Recycling Center #1 charges $0.75 per visit but lets you use their equipment for free. Recycling Center #2 charges you $0.25 per visit and $5.00 to use their equipment. How many times would you need to visit the recycling center for the total cost to be the same?

Answer:

[tex]\displaystyle \lim_{x \to 2} \frac{\sqrt{x + 7} - 3 \sqrt{2x - 3}}{\sqrt[3]{x + 6} - 2 \sqrt[3]{3x - 5}} = \boxed{ \frac{34}{23} }[/tex]

General Formulas and Concepts:
Calculus

Limits

Limit Rule [Variable Direct Substitution]:
[tex]\displaystyle \lim_{x \to c} x = c[/tex]

Limit Property [Multiplied Constant]:
[tex]\displaystyle \lim_{x \to c} bf(x) = b \lim_{x \to c} f(x)[/tex]

Limit Property [Multiplication]:
[tex]\displaystyle \lim_{x \to c} f(x)g(x) = \lim_{x \to c} f(x) \lim_{x \to c} g(x)[/tex]

Step-by-step explanation:

*Note:

The problem is too big to fit all work. I will assume that you know how to do basic algebra and calculus.

Step 1: Define

Identify given limit.

[tex]\displaystyle \lim_{x \to 2} \frac{\sqrt{x + 7} - 3 \sqrt{2x - 3}}{\sqrt[3]{x + 6} - 2 \sqrt[3]{3x - 5}}[/tex]

Step 2: Find Limit (Algebriaclly)

Rationalize the function and apply basic limit techniques listed under "Calculus":

[tex]\displaystyle\begin{aligned}\lim_{x \to 2} \frac{\sqrt{x + 7} - 3 \sqrt{2x - 3}}{\sqrt[3]{x + 6} - 2 \sqrt[3]{3x - 5}} & = \lim_{x \to 2} - \frac{17(x - 2)}{\big( \sqrt[3]{x + 6} - 2 \sqrt[3]{3x - 5} \big) \big( \sqrt{x + 7} + 3 \sqrt{2x - 3} \big)} \\& = -17 \lim_{x \to 2} \frac{1}{\sqrt{x + 7} + 3 \sqrt{2x - 3}} \lim_{x \to 2} \frac{x - 2}{\sqrt[3]{x + 6} - 2 \sqrt[3]{3x - 5}}\end{aligned}[/tex]

[tex]\displaystyle\begin{aligned}\lim_{x \to 2} \frac{\sqrt{x + 7} - 3 \sqrt{2x - 3}}{\sqrt[3]{x + 6} - 2 \sqrt[3]{3x - 5}} & = - \frac{17}{6} \lim_{x \to 2} \frac{x - 2}{\sqrt[3]{x + 6} - 2 \sqrt[3]{3x - 5}} \\\end{aligned}[/tex]

Let's now focus on just the rationalization of the function within the limit:

[tex]\displaystyle\begin{aligned}\frac{x - 2}{\sqrt[3]{x + 6} - 2 \sqrt[3]{3x - 5}} & = \frac{(x - 2) \bigg[ (x + 6)^\Big{\frac{2}{3}} + 2 \sqrt[3]{x + 6} \sqrt[3]{3x - 5} + 4(3x - 5)^\Big{\frac{2}{3}} \bigg] }{\Big( \sqrt[3]{x + 6} - 2 \sqrt[3]{3x - 5} \Big) \bigg[ (x + 6)^\Big{\frac{2}{3}} + 2 \sqrt[3]{x + 6} \sqrt[3]{3x - 5} + 4(3x - 5)^\Big{\frac{2}{3}} \bigg] } \\& = - \frac{1}{23} \bigg[ (x + 6)^\Big{\frac{2}{3}} + 2 \sqrt[3]{x + 6} \sqrt[3]{3x - 5} + 4(3x - 5)^\Big{\frac{2}{3}} \bigg] \\\end{aligned}[/tex]

Substitute in the simplified function and continue:

[tex]\displaystyle\begin{aligned}\lim_{x \to 2} \frac{\sqrt{x + 7} - 3 \sqrt{2x - 3}}{\sqrt[3]{x + 6} - 2 \sqrt[3]{3x - 5}} & = - \frac{17}{6} \lim_{x \to 2} \frac{x - 2}{\sqrt[3]{x + 6} - 2 \sqrt[3]{3x - 5}} \\& = \frac{17}{138} \lim_{x \to 2} \bigg[ (x + 6)^\Big{\frac{2}{3}} + 2 \sqrt[3]{x + 6} \sqrt[3]{3x - 5} + 4(3x - 5)^\Big{\frac{2}{3}} \bigg] \\& = \frac{17}{138} (12) \\& = \boxed{ \frac{34}{23} } \\\end{aligned}[/tex]

∴ we have evaluated the following limit without L' Hopital's Rule.

Step 3: Geometrical Interpretation

We can also graph the given function and approximate the limit. Please refer to the attachment for visualization.

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Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Limits

Answer:

  34/23

Step-by-step explanation:

The limit can be evaluated by simplifying the expression in such a way as to cancel the offending zeros.

__

Define the following (to help keep the algebra cleaner):

  [tex]a=\sqrt{x+7},\ b=3\sqrt{2x-3}\\\\c=\sqrt[3]{x+6}.\ d=2\sqrt[3]{3x-5}[/tex]

Then the desired limit is ...

  [tex]\displaystyle \lim_{x\to 2}\dfrac{a-b}{c-d}[/tex]

We observe that ...

  [tex](a-b)(a+b)=\underline{a^2-b^2}=(x+7)-(3^2(2x-3))=-17x+34=\underline{-17(x-2)}\\\\(c-d)(c^2+cd+d^2)=\underline{c^3-d^3}=(x+6)-(2^3(3x-5))=-23x+46=\underline{-23(x-2)}\\\\\displaystyle\lim_{x\to2}{(a+b)}=3+3=6\\\\\lim_{x\to2}{(c^2+cd+d^2)}=4+4+4=12[/tex]

In the following expression, the factors (x-2) cancel, so, our limit is ...

  [tex]\displaystyle\lim_{x\to2}\dfrac{a-b}{c-d}=\lim_{x\to2}\dfrac{(a^2-b^2)(c^2+cd+d^2)}{(c^3-d^3)(a+b)}=\lim_{x\to2}\dfrac{-17(x-2)(12)}{-23(x-2)(6)}=\boxed{\dfrac{34}{23}}[/tex]

-2.15

-2.154 should be your answer!! have a blessed day x

Answer:

  A = 90°

  B = 37°

  C = 53°

Step-by-step explanation:

Given three sides of a triangle, the Law of Cosines can be used to find the angles.

__

For angle A (the largest), we can use ...

  a² = b² +c² -2bc·cos(A)

Solving for A, we get ...

  A = arccos((b² +c² -a²)/(2bc)) = arccos((3² +4² -5²)/(2·3·4)) = arccos(0)

  A = 90°

We can use a similar equation for angle B:

  B = arccos((a² +c² -b²)/(2ac)) = arccos((25 +16 -9)/(2·5.4)) = arccos(4/5)

  B ≈ 37°

The sum of angles is 180°, so ...

  C = 180° -90° -37°

  C = 53°

We will solve these through graphical way .

Find some coordinates

And

Find some coordinates

Graph both

The intersection points are

Answer:

45 cubic units

Step-by-step explanation:

Vtotal = 1/2 (2.5*3) * 6 *2

where

2.5 is the base

3 is the height

6 is the length

2 is the number of prisms

Answer:

The correct answer is 90 cubic units which is Option A.

Hope this helps you!!

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[tex]\qquad\qquad\huge\underline{{\sf Answer}}[/tex]

let's evaluate ~

[tex]\qquad \sf  \dashrightarrow \: - 6 - 5(6x - 4)[/tex]

[tex]\qquad \sf  \dashrightarrow \: - 6 - [(5 \cdot6x) - (5 \cdot4)][/tex]

[tex]\qquad \sf  \dashrightarrow \: - 6 - (30x - 20)[/tex]

[tex]\qquad \sf  \dashrightarrow \: - 6 -30x + 20[/tex]

[tex]\qquad \sf  \dashrightarrow \: - 30x + 14[/tex]

I hope you understood the whole procedure ~

Answer:

This one is just 15+15+7+7 then divide by the preciptical

Step-by-step explanation:

add 15+15+7+7=44/2=22

Answer:

only one.

We can construct onle one triangle with unique measurements.

Answer:

1 triangle can be constructed

Step-by-step explanation:

Answer:

2x=x+30 [vertically opposite angle]

or,2x-x=30

:.x=30

Answer:

2x = x+30

2x - x = 30

X = 30

And goodluck

Answer:

1.36 centimeters

Step-by-step explanation:

Given,

side length ( a ) = 17 mm

To find : Perimeter of regular octagon ( in cm )

Octagon has 8 sides.

Perimeter of regular octagon

= 8 x a

= 8 x 17

= 136 mm

1 mm = 0.1 cm

136 mm

= 136 x 0.1

= 1.36 centimeters ( cm )

Hence,

Perimeter of regular octagon is 1.36 centimeters.

Answer:

13.6 cm

Step-by-step explanation:

17 mm = 1.7 cm

Perimeter of octagon

= 8 x 1.7 cm

= 13.6 cm

This is one of the many possible responses:

Different kinds of balls bounce to different heights when dropped on the same floor.

Sugar dissolves faster in hot milk than in cold milk.

Plants grow more slowly when they are not near a window.

Answer:

0

Step-by-step explanation:

slope = y2-y1/x2-x1

X1= 3 y1= -2

X2 = 9 y2= -2

slope= -2--2/9-3

slope= -2+2/6

slope= 0/6

slope= 0

Answer:

last one

Step-by-step explanation:

after solving that, i got the same answer from the orginail one:

x=1,x=8

The proportion is equivalent to the original equation is (x+1)/2x = x²-7x+10 / 4x [option c]

Proportion is a mathematical comparison between two numbers. According to proportion, if two sets of given numbers are increasing or decreasing in the same ratio, then the ratios are said to be directly proportional to each other.

Given is an equation, 1/2 + 1/2x = x²-7x+10 / 4x

We need to determine the proportion equivalent to this,

So, taking the LHS,

1/2 + 1/2x

= 2+2x / 4x

= 2(x+1) / 4x

= x+1 / 2x

Hence, the proportion is equivalent to the original equation is (x+1)/2x = x²-7x+10 / 4x [option c]

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Answer:

the answer is 117

sorry if wrong but that is what I got

Step-by-step explanation:

[tex] = - 39 \div \frac{1}{ 3} \\ [/tex]

[tex] = - \frac{39}{1} \times \frac{3}{1} \\ [/tex]

[tex] = - \frac{117}{1} \\ [/tex]

[tex] = - 117 [/tex]

Answer:

  x²/2166784 +y²/2159989 = 1

Step-by-step explanation:

The relationship between the semi-axes and the eccentricity of an ellipse is ...

  e = √(1 -b²/a²)

In order to write the desired equation we need to find 'b' from 'e' and 'a'.

__

Squaring the equation for eccentricity gives ...

  e² = 1 -b²/a²

Solving for b², we have ...

  b²/a² = 1 -e²

  b² = a²(1 -e²)

Using the given values, we find ...

  b² = 1472²(1 -0.056²) = 2166784(0.996864) ≈ 2159989

The desired equation is ...

  x²/2166784 +y²/2159989 = 1

One Opposite side pair of a trapezoid has equal length

A is true

<J is supplement to <K

JK and LM are not paralle

<J is congruent to <M may be true

May not be

Answer:

A.  [tex]\sf \overline{KL}[/tex] is parallel to [tex]\sf \overline{JM}[/tex]

D.  ∠J is supplementary to ∠K

Step-by-step explanation:

Properties of a trapezoid:

Although the sides of the shape do not have any arrows to indicate that any are parallel, we know that a trapezoid has one pair of parallel sides, so:

When two angles sum to 180° they are called supplementary angles.

As the two angles that are on the same leg sum to 180° then:

Answer:

xy represents the diameter of the circle

The solution to the amount of light bulbs Julia needs to sell to meet her quota is represented by the shaded area in graph X.

An equation is an expression that shows the relationship between two or more variables and numbers.

Let x represent the incandescent light bulbs and y represent the fluorescent light bulbs. Therefore:

x + y > 100   (1)

Also:

2x + 5y ≥ 350    (2)

The solution to the amount of light bulbs Julia needs to sell to meet her quota is represented by the shaded area in graph X.

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He had bank fees $15 that were not in his check register.

The correct option is (B)

Balancing statement is a final statement of your income tax liability for a tax year. It is issued by Revenue when your income tax liability has been reviewed and it shows: your total income from all sources. tax reliefs and credits claimed. amounts of income tax

Balance in check register be,

=Initial Balance - ( Expense on Food Market + Frank's auto parts + Barber shop ) + Deposit

Initial Balance = 900

Expense on Food Market= 67.25

Frank's auto parts =36.80

Barber shop =15

Deposit =650.45

So,

Balance = 900 - (67.25 + 36.80 + 15) + 650.45

              = 1550.45-119.05

             = $1431.40

Balance on Account statement be,

=  Opening balance - ( checks amount on different dates ) + deposit - fees

= 900 - (67.25 + 36.80 + 15) + 650.45  - 15

= 1550.45-134.05

= $1416.40

As, the balances in bank statement and in checking register are not matched. This is because the fees $15 paid on 9/30 is not calculated by Jerry in his check register

thus,

= $1431.40 - $15.00 (Fees)

= $1416.40

Now the balance on account and checking statements matched.

So, he had bank fees were not in Check Register.

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Answer:

8y

Step-by-step explanation:

The equation of the graph is given by y = cos(0.4x) and has an amplitude of 1

An equation is an expression that shows the relationship between two or more numbers and variables.

Since the curve crosses the y-axis at y = 1, hence it is a cosine graph. It has a maximum of 1 and a minimum of negative 1, therefore the amplitude is 1

Also, the angular frequency = 2π * 1/5π = 0.4

The equation of the graph is given by y = cos(0.4x) and has an amplitude of 1

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Answer:

C

Step-by-step explanation:

Edge 2022

Answer:

The answer is 20

Step-by-step explanation:

Now, we know that their ages are consecutive integers.

Let their ages be (x), (x + 1), (x + 2) and (x + 3). Where, Mariana's age is x.

Here, The sum of their ages is 86.

(x) + (x + 1) + (x + 2) + (x + 3) = 86

x + x + 1 + x + 2 + x + 3 = 86

4x + 6 = 86

4x + 6 – 6 = 86 – 6

4x = 80

4x/4 = 80/4

x = 20

We assumed that Mariana's age is x and the value of x is 20.

Thus, The age of Mariana is 20.

Answer:

D

Step-by-step explanation:

3(25-(12))=39

3(13)=39

39=39

When added together the 4 angles need to equal 360

x = 360 -66-95-114

X = 85

The graph simply shows the distribution with the area corresponding to the information about the lawyer representing a Hispanic individual in a criminal trial.

A graph is a diagram showing the relation between variable quantities, typically of two variables, each measured along with one of a pair of axes at right angles.

Here, the probability that a juror is hispanic = p = 0.55 and the number of jurors = n =12

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[tex]\huge\underline\mathsf \colorbox{cyan}{-1/2}[/tex]

Step-by-step explanation:

The relationship between the variable x and y will be linear. Then the correct option is A.

A linear system is one in which the parameter in the equation has a degree of one. It might have one, two, or even more variables.

The graph is given below.

At x = 2, the value of y will be 6.

At x = 3, the value of y will be 4.

At x = 4, the value of y will be 2.

Then the relationship between the variable x and y will be linear.

Then the equation of the line will be

y = -2x + 10

Thus, the correct option is A.

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