 \pi}+\left(15-\frac{0}{4}\right)^2 \\& A(0)=(15)^2 \\& A(0)=225 \\& A(60)=(60)^2 \frac{1}{4 \pi}+\left(15-\frac{60}{4}\right)^2 \\& A(60)=(60)^2 \frac{1}{4 \pi} \\& A(60)=286.47\end{aligned}$$[/tex]

So, at x=60, A(x) is maximum. That is in the case, when total length of the wire is used for circle.

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JKL triangles could not be similar to triangle abc if triangle a b c with right angle b. Side a b is 12 cm. Side b c is 5 cm. Side a c is 13 cm.

A triangle is a three-sided polygon, which has three vertices. The three sides are connected with each other end to end at a point, which forms the angles of the triangle. The sum of all three angles of the triangle is equal to 180 degrees.

This is because the lengths of each side JKL is a factor to ABC

respectively .

AB=12,JK=36

AC=13,JL=39

BC=5,KL=25

JKL triangles could not be similar to triangle abc if triangle a b c with right angle b. Side a b is 12 cm. Side b c is 5 cm. Side a c is 13 cm.

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Triangles that cannot be similar to triangle ABC are those that do not meet the necessary conditions for similarity, which are:

Corresponding angles are congruent

Corresponding side lengths are in proportion

Given the information provided, triangle ABC is a right triangle with a right angle at vertex B, and side lengths of 12 cm, 5 cm, and 13 cm.

A triangle with a right angle is already a special case of triangle, as the Pythagorean theorem must apply. Therefore any triangle that does not obey this theorem can not be similar.

So, any triangle that doesn't obey pythagorean theorem, not having the same angles or ratios of side lengths can't be similar to ABC triangle.

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The number of students that can fit in a bus and each van is 45 and 12 respectively

You should know that when two equations are solved together in two variables, they are simultaneously solved.

The given parameters that can help us to get the number of students is

These can be expressed in equation form as follows

2x +2y = 114 .....................1

x + 3y = 81 .........................2

Making s  the subject of the formular in equation 2

x=81-3y

substitute 81-3y in equation 1 to have

2(81-3y) + 2y = 114

162 -6y +2y = 114

-4y = 114-162

-4y = -48

Making y the subject of the relation we have

y = 48/4

y=12

= 12 vans

Recall that

x= 81-3y

Substitute 12 for y in the equation to have

x=81-3(12)

x=81-36

x=45

= 45 buses

In conclusion 45 buses and 12 vans will fit made available

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On solving the provided question, we can say that - the the range of the following will be (3, 6.5 )

Finding the variable's greatest observed value (maximum) and deducting the least observed value will yield the range (minimum). Limits of variation or potential range: a variety of steel costs; several styles; The size or scope of an action or operation: insight. how far a weapon's projectile can or will travel. The number in a list or set between the lowest and maximum is referred to as a range. Line up all the numbers before locating the region. After that, take away (get rid of) the lowest number from the greatest number. The solution provides the list's range.

here,

we have

(1,8); (3, 6.5); (3.5, 3); (8, 2)

range of the following will be (3, 6.5 )

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The equation you put is in slope-intercept form (y = mx + b).

m = slope, so 14.25 would be the slope.

Hope this helps.

Answer;

14.25

Step-by-step explanation;

Well take a look at what is right before x that is meant to tell you the slope when the equation is in y = mx+c form, which it is right here , y=14.25x is in y=mx+c form but c=0 .

Answer:

-4

Step-by-step explanation:

Let x = the unknown number

Twice of x would = 2x

Write out the equation

2x + 13 = 5

Solve

2x = -8

x = -4

Answer:

864

Step-by-step explanation:

The time Billy should get up to be right on time for school is 6:52 am.

The formula to calculate the time Billy should get up to be right on time for school is as follows: Time Billy Should Get Up = School Start Time - (Time to Get Dressed + Time to Eat Breakfast + Time to Walk to School). In this case, the formula is: Time Billy Should Get Up = 8:35 am - (32 minutes + 13 minutes + 17 minutes). In order to solve this equation, first we must convert the minutes to hours. 32 minutes is equal to 0.53 hours and 13 minutes is equal to 0.22 hours. 17 minutes is equal to 0.28 hours. Thus, the formula is: Time Billy Should Get Up = 8:35 am - (0.53 hours + 0.22 hours + 0.28 hours). Now, we can solve for the time Billy should get up. 8:35 am minus 0.53 hours is 7:42 am. Then, 7:42 am minus 0.22 hours is 7:20 am. Finally, 7:20 am minus 0.28 hours is 6:52 am. Therefore, the time Billy should get up to be right on time for school is 6:52 am.

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Answer : 9/16

so sorry if I’m wrong

Answer:

27720

Step-by-step explanation:

multiple all of them 11x14x9x20 = 27720

One of the options is c=d because matching angles are made and the same corners are congruent when a line joins two parallel lines.

In geometry, parallel lines are coplanar, straight lines that don't intersect anywhere. Parallel planes are those in the same three-dimensional space that never cross one another. Curves do not touch or intersect when they are parallel to one another and keep a predetermined minimum distance between them. Lines in a plane are considered to be parallel if their spacing is constant. Objects cannot cross parallel lines. Perpendicular lines are those that intersect at a precise angle of 90 degrees.

Due to vertical angles, A = d is also a possible answer. The outcome is that b+d=180 since b+c=180 and c=d, and b+d=180 as a result. Therefore, the answer is B, C, E, where a=d, c=d, and b+d=180.

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You can find the complex roots of a 6th-degree polynomial by using approaches such as the rational root theorem.

The most common approach is to use the Rational Root Theorem, which can be used to determine the possible rational roots of the polynomial - these are the roots that can be expressed as a fraction with an integer numerator and an integer denominator.

Once you have identified these possible rational roots, you can then use the Synthetic Division algorithm to divide the polynomial with each of the rational roots to determine the actual roots of the equation.

Another approach is to use the Sturm’s Theorem, which can be used to determine the number of real and complex roots of the equation.

Finally, you can also use numerical methods, such as the Newton-Raphson method, to calculate the complex roots of the polynomial.

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Answer:

387 in³

Step-by-step explanation:

Volume of the solid 8" cube = 8³ = 512

The "empty space" volume of the 5" hole = 5³ = 125

Subtract the volume of the hole from the solid cube:

512 - 125 = 387 in³  (that's what is left)

A. 3/2 is not a proper fraction because it is greater than 1.

A proper fraction is a fraction where the numerator (the top number) is less than the denominator (the bottom number). In the fraction 3/2, the numerator (3) is greater than the denominator (2), which means it is not a proper fraction. This can be expressed as a formula using inequality symbols:

Numerator < Denominator

3 < 2

Since 3 is not less than 2, 3/2 is not a proper fraction.

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The largest possible value of δ that guarantees |cos(x) + 2 sin(3x) − 1| < 0.

What is trignomentry ?

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles, particularly right triangles. It is used to study the properties of angles,sines, cosines and tangents and circles.

We can start by using the Triangle Inequality to find an upper bound on |cos(x) + 2 sin(3x) − 1| in terms of |x − 0.5|:

|cos(x) + 2 sin(3x) − 1| <= |cos(x) − 1| + 2 |sin(3x)|

We know that |cos(x) − 1| <= |x − 0.5| and |sin(3x)| <= |3x|, since |sin(x)| <= |x| for all x. So we have ,

|cos(x) + 2 sin(3x) − 1| <= |x − 0.5| + 2|3x|

Now we can set this inequality equal to 0.5 and solve for |x − 0.5|:

|x − 0.5| + 2|3x| <= 0.5

|x − 0.5| <= 0.5 − 2|3x|

To find the largest possible value of δ, we need to find the maximum value of the right side of the inequality,

δ = 0.5 - 2|3x|

The maximum value of the right side of the inequality occurs when |3x| is at its smallest. Since |3x| >=0 for all x, the maximum value of δ is 0.5.

So , the largest possible value of δ that guarantees |cos(x) + 2 sin(3x) − 1| < 0.

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The solution to the question is 1.

Logarithm is nothing but another way of expressing exponents and can be used to solve problems that cannot be solved using the concept of exponents only.

Squaring both sides of the log

= (2 (√log2(3))²  - (3 (√log3(2))²

= 4log2(3) - 9log3(2)

= 4(3)log2 - 9(2)log3

=  12log2 - 18log3

= 12log2/18log3

= 2log2/3log3

= 1log2² / 1log3³

= 1/1

= 1

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By working with the given percentage, we will find that there are 1,050 students in the school.

First, we know that (66 + 2/3)% = (66.66...%) of the students are girls.

Then the remaining 33.33% are boys.

And we know that there are 350 boys, so the 33.33...% of the students is equal to 350.

So we can write the equations for percentages:

350 = 33.33%

X = 100%

Where X is the total number of students, to find X, we can take the quotient between these two equations:

X/350 = 100%/33.33%

X = 350/0.333... = 1,050

So there are 1,050 students on that school.

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The way that mutual funds are similar to common stocks is D. Investing in a mutual fund gives the investor a proportionate ownership in that fund.

Mutual funds are investment vehicles that pool together money from many different investors and use that money to buy a diversified portfolio of stocks, bonds, or other securities.

Just like when you buy a single stock, when you invest in a mutual fund, you become a shareholder of the fund and own a proportionate share of the underlying securities in the fund.

The correct option is D

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In a shortcut method, 1500 divided by 50 equals 30.

Before we start to count the division, we should know that the number 500 is called the divisor or the denominator, and the number 1500 is called the dividend or the numerator.

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The mean quiz score is 5.42.

The median quiz score is 5.5.

Mean is the average of a set of numbers. It is determined by adding the set of scores together and dividing it by the total scores in the dataset.

Mean = sum of the numbers / total number

Mean = (3 + 4.5 + 6.5 + 00 + 8.5 + 10) / 6 = 5.42

Median is the number at the center of the dataset when the scores are arranged in either an ascending or descending order.

The scores arranged in ascending order - 0, 3, 4.5,  6.5, 8.5, 10

Median = (4.5 + 6.5) / 2 = 5.5

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Answer: 29/48

Step-by-step explanation:

If she spent 5/12 of 1200, she's left with 7/12 of 1200 which is 700. If she spends 1/4 of that, she has 3/4 of it left. 3/4 x 700 is 525. 525 + 250 is equal to 725. To simplify 725/1200, divide both sides by 25 to get 29/48.

Answer:

The first answer

Step-by-step explanation:

In this problem, i represents that number of items that Vance sells. Since he sold between 300 and 310 items, i can be any integer between those values. This immediately rules out the second option. Since you cannot sell only a part of an item, the third and fourth answer choices are also incorrect. Therefore, the first option is the correct answer.

The coordinates of the circumcenter of the triangle are (5,8).

Given, Three points of the circle,

A(-5,3), B(5, 8), C(-5.8).

Mid point of AB = (5-5 / 2, 8+3/2) = (0, 11/2)

Slope of AB = y2 - y2 / x2 - x1

= 8-3 / 5 + 5

= 5/10 = 1/2

Equation of AB = y - yo = m (x - xo)

y - 11/2 = 1/2 (x - 0)

2y - 11 = x                  .....(1)

Similarly finding the equation of AC and BC -

Mid points of BC = (0,8)

slope of BC = 0

equation of BC = y = 8                   ....(2)

On solving (1) ans (2) simultaneously,

2y - 11 = x

y = 8

16 - 11 = x

x = 5

The coordinates of the circumcenter of the triangle are (5,8).

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The equivalent equation after combining like terms is x-7=1. To solve this equation, simply add 7 to both sides of the equation and you will get x=8.

An equivalent equation is an equation that is mathematically equal to another equation, meaning that both equations have the same solutions. For example, the equation "2x + 5 = 11" is equivalent to the equation "2x = 6". Both equations have the same solution, x = 3.

This is done by subtracting 2x from both sides, and then adding 9 to both sides. Doing so eliminates the -2x and -9 terms, leaving just x on one side, and 7 on the other. Combining like terms is a useful way to simplify equations, and it is important to understand how to do it in order to solve more complex equations.

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Answer:

n<-21

Step-by-step explanation:

31-n>52

-31 on both sides

-n>21

flip

n<-21

[tex]31-n > 52[/tex]

Simplify:

[tex]-n+31 > 52[/tex]

Subtract 31 from both sides:

[tex]-n+31-31 > 52-31[/tex]

[tex]-n > 21[/tex]

Divide both sides by -1:(to remove the negative from the variable)

[tex]\dfrac{-n}{-1} > \dfrac{21}{-1}[/tex]

Reverse the inequality sign since we are dividing by a negative number.

[tex]n < -21[/tex]

Answer:

6 : 16, 12 : 32,  18 : 48 etc

Step-by-step explanation:

There are many equivalent ratios

3:8

3x2=6

8x2=16

so one ratio will be 6:16

Answer:

9:12

Step-by-step explanation:

...

A figure with 7 sides is a heptagon

A heptagon is a seven-sided shape. A heptagon is a closed object with straight lines and angles that is a type of polygon. (n-2) * 180 is the total number of angles in a heptagon, where n is the number of sides. There are 7-2 = 5 angles in a heptagon, each of which measures (180-2) = 128.57 degrees. Heptagons can be regular or irregular, with cyclic or dihedral symmetry groups.

It's also worth mentioning that the number of sides in a form is commonly referred to as its "order" in mathematics, hence a heptagon is often referred to as a "seven-gon."

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Scaler projection of b onto a is 3

vector projection of b onto a is (-24/17 , 45/17)

Scaler projection of b onto a is given by (a.b)/|a|

a.b = ( -8,15) . (3 ,5)

=> -24 + 75

=> 51

|a| = √-8^2 +15^2

=> √64+255

=> √289

=> 17

so Scalar projection of b onto a is given as

(a.b)/|a|

=> 51/17

=>3

so Scalar projection of b onto a is 3

The vector projection of b onto a is given by  (a.b)/[tex]|a|^2[/tex] * a

=> we know that a.b =51

and |a| = 3

so the vector projection is given by :

=> 51/289 (-8,15)

=>  (-24/17 , 45/17)

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