find a number such that 14 less than twice the number is 66

Answer:

Equation is [tex]a_n = 2.50n + 2.50[/tex]

This is discrete

=======================================================

Explanation:

There are n = 12 months in a year.

[tex]a_1[/tex] = first term = x

d = common difference = 2.50, which is the amount the allowance is increasing month to month.

[tex]S_n = \text{Sum of the first n terms of an arithmetic sequence}\\\\S_n = (n/2)*(2a_1 + d(n-1))\\\\S_{12} = (12/2)*(2\text{x} + 2.50(12-1))\\\\S_{12} = 6(2\text{x} + 27.5)\\\\S_{12} = 12\text{x} + 165\\\\[/tex]

The sum of the first n = 12 terms of the arithmetic sequence is 12x+165, where x is the first term (i.e. the allowance on January 1st).

We're told that this total is $225. We'll set the value of [tex]S_{12}[/tex] equal to 225 and solve for x.

[tex]S_{12} = 225\\\\12\text{x} + 165 = 225\\\\12\text{x} = 225 - 165\\\\12\text{x} = 60\\\\\text{x} = 60/12\\\\\text{x} = 5[/tex]

Therefore, you got $5 on January 1st. Then 5+2.50 = 7.50 dollars on February 1st. Then 7.50+2.50 = 10 dollars on March 1st. And so on, until reaching December. All of those dollar amounts then should add to $225 to help confirm the answer.

This equation is discrete because the terms are finite from n = 1 to n = 12. We can't have a midpoint of say between months 3 and 4. It doesn't make sense to have month 3.5 for instance. There's only a set amount of payments.

--------

Now that we know [tex]a_1 = 5[/tex] is the first term, we can then determine the value of any allowance value for any given value of n.

[tex]a_n = \text{nth term of arithmetic sequence}\\\\a_n = a_1 + d(n-1)\\\\a_n = 5 + 2.50(n-1)\\\\a_n = 5 + 2.50n-2.50\\\\a_n = 2.50n + 2.50\\\\[/tex]

Let's see what the allowance would be for month n = 2

[tex]a_n = 2.50n + 2.50\\\\a_{2} = 2.50(2) + 2.50\\\\a_{2} = 5 + 2.50\\\\a_{2} = 7.50\\\\[/tex]

The allowance for month n = 2  (aka February) is $7.50, which was mentioned earlier in the previous section. I'll let you confirm the other values of n.

Keep in mind that n is a positive whole number in the set {1,2,3,...,10,11,12}. It might help to make a table of values.

Yes, there is enough weight of plastic pellets in a full barrel for the machine to make trash bags for 5 minutes.

It is given that the factory makes trash bags using plastic pellets. The plastic pellets are placed in barrels. The barrels containing the plastic pellets are then poured into a machine. When the machine is operating, the change from a barrel of pellets to the trash bags is 3 kg per minute. Each full barrel contains 18 kg of plastic pellets.

We need to check if there are enough pellets in a full barrel for the machine to make trash bags for 5 minutes. Let the time for which the machine can make the bags from a single barrel of plastic pellets be denoted by the variable "t". The time taken by the machine to fully convert a barrel full of plastic pellets into trash bags is the ratio of the total weight of plastic pellets in the barrel to the rate of conversion of the plastic pellets into trash bags.

t = 18/3

t = 6

Hence, the machine can convert a barrel of plastic pellets into trash bags in a time duration of 6 minutes. So, there are enough pellets in a full barrel for the machine to make bags for 5 minutes.

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Answer:

-2x + 15.

Step-by-step explanation  Perpendicular lines have slopes that are opposite reciprocals of each other. Therefore, the slope of the line is -2. When we solve for the y-intercept, the result is 15.

Answer:

  28

Step-by-step explanation:

You require a teacher : children ratio of at least 1 : 4 for the activity of climbing, and you want to know the greatest number of children that can go climbing with 7 teachers.

We can describe this problem by the inequality ...

  teachers/children ≥ 1/4

  7/children ≥ 1/4 . . . . . for 7 teachers

multiplying by (4·children) gives ...

  28 ≥ children

At most, 28 children can go climbing.

The erosion occurs at a rate of 2/1000 m/years. Over 1,000,000 years, the erosion will be 2,000 meters.

Rate is a ratio between one quantity to another quantity. Rate is also a measure of how fast a quantity changes when another quantity changes. For instance, velocity is a rate to measure the change of distance per amount of travelled time.

Suppose we have two quantities: x and y, with rate of change in y per change in x is denoted by dy/dx, then

y = dy/dx · x

In the given problem:

y = length of erosion

t = time

dy/dt = 2 meters / 1000year = 2/1000 meters/year

Then,

y = dy/dt · t

  = 2/1000 · 1,000,000 = 2,000 meters

Hence, over  1,000,000 years, the regional erosion = 2,000 meters

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The simplified value of the expression (x³ + 9x²+ 26x + 24), when divided by (x + 3), is: the quotient will be x^2 + 6x + 8 and the remainder will be 0. (x + 3) will be a factor of (x³ + 9x²+ 26x + 24).

Let us solve the given algebraic expression by the long division method:

We are given the expression:

(x³ + 9x²+ 26x + 24) and (x + 3)

We need to perform the division of the given algebraic expression:

x + 3  ) x^3 + 9x^2 + 26x + 24 ( x^2 + 6x + 8

           x^3 + 3x^2

         -         -                            

                    6x^2 + 26x + 24

                    6x^2 + 18x

                     -        -                  

                                 8x + 24

                                 8x + 24

                               -       -          

                                     0

So,

quotient = x^2 + 6x + 8

remainder =  0

As the remainder is 0.

So, (x + 3) will be a factor of (x³ + 9x²+ 26x + 24)

Thus, the simplified value of the expression (x³ + 9x²+ 26x + 24), when divided by (x + 3), is: the quotient will be x^2 + 6x + 8 and the remainder will be 0. (x + 3) will be a factor of (x³ + 9x²+ 26x + 24).

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Answer:

750 miles

Step-by-step explanation:

20 gallons -------------- 300 miles

1 gallon ---------------- 300/20 = 15 miles

50 gallons -------------- 15x50 = 750 miles

The equation in slope intercept form of the line that passes through the y-intercept (0, 1) and has a slope of 4 is y = 4x + 1

The line is passing through the y intercept = (0, 1)

Here at x = 0, the values of y = 1

Then the y intercept of the line is 1

The slope of the line = 4

The slope of the line is the change in y coordinate with respect to the change in x coordinates of the function

The slope intercept form is

y = mx + b

Where m is the slope of the line

b is the y intercept

x is the x coordinate

y is the y coordinate

Then the equation in the slope intercept form

y =4x + 1

Hence, the equation in slope intercept form of the line that passes through the y-intercept (0, 1) and has a slope of 4 is y = 4x + 1

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Answer:

yes

Step-by-step explanation:

A cat is a domestic species of small carnivorous mammal. It is the only domesticated species in the family Felidae and is often referred to as the domestic cat to distinguish it from the wild members of the family. A cat can either be a house cat, a farm cat or a feral cat; the latter ranges freely and avoids human contact.

So, it will be considered as a yes.

The value o f c based on The Mean Value Theorem is 1

The Mean Value Theorem stated that if f(x) is a continuous on [a,b] and differentiable on (a,b) then there is a c such that:

f'(c) = [tex]\frac{f(b)-f(a)}{b-a}[/tex] ... (i)

Based on the question, we got equations:

f(x) = 1 + x + x² ... (ii)

By using derivative, we get:

f'(x) = 1 + 2x ... (iii)

From the question, we get:

f(a) = f(0)

f(b) = f(2)

Then we need to find the value of both f(0) and f(2):

f(0) = 1 + (0) + (0)²

f(0) = 1 ... (iv)

f(2) = 1 + (2) + (2)²

f(2) = 7 ... (v)

We also need to find the equation for f'(c) based on equation (iii):

f'(x) = 1 + 2x

f'(c) = 1 + 2c ... (vi)

We will input equations (iv) and (v) into equation (i) to find another equation of f'(c):

f'(c) = [tex]\frac{f(2)-f(0)}{2-0}[/tex]

f'(c) = [tex]\frac{7-1}{2-0}[/tex]

f'(c) = [tex]\frac{6}{2}[/tex]

f'(c) = 3 ... (vii)

Then we will subtitute the equation (vi) with equation (vii) to find the value of c:

f'(c) = 1 + 2c

3 = 1 + 2c

2c = 2

c = 1... (viii)

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The expression for the rectangular prism's base area is 11x² + 7x + 5.

Given;

The volume of rectangular prism = 33x³ + 43x² + 29x + 10

The height of the prism = 3x + 2

We know that, the area of the prism = length * breadth

where as, volume = length * breadth * height

So, area = volume / height

Using synthetic division, dividing volume by height, we get;

3x+2  ) 33x³ + 43x² + 29x + 10 ( 11x² + 7x + 5

            33x3 + 22x2

           ---------------------------------

                        21x² + 29x

                        21x² + 14x

          ----------------------------------

                                  15x + 10

                                   15x + 10

          ----------------------------------

                                       x

Quotient is 11x² + 7x + 5 which is equal to expression for the area of the base of the rectangular prism.

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Answer:

x = 4 or x = 6

Step-by-step explanation:

to solve the inequality

solve both

x - 3 = 9 - 2x OR x - 3 = - (9 - 2x)

x - 3 = 9 - 2x ( add 2x to both sides )

3x - 3 = 9 ( add 3 to both sides )

3x = 12 ( divide both sides by 3 )

x = 4

OR

x - 3 = - (9 - 2x)

x - 3 = - 9 + 2x ( subtract 2x from both sides )

- x - 3 = - 9 ( add 3 to both sides )

- x = - 6 ( multiply both sides by - 1 )

x = 6

Answer: $56.25

Step-by-step explanation:

x = cost to make the shirt

x + 75(0.25) = 75

x = 75 - 18.25 = 56.25

Answer:

5

Step-by-step explanation:

Since we have given that

Using ratios we know that 40 hamburgers will be served.

So, a number of hamburgers that will be served are:

So, the number of hamburgers will be:

Therefore, using ratios we know that 40 hamburgers will be served.

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Correct question:
A concession stand serves hot dogs and hamburgers at a ratio of 3 to 2. Based on this ratio, how many hamburgers will be served if there are a total of 60 hot dogs served?

Answer:150 minutes easy?

Step-by-step explanation:

Answer:87,600

Step-by-step explanation:

a) 49 square tiles are needed to make pattern number 7

b) 84 circular tiles are needed to make pattern number 20

c) When the pattern number is odd, the total number of tiles (square

and circular) needed to make the pattern is B) always odd

A sequence in mathematics is an enumerated collection of objects in which repetitions are permitted and order is important. It, like a set, has members (also called elements, or terms). The length of the sequence is defined as the number of elements (which could be infinite). Unlike a set, the same elements can appear multiple times in a sequence at different positions, and the order does matter. Formally, a sequence can be defined as a function from natural numbers (the sequence's positions) to the elements at each position. The concept of a sequence can be extended to include an indexed family, which is defined as a function from an arbitrary index set.

a) Squares tiles needed for 7 pattern =n²

n²=7²

n²=49

b) Circular tiles for 20th pattern=4(n+1)

=4(20+1)

=4(21)

=84

c) Total tiles(square and circle)

=n²+4(n+1)

for n=odd (let n=1,3,5)

n=1; 1+4(1+1)=9

n=2;  9+4(3+1)=25

n=5;  25+4(5+1)=49

Therefore, It is always odd

So, the answer is option B

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Answer: love=an intense feeling of deep affection

Step-by-step explanation:

Answer:

Step-by-step explanation:

Down payment = $150

Monthly Payment = $48.5

Buyer price = $1250

Number of months = 30  (2 1/2 years)

The total amount of monthly payments = Monthly payments × Number of months

                                                                  = 48.5 × 30

                                                                  = 1447.50

The total cost = Total amount of monthly payments + Down payments

                       = 1447.50 + 150

                       = 1597.50

Finance charge =  Total cost  - Buyer price

                          = 1597.50 - 1250

                          = 347.50

Answer:

4 x y

Step-by-step explanation:

y x y x y x y = y

2 x 2 = 4

4 x y

The word problem of swimming pool resulted to a simultaneous equation and the solution is

the number of adults in the pool is 242

the number of children in the pool is 317

The problem is a simultaneous equation of two unknowns with two equations.

The equations are formed as follows

let the number of adults in the pool be x

let the number of children in the pool be y

559 people used the public swimming pool

559 = x + y

The daily prices are $1.25 for children and $2.50 for adults. The receipts for admission totaled $1001.25

1.25y + 2.50x = 1001.25

Hence:

x + y = 559

1.25y + 2.50x = 1001.25

solving the equation gives

y = 317

x = 242

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a.) Expression for the amount of money is 75+18n

b.) money spend in 12 months is $291

From the question, we have

Buy a network router for $75.00. The fee for internet service is $18.00 per month.

a.) Expression for the amount of money (y) is,

y = 75+18n

b.) money spend in 12 months = 75+18*12

= $291

Addition:

The phrase "the addition" refers to the combining of two or more integers. Three is written as three plus three since the plus sign (+) signifies the sum of two integers. The frequency with which the plus symbol (+) is used is likewise under your control. for example, 3 + 3 + 3 + 3. The teaching of addition in math to children is crucial. In primary school, students learn fundamental addition sums, such as one-digit facts, math addition sums, and double-digit arithmetic addition sums. One of the most fundamental and exciting Math concepts for primary school children is sums in addition. Because math is a fascinating subject, children begin learning basic mathematical concepts like adding numbers in their early years.

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Step-by-step explanation:

[tex] {x}^{2} + 5x + 6[/tex]

[tex] {x}^{2} + 6x - x + 6[/tex]

[tex]x(x + 6) - 1(x + 6)[/tex]

[tex](x - 1)(x + 6)[/tex]

The slopes are similar in each segment of the line because the ratios, 8/2 and 4/1 are similar.

The first step to understanding slope of a line is that; it represents the rate of change in the vertical axis to that in the horizontal axis.

Consequently, it follows that the slope of the first segment can be computed by drawing a triangle with vertical side length of 4 and a horizontal side length of 1.

Hence, the slope of the first segment (from the triangle is) 4 / 1.

Also, for the second segment; a triangle with vertical length; 8 and a horizontal length, 2. Hence, the slope of the segment is; 8 / 2.

Ultimately, since the ratios, 8 / 2 and 4 / 1 are similar, it follows that the two triangles are similar and consequently, the slope is same in each segment of the line.

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Using the volume formula, the height of the cylinder is [tex]h = \frac{v}{\pi r^{2} }[/tex]

A cylinder is a three-dimensional shape consisting of two parallel circular bases, joined by a curved surface.

In other words, a cylinder is a geometric solid with two circular bases joined by a curved surface.

The volume of a cylinder can be represented as follows:

The formula to calculate volume of a cylinder is given by the product of base area and its height

where

Therefore, the question asked us to solve for height, h.

Hence, let's make height the subject of the formula.

v = πr²h

divide both sides by πr²

[tex]\frac{v}{\pi r^{2} }=\frac{\pi r^{2} h}{\pi r^{2} }[/tex]

Therefore,

[tex]h = \frac{v}{\pi r^{2} }[/tex]

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The smallest positive integer a such that the intermediate value theorem guarantee a zero exist between 0 and 1

given that

polynomial f(x) = 2[tex]x^{3}[/tex] - [tex]x^{2}[/tex]- 4x+4

first derivation of f(x)

f'(x) = 6[tex]x^{2}[/tex] - 2x - 4

      = 6[tex]x^{2}[/tex] - 6x +4x -4

      = 6x(x-1) + 4(x-1)

     = (6x+4)(x-1)

x = -0.6,1

second derivation of f(x)

f''(x) = 12x-2

now substitute x values in f''(x)

x=1

f"(x) = 12(1) - 2 = 12-2= 10

x= -0.61

f"(x) = 12(-0.61) -2 = -7.32-2= -9.32

a=1

the intermediate value theorem guarantee a zero exist between 0 and 1

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The restaurant delivery service cover the area 113.04 miles².

What is Circle?

The circle is a closed two dimensional figure , in which the set of all points is equidistance from the center.

Given that;

A fast-food restaurant offers delivery service anywhere within a 6-miles radius.

Now,

The area cover by delivery service = πr²

Here, r = 6 miles

Thus, The area cover by delivery service = πr²

                                                                 = 3.14 x 6²

                                                                 = 3.14 x 36

                                                                 = 113.04 miles²

Therefore, The restaurant delivery service cover the area 113.04 miles².

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