Is the area between y = 1/x and y = 1/x^2 on the interval from x = 1 to finite or infinite?

Answer:

[tex]E=\frac{9r}{i}[/tex]

Step-by-step explanation:

We have been given an equation [tex]Ei=9r[/tex], where [tex]E[/tex] represents ERA, [tex]i[/tex] represents number of innings pitched, and [tex]r[/tex] represents number of earned runs allowed.

To solve the given equation for [tex]E[/tex], we need to separate [tex]E[/tex] on one side on equation.

To separate [tex]E[/tex] on one side on equation, we will divide both sides of equation by [tex]i[/tex].

[tex]\frac{Ei}{i}=\frac{9r}{i}[/tex]

[tex]E=\frac{9r}{i}[/tex], where [tex]i\neq 0[/tex]

Therefore, our required equation would be [tex]E=\frac{9r}{i}[/tex].

Answer: 1.044 ounces

Step-by-step explanation:

Given : One ounce of 14 karat gold  = 0.58 ounce of pure gold.

i.e. Weight of pure gold in 1 ounce of 14 karat gold = 0.58 ounce of pure gold.

Then, Weight of pure gold in 1.8 ounces of 14 karat gold  = 1.8 x (Weight of pure gold in 1 ounce of 14 karat gold)

Weight of pure gold in 1.8 ounces of 14 karat gold = 1.8 x ( 0.58) ounces of pure gold.

It implies 1.8 ounces of 14 karat gold  = 1.8 x ( 0.58) ounces of pure gold.

= 1.044 ounces of pure gold.

Therefore, If a 14 karat gold necklace weighs 1.8 ounces that means it contains 1.044 ounces of pure gold.

Answer:

Factors are (x +1) ( x -7) .

Step-by-step explanation:

Given  :  x² – 6x – 7.  

To find : Factor.

Solution : We have given  x² – 6x – 7.  

On factoring

x² – 7x + 1x – 7.  

On taking x common from first two terms and 1 from last two terms.

x ( x -7) +1( x -7)

On grouping

(x +1) ( x -7)

Factors are (x +1) ( x -7) .

Therefore, Factors are (x +1) ( x -7) .

To find the numbers with the largest and smallest sums of their squares, we can use the fact that the sum of two numbers is constant. By rearranging the equation and finding the maximum or minimum of a quadratic function, we can determine the numbers that yield the desired sums of squares.

To find the numbers, let's call them x and y. We know that the sum of two nonnegative numbers is 20, so we can write the equation x + y = 20. To find the numbers that yield the largest sum of their squares, we can use the fact that the sum of two numbers is constant. Let's rearrange the equation as y = 20 - x and substitute it into the equation for the sum of squares: x^2 + (20 - x)^2.

Expanding and simplifying the equation, we get f(x) = 2x^2 - 40x + 400. Since this is a quadratic function, we can find the maximum by finding the vertex. The x-coordinate of the vertex can be found using the formula -b/2a, where a = 2 and b = -40. Plugging in the values, we get x = 10. Substituting this value back into the equation for y, we find that y = 20 - 10 = 10. Therefore, the numbers that yield the largest sum of their squares are 10 and 10.

Similarly, to find the numbers that yield the smallest sum of their squares, we can find the minimum of the quadratic function. The x-coordinate of the minimum can also be found using -b/2a. Plugging in the values, we get x = 20/2 = 10. Substituting this value back into the equation for y, we find that y = 20 - 10 = 10. Therefore, the numbers that yield the smallest sum of their squares are also 10 and 10.

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Answer:

The correct option is C) rational numbers

Step-by-step explanation:

Consider the provided number 1.68

Natural numbers: The natural numbers are 1, 2, 3, 4, 5,…

Whole number: Whole numbers are the set of natural numbers but starts with 0. i.e. 0, 1, 2, 3, 4, 5,…

Integers: Integers are the set of whole number and the negatives of the natural numbers, i.e, … ,-2, -1, 0, 1, 2, …

Rational number: A number is said to be rational, if it is in the form of p/q. Where p and q are integer and denominator is not equal to 0.

Irrational number: A number is irrational if it cannot be expressed be expressed by dividing two integers. The decimal expansion of Irrational numbers are neither terminate nor periodic.

Real numbers: All rational and Irrational numbers are called real number.

Now consider the provided number 1.68

The number has 2 digit after decimal point its means it is neither natural nor whole number, also it is not an integer.

The number 1.68 can be written in the form of p/q. Thus, it is a rational number.

As it is rational it can't be irrational. The number 1.68 is real number.

Hence, the correct option is C) rational numbers.

To translate the function f(x) = x² to the left 8 units and up 2 units, the function rule for g(x) is g(x) = (x - 8)² + 2.

To translate the function f(x) = x² to the left 8 units and up 2 units, we can use the transformation rules. The leftward translation can be achieved by subtracting 8 from the variable x in the function, resulting in g(x) = (x - 8)². The upward translation can be achieved by adding 2 to the function, resulting in g(x) = (x - 8)² + 2.

Therefore, the function rule for g(x) given f(x) = x² is g(x) = (x - 8)² + 2.

Final answer:

To find the function rule for g(x) when the graph of f(x)=x² is translated left by 8 units and up by 2 units, the new function is g(x) = (x + 8)² + 2.

Explanation:

The question pertains to transforming a function graphically. Given the original function f(x) = x² (a parabola opening upwards with its vertex at the origin), we are to translate this function to the left by 8 units and up by 2 units to get the new function g(x). To translate a function to the left by a units, you replace x with x + a. Similarly, to translate a function up by b units, you add b to the function.

Therefore, to translate the original function f(x) = x² to the left by 8 units and up by 2 units, the new function g(x) would be:

g(x) = (x + 8)² + 2

Answer: If my mom does not have to work, then I do not babysit my little sister.

Step-by-step explanation:

The contrapositive of a statement of the form " If a then b" is given by :-

"If not a then not b."

The given conditional statement :" If my mom has to work, then I babysit my little sister.”

Then the contrapositive statement of the conditional statement will be

"If my mom does not have to work, then I do not babysit my little sister.”

The center of mass of the lamina with the given density p=5 and the shape shown in the image is (0, -0.529).

To calculate the moments Mx and My:

We first need to identify the functions f(x) and g(x) that define the boundaries of the lamina. In this case, we have:

[tex]f(x) = \sqrt{(1 - x^2)}[/tex]

g(x) = -2

We then use the following formulas to calculate the moments Mx and My:

[tex]M_x = \frac{1}{2} p \int (f(x)^2 - g(x)^2) dx\\M_y = p \int x \times (f(x)) dx[/tex]

where p is the density of the lamina.

Substituting the values of f(x), g(x), and p into the above formulas, we get:

[tex]M_x = \frac{1}{2} \times 5 \times \int ({\sqrt{(1 - x^2}})^2 - (-2)^2) dx\\\\M_x = \frac{1}{2} \times 5 \times \int (1 - x^2 + 4) dx\\\\M_x = \frac{1}{2} \times 5 \times (x - \frac{x^3}{3} + 4x) dx\\\\M_x = \frac{5}{2} (x^2 - \frac{x^4}{3} + 8x) |_{-2}^1\\\\M_x = \frac{5}{2} ((1 - \frac{1}{3} + 8) - (-4 - \frac{16}{3} - 16))\\\\M_x = -\frac{50}{3}\\\\M_y = 5 \times \int x \times ({\sqrt{(1 - x^2)}}) dx[/tex]

This integral is difficult to solve analytically, so we can use numerical methods to approximate its value. Using the trapezoidal rule, we get:

[tex]M_y \approx 5 \times 6.3 \approx 31.2[/tex]

To calculate the center of mass (x_bar, y_bar):

We use the following formulas to calculate the center of mass (x_bar, y_bar):

[tex]x_{bar} = \frac{M_y}{M}\\\\y_{bar} = \frac{M_x}{M}[/tex]

where M is the total mass of the lamina, which is given by:

M = p * A

where A is the area of the lamina.

In this case, the area of the lamina is given by:

[tex]A = \int_{-2}^1 {\sqrt{(1 - x^2)}} dx[/tex]

This integral is also difficult to solve analytically, so we can use numerical methods to approximate its value. Using the trapezoidal rule, we get:

A ≈ 6.3

Therefore, the total mass of the lamina is:

M = p * A = 5 * 6.3 ≈ 31.5

Substituting the values of [tex]M_x, M_y[/tex], and M into the formulas for x_bar and y_bar, we get:

[tex]x_{bar} = \frac{M_y}{M} = \frac{31.2}{31.5} \approx 0\\\\y_{bar} = \frac{M_x}{M} = -\frac{50}{3} :31.5 \approx - 0.529[/tex]

Therefore, the center of mass of the lamina is (0, -0.529).

Answer: 5/6 oz is the answer

Step-by-step explanation:

An employee does not need to stay a specific number of years to make a salary of over $35,000 per year; instead, they must complete a particular level of education. Once they have completed high school, they can earn a median annual income of $40,612, which exceeds the $35,000 threshold.

To answer the question: What is the minimum number of years an employee would have to stay to make a salary of over $35,000 per year? we will need to consider the wage data provided.

According to the information, with further education, the median weekly earnings significantly increase:

Therefore, an individual needs to complete at least a high school diploma to earn an annual salary above $35,000. No additional years of employment beyond obtaining the appropriate level of education are needed to reach over $35,000 in annual salary considering the provided median incomes for each educational level.

Answer:

[tex]4 {x}^{2} - 81 = 0 \\ 4 {x}^{2} = 81 \\ {x}^{2} = \frac{81}{4} \\ x = \sqrt{ \frac{81}{4} } \\ x = \frac{9}{2} \\ {x = 4.5}[/tex]

Answer:

Distance of Carroll from the starting point is 7.2 km.

Step-by-step explanation:

Carroll bikes 1 km east, then 4 kilometers North, and 5 kilometers east again.

We have to calculate the distance from the starting point of Carroll.

Here we can find the distance finally from the starting point with the help of Pythagoras theorem.

Diagonal² = (Side 1 of the rectangle)² + (Side 2 of the rectangle)²

Diagonal² = 4² + 6²

                = 16 + 36

                = 52

Diagonal = √52

               = 7.21

               ≈ 7.2 kilometers

Therefore, distance of Carroll form her starting point is 7.2 kilometers.

Answer:

D. [tex]x\geq6[/tex]

Step-by-step explanation:

We are asked to find the inequality that represents an unknown number x is no less than 6.      

Since we know that no less than 6 means greater than or equal to 6.

So our unknown number x should be greater than or equal to 6.

We can represent this information in an inequality as:

[tex]x\geq6[/tex]

Upon looking at our given choices we can see that option D is the correct choice.

Answer:

Given an equation:  [tex]3x^3-21x^2-27x[/tex]

A given equation is trinomials have three terms( i.e, [tex]3x^3[/tex] , [tex]21x^2[/tex] and [tex]27x[/tex] )

Factoring is the division of the polynomial terms to the simplest forms.

A greatest common factor (GCF) identifies a factor that all terms within the polynomial have in common.

[tex](3x)(x^2) - (3x)(7x) - (3x)(9)[/tex]

now, 3x can be removed from the polynomial to simplify the factoring process.

i.e,

[tex]3x(x^2-7x-9)[/tex]

Therefore, the factor completely of  [tex]3x^3-21x^2-27x[/tex]  is,[tex]3x(x^2-7x-9)[/tex]

To factor the expression 3x³ - 21x² - 27x, we first factor out the common term 3x, which leaves us with 3x(x² - 7x - 9). The quadratic expression does not factor any further.

The expression 3x³ - 21x²- 27x can be factored by looking for common factors in each term. First, we can see that each term has a factor of 3x. We can factor out this common term which gives us:3x(x² - 7x - 9)

Next, we can factor the quadratic expression within the parentheses. However, this quadratic expression does not factor neatly, and it seems we've reached the simplest form by just factoring out 3x. Therefore, the completely factored form of the expression is:3x(x² - 7x - 9)

For this case we have the following polynomials:

[tex] -x ^ 2 + 9

-3x ^ 2 - 11x + 4
[/tex]

Adding the polynomials we have:

[tex] (-x ^ 2 + 9) + (-3x ^ 2 - 11x + 4)
[/tex]

Rewriting we have:

[tex] x ^ 2 (-1-3) - 11x + (9 + 4)
[/tex]

Therefore, the result of the sum is:

[tex] -4x ^ 2 - 11x + 13
[/tex]

Answer:

the sum of the polynomials is:

[tex] -4x ^ 2 - 11x + 13
[/tex]

option 2

To sketch the graph of the inequality y > -2x - 2, draw a dashed line for y = -2x - 2 and shade the area above this line, as this area represents the solution set of the inequality.

To sketch the graph of the linear inequality y > -2x - 2, you would begin by drawing the line y = -2x - 2 as if it were an equation. This line serves as the boundary between the solutions of the inequality and the non-solutions. Since the inequality is greater than, not greater than or equal to, you'll use a dashed line to indicate that points on the line are not included in the solution set.

Next, since the inequality is y > -2x - 2, you will shade the area above the line to indicate that all the points in this area satisfy the inequality. That shaded region represents all the possible solutions to the inequality. Always remember when graphing inequalities to pick a point, often(0,0) if it's not on the line, and check if it satisfies the inequality to ensure correct shading area.

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Answer-

The initial value of the linear relationship is 5.

Solution-

The initial value, also know as y-intercept, is the output value when the input of a linear function is zero. It is the y-value of the point where the line crosses the y-axis or x=0 line.

As the line passes through the y-axis at 5, i.e 5 is the y intercept of the line.

Therefore, the initial value of the linear relationship is 5.