Select the correct location on the image.
Marta simplified this expression.
4 logs z logs 22
log, 3x
In which step did she incorrectly apply a property of logarithms?
-
Step 1
Step 2
Step 3
Step 4
=
4 log5 x + log5 2x - log, 3x
log, 4x + logs 2x - log5 3x
log, 8x²- log5 3x
= logs (3x)
logs (3x)
||

Using Geometric Mean Theorem, In the given triangle, The length of altitude is 12.

The geometric mean theorem states that in a right triangle, the length of the altitude from the right angle to the hypotenuse is the geometric mean of the lengths of the two segments of the hypotenuse

In other words, the square of the length of the altitude from the right angle to the hypotenuse is equal to the product of the lengths of the two segments of the hypotenuse. And the length of the altitude itself is equal to the square root of this product.

Now,

As we know

using Geometric Mean Theorem,

hypotenuse=18+8=26

let base=a

then   26/a=a/8

a²=208

Now, using Pythagoras theorem

8²+(x+9)²=a²

(x+9)²=208-64

x+9=√144

x+9=12

Hence, The length of altitude is 12.

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Answer:

The slope is "m"

m= -5

Step-by-step explanation:

x-- Axis interception points of x- 2x - 4x +8 ( 8 over 5, 0)

y-- Axis interception points of x - 2x - 4x +8 (8,0)

x-2x-4x+8

y=-5x+8

Men will have completed oil changes in hours Therefore, Will and Gabriel will each have done 12 oil changes after 4 hours.

Hours is a unit of time measurement. It is used to measure a specific amount of time and is usually denoted by the symbol “h”. There are 24 hours in a day, 60 minutes in an hour and 60 seconds in a minute. Hours are used to measure both short and long periods of time. Commonly, hours are used to measure the length of a workday, the length of a school day, or the length of a movie. Hours are also used to measure how long a person has been alive, how long an event has been going on, or how long an item has been in use.

Let W be the total number of oil changes Will has completed, and G be the total number of oil changes Gabriel has completed.

System of equations:

W = 8 + 2t

G = 3t

Since they will be tied at some point during the day, W = G.

Substituting W into G's equation:

8 + 2t = 3t

Solving for t:

2t = 8

t = 4

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De acuerdo con la información, podemos inferir que el volumen total de la figura sería 1,476.58 cm³

To find the volume of the figure we must divide it into different sections because it has cones, pyramids, hemispheres, cylinders, and rectangular cubes. Below is the procedure:

Volume of rectangular segments:

Volume of the pyramids:

First we must calculate the height of the pyramids.

Cone volume:

First we need to calculate the height of the cone.

Cylinder volume:

Volume of the hemisphere:

Finally we just have to add all the values and we will obtain the total volume of the figure:

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Answer:

Step-by-step explanation:

If we use the substitution u = 4x - 3, then du = 4 dx. Solving for dx, we get dx = du/4. Also, we can solve for x in terms of u: x = (u + 3)/4.

Substituting these values into the integral, we get:

∫ x (4x - 3)^10 dx = ∫ [(u + 3)/4] * u^10 * (du/4) = (1/16) ∫ (u + 3) * u^10 du = (1/16) ∫ (u^11 + 3u^10) du = (1/16) [(u^12)/12 + (3u^11)/11] + C = (1/192) * u^12 + (3/176) * u^11 + C

Substituting back for u = 4x - 3, we get:

(1/192) * (4x - 3)^12 + (3/176) * (4x - 3)^11 + C

So the integral of x(4x - 3)^10 dx is equivalent to (1/192) * (4x - 3)^12 + (3/176) * (4x - 3)^11 + C.

The integral of the function ∫x ( 4x - 3 )¹⁰ is given by A=( 1/16 ) ∫(u¹¹ + 3u¹⁰ )du

The mathematical procedure known as the integral of a function depicts the accumulation of a quantity over a period of time or the region beneath a curve. It is used in calculus and is represented by the symbol ∫

In math, the integral of a function f(x) with respect to x is written as f(x) dx, where f(x) is the function being integrated and dx is the change in the variable x that is infinitesimally small. The antiderivative or integral of the function is the outcome of the integration.

Given data ,

Let the value of the function be f ( x ) = ∫x ( 4x - 3 )¹⁰

On substituting the value of u = 4x - 3 , we get

du/dx = 4

So , On replacing x with (u + 3)/4, and dx with du/4 in the integral:

∫x(4x - 3)¹⁰ dx = ∫((u + 3)/4)(4(u - 3))¹⁰ du

On further simplifying , we get

∫((u + 3)/4)(4(u - 3))¹⁰ du = ∫(1/4)(4(u + 3))(u - 3)¹⁰ du

Hence , the integral is  A = ( 1/16 ) ∫(u¹¹ + 3u¹⁰ )du

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The complete question is attached below :

Using the substitution u = 4x - 3, the integral of x ( 4x - 3 ) ^ 10 dx is equivalent to which of the following?

it will take approximately 22.56 years for the amount to triple.

The given information for this problem is that there is an initial investment of $600 in an account with an annual interest rate of 4%. The task is to determine after how many years the amount will triple.Using the compound interest formula, we can find the amount in the account after t years:A = P(1 + r/n)nt Where,A = final amount in the account, P = initial amount in the account r = annual interest rate ,n = number of times the interest is compounded per year ,t = time in years.

From the problem statement, we know that the initial amount, P, is $600 and the annual interest rate, r, is 4%. Let's assume that the interest is compounded annually, i.e., n = 1.Substituting these values in the formula, we get:A = $600(1 + 0.04/1)1t Simplifying this expression,A = $600(1.04)t.

Taking the ratio of the final amount to the initial amount, we get: 3P = $600 × 3 = $1800. Therefore,A/P = 3 = (1.04)t.Dividing both sides by P, we get:3 = (1.04)t ln(3) = ln(1.04)t. Using the logarithmic property, we can bring down the exponent to the front:ln(3) / ln(1.04) = t Using a calculator, we get ≈ 22.56. Therefore, it will take approximately 22.56 years for the amount to triple.

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Score of 62.816 is the lowest score that can be earned in the California Police Officer Standards and Training test to be eligible to be hired by the agency.

We have a normal distribution of scores for the california peace officer standards and training Let X be the random variable representing scores for the California Police Officer Standards and Training test, X ~ Normal(50, 10²).

Mean = 50

Standard deviations = 10

Let x be the lowest score that can be earned to be eligible to be hired by the agency that is x is the lowest score than can be earned to get in top 10%. Therefore, P(X > x) = 0.10

P((X- 50)/10 > (x- 50)/10) = 0.10 (converting Normal variate to Standard Normal variate)

P(Z > (x - 50)/10) = 0.10 --(1)

From the Standard Normal Distribution table, P(Z > 1.2816) = 0.10 -- (2)

From comparing the equation (1) and (2),

=> (x- 50)/10 = 1.2816

=> x-50 = 12.816

=> x = 62.816

Thus, the required lowest score is 62.816.

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The heights of the tips of the propeller blades at π/6, 5π/6, and 3π/2 radians are 15.5 feet, 11.5 feet, and 10 feet, respectively.

What is the equation of a circle in parametric form?

The equation of a circle in parametric form is given by x=acosθ,y=asinθ.

We can use the equation for a circle in standard form:

(x - h)² + (y - k)² = r²

where (h, k) is the center of the circle and r is the radius.

In this case, the center of the propeller is (0, 13) and the radius is 3. So the equation of the circle is:

x² + (y - 13)² = 9

We can use this equation to find the x and y coordinates of the tip of each propeller.

For the blade at π/6 radians, we can use the parametric equations for a circle:

x = r cos(t) + h

y = r sin(t) + k

where t is the angle in radians.

Substituting r = 3, h = 0, k = 13, and t = π/6, we get:

x = 3 cos(π/6) + 0 = 3√3/2

y = 3 sin(π/6) + 13 = 13 + 3/2 = 15.5

So the height of the tip of the propeller blade at π/6 radians is 15.5 feet.

Using the same method for the blades at 5π/6 and 3π/2 radians, we get:

Blade at 5π/6 radians:

x = 3 cos(5π/6) + 0 = -3√3/2

y = 3 sin(5π/6) + 13 = 13 - 3/2 = 11.5

Height of tip: 11.5 feet

Blade at 3π/2 radians:

x = 3 cos(3π/2) + 0 = 0

y = 3 sin(3π/2) + 13 = 10

Height of tip: 10 feet

Therefore, the heights of the tips of the propeller blades at π/6, 5π/6, and 3π/2 radians are 15.5 feet, 11.5 feet, and 10 feet, respectively.

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  The decimal values:

  - Positive ratio: 0.4
  - Negative ratio: 0.4
  - Zero ratio: 0.2

To calculate the ratios of positive, negative, and zero elements in an array of integers,

Follow these steps:

1. Initialize three variables, 'positive', 'negative', and 'zero', to count the number of positive, negative, and zero elements in the array.

2. Loop through the array of integers and for each element, check if it is positive, negative, or zero. Increment the corresponding count variable accordingly.

3. Calculate the decimal value of each fraction (ratio) by dividing the count of positive, negative, and zero elements by the total number of elements in the array.

4. Print the decimal value of each fraction on a new line with places after the decimal.

Here's an example with the array [1, -2, 0, 3, -1]:

1. Initialize positive = 0, negative = 0, zero = 0.

2. Loop through the array:
  - 1 is positive, so positive = 1.
  - -2 is negative, so negative = 1.
  - 0 is zero, so zero = 1.
  - 3 is positive, so positive = 2.
  - -1 is negative, so negative = 2.

3. Calculate the ratios:
  - Positive ratio = positive / total elements = 2 / 5 = 0.4.
  - Negative ratio = negative / total elements = 2 / 5 = 0.4.
  - Zero ratio = zero / total elements = 1 / 5 = 0.2.

4. Print the decimal values:
  - Positive ratio: 0.4
  - Negative ratio: 0.4
  - Zero ratio: 0.2

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The length of the minute hand of the clock is given as follows:

10.6 inches.

Suppose that we have two points of the coordinate plane, and the ordered pairs have coordinates [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex].

The shortest distance between them is given by the equation presented as follows, derived from the Pythagorean Theorem:

[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

The ordered pairs for this problem are given as follows:

(0,0) and (7.5, 7.5).

Hence the distance between the two points, representing the length of the minute hand, is given as follows:

d = sqrt(7.5² + 7.5²)

d = 10.6 inches.

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The required medians of the triangle intersect at the point [tex]\left(4,\frac{10}{3}\right)$[/tex].

The medians of a triangle intersect at a point known as the centroid. To find the centroid of a triangle, we need to find the average of the x-coordinates and the average of the y-coordinates of its vertices.

Let the vertices of the triangle be A(0,0), B(5,0), and C(7,5). Then the midpoint of AB is

[tex]\left(\frac{0+5}{2},\frac{0+0}{2}\right) = (2.5,0)$[/tex],

the midpoint of BC is [tex]\left(\frac{5+7}{2},\frac{0+5}{2}\right) = (6,2.5)$[/tex], and the midpoint of CA is

[tex]\left(\frac{0+7}{2},\frac{0+5}{2}\right) = (3.5,2.5)$[/tex]

Therefore, the centroid of the triangle is:

[tex]$\begin{align*}\left(\frac{0+5+7}{3},\frac{0+5+5}{3}\right) &= \left(\frac{12}{3},\frac{10}{3}\right) \&= \left(4,\frac{10}{3}\right)\end{align*}[/tex]

So the medians of the triangle intersect at the point [tex]\left(4,\frac{10}{3}\right)$[/tex].

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a.  The cost of renting the gym and paying for supplies, which is $300. b. $10, c. y = 10x - 300 d. $200 e.  30 tickets f. the slope of the line is 1/6.

The slope of a line is the ratio of the change in the y-coordinate to the change in the x-coordinate. It represents the rate of change of the line. A positive slope indicates that the line is increasing, while a negative slope indicates that the line is decreasing. A slope of zero indicates that the line is horizontal.

The y-intercept is the point where the line crosses the y-axis. It represents the value of y when x is equal to zero.

Linear equations can be solved algebraically or graphically. In algebra, the goal is to isolate the variable and find its value. In graphing, the equation is plotted on a coordinate plane and the solution is the point where the line intersects with the x- or y-axis.

a. The constant in this situation is the cost of renting the gym and paying for supplies, which is $300.

b. The rate in this situation is the cost per ticket, which is $10.

c. Let x be the number of tickets sold and y be the total profit. The linear equation that represents this situation is:

y = 10x - 300

d. If Amanda sells 50 tickets, then the profit is:

y = 10(50) - 300 = $200

e. To break even, the profit must be 0. So we can set y = 0 and solve for x:

0 = 10x - 300

10x = 300

x = 30

Therefore, Amanda needs to sell 30 tickets to break even.

f. The graph of the linear equation is:

graph{y=10x-300 [-10, 50, -500, 500]}

To calculate the slope of a line that goes through points (10, 6) and (-2, 4), we can use the slope formula:

slope = (y2 - y1) / (x2 - x1)

Substituting the given values, we get:

slope = (4 - 6) / (-2 - 10) = -2 / -12 = 1/6

Therefore, the slope of the line is 1/6.

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The total number of t-shirts principal ordered in $1520 for the school store is given 214.

Let us consider the number of t-shirts the principal ordered be equal to 'x'.

The cost of each t-shirt is equal to $7.

⇒ The total cost of 'x' t-shirts =  7x dollars.

Shipping cost is a flat rate for any order, regardless of the number of t-shirts =  $22

Total cost of the order (including shipping) =  $1520.

Required equation is equal to,

7x + 22 = 1520

Subtract 22 from both sides of the equation we get,

⇒ 7x = 1498

Divide both sides of the equation by 7 we get,

⇒ x = 214

Therefore, the number of t-shirts ordered by  principal for the school store is equal to  214.

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An algebraic expression for representing the Nora's income in form of Tonya's income is given by  y = ( x / 4 ) .

Let us consider 'x' represents the Tonya's income.

And variable 'y' represents the Nora's income.

Tonya income is equal to four times of Nora's income.

This implies,

Nora's income is equal to one fourth times of Tonya's income.

⇒ y = ( x / 4 )

Rewrite an algebraic expression to represents Tonya's income in terms of Nora's income we have,

Simplify by multiplying both the sides of the algebraic expression by 4 we get,

⇒ x  = 4y

Therefore, an algebraic expression to represents the Nora's income in terms of Tonya's income is equal to y = ( x / 4 ).

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Answer: 1.5

Step-by-step explanation:

As per the surface area, the height of the cylindrical gallon tub of ice cream is approximately 2.625 inches. (option d).

In this problem, we have been given the diameter of the tub, which is 8 inches. The radius of the tub can be calculated by dividing the diameter by 2, which gives us a radius of 4 inches.

We have also been given the total surface area of the tub, which is 232.36 square inches. Using the formula for surface area, we can write:

232.36 = 2π(4)² + 2π(4)h

232.36 = 100.48 + 25.12πh

Subtracting 100.48 from both sides, we get:

131.88 = 25.12πh

Dividing both sides by 25.12π, we get:

h = 131.88 / (25.12π)

Using the value of π as 3.14, we can simplify this expression to:

h = 131.88 / 79.104

h ≈ 2.6256 inches

Hence the correct option is (d).

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Answer:

To solve this problem, we can use trigonometry and differentiation. Let θ be the angle of elevation formed by the lines from the top and bottom of the building to the tip of the shadow. Then, we have:

tan(θ) = height of building / length of shadow

Differentiating both sides with respect to x, we get:

sec^2(θ) dθ/dx = (-1 / length of shadow^2) (d length of shadow / dx) height of building

Substituting the given values, we get:

sec^2(θ) dθ/dx = (-1 / x^2) (d x / dx) 235

At x = 286, we have:

length of shadow = x + 235 tan(θ)

Differentiating this expression with respect to x, we get:

d length of shadow / dx = 1 + 235 sec^2(θ) dθ/dx

Substituting this into the previous equation and simplifying, we get:

dθ/dx = - x^2 / (235 (x + 235 tan(θ)))

At x = 286, we have:

length of shadow = 286 + 235 tan(θ)

tan(θ) = height of building / length of shadow = 235 / (286 + 235 tan(θ))

Solving for tan(θ), we get:

tan(θ) = 235 / (286 + 235 tan(θ))

tan(θ) (286 + 235 tan(θ)) = 235

235 tan^2(θ) + 286 tan(θ) - 235 = 0

Using the quadratic formula, we get:

tan(θ) = 0.470835 or -1.00084

Since the angle of elevation is positive, we take:

tan(θ) = 0.470835

Substituting this into the expression for dθ/dx, we get:

dθ/dx = - 286^2 / (235 (286 + 235 (0.470835)))

Simplifying this expression, we get:

dθ/dx ≈ -0.00074675 radians per foot (rounded to five decimal places)

Therefore, the rate of change of the angle of elevation at x = 286 feet is approximately -0.00074675 radians per foot.

Answer:

The information that suits the display on a double-line graph is:Option: B and Option: C

B. number of Democrats and Republicans in the Senate during the past decade.

C. monthly sales of two different types of cars during a one-year period.

Step-by-step explanation:

A double-line graph consist of two axes.

It shows the occurrence and the categories  which are being compared over the time. i.e. it is used to compare two sets of the data.

Hence, the correct option is:

B. number of Democrats and Republicans in the Senate during the past decade.

C. Monthly sales of two different types of cars during a one-year period.

( C. Since the x-axis represent the number of months and the y-axis represent the number of cars sold.

B. Since, the decade are represented by the x-axis and the number of people are represented by y-axis )

whereas in option A the height and weight are represented by a different units.

Step-by-step explanation:

To indicate values in a data set that are above or below the average monthly order totals use AVG([Order Total]). Thus, option (D) is correct.

AVG([Order Total]): This expression calculates the average of the "Order Total" values in your data set. The average is simply the sum of all the order totals divided by the number of data points.

SUM([Order Total]) calculates the total sum of all order totals.

AVG([Order Total]) then divides that sum by the number of data points to find the average.

Now, to determine which values are above or below the average monthly order totals:

Thus, option (D) is correct.

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The question attached here seems to be incomplete, the complete question is:

Which calculation would you use to indicate values in a data set that are above or below the average monthly order totals?

Select an answer:

AVG(SUM [Order Total])

SUM({Order Total])

SUM(AVG [Order Total])

AVG([Order Total]

The error made by the student is in simplifying the expression inside the square root.

What is the quadratic equation?

The solutions to the quadratic equation are the values of the unknown variable x, which satisfy the equation. These solutions are called roots or zeros of quadratic equations. The roots of any polynomial are the solutions for the given equation.

The student made an error in simplifying the expression inside the square root.

The expression should be simplified as follows:

25 + 4(2)(-12) = 25 - 96 = -71

So the correct expression inside the square root is -71, not 121.

Therefore, the correct solution using the Quadratic Formula is:

x = (-(-5) ± √((-5)² - 4(2)(-12)))/(2(2))

x = (5 ± √(25 + 96))/4

x = (5 ± √121)/4

x = (5 ± 11)/4

Hence, the solutions are x = -3 and x = 4/2, which simplifies to x = 2.

Therefore, the error made by the student is in simplifying the expression inside the square root.

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The solution to the system of equations is (x, y) = (2, 1).

A system of equations is a set of two or more equations that are to be solved simultaneously, meaning that the values of the variables that satisfy each equation in the system must be found. The solution to a system of equations is the set of values for the variables that satisfy all the equations in the system.

To solve the system of equations:

8x - 5y = 11 ...(1)

4x - 3y = 5 ...(2)

We can use the elimination method to eliminate one of the variables. We want to eliminate the variable "y", so we need to multiply equation (2) by -5/3, which will give us:

-5/3(4x - 3y) = -5/3(5)

-20x/3 + 5y = -25/3 ...(3)

Now we can add equations (1) and (3) to eliminate "y":

8x - 5y + (-20x/3 + 5y) = 11 - 25/3

Combining like terms, we get:

(24x - 15y - 20x + 15y)/3 = 8/3

Simplifying, we get:

4x/3 = 8/3

Multiplying both sides by 3, we get:

4x = 8

Dividing both sides by 4, we get:

x = 2

Now we can substitute x = 2 into equation (1) or (2) to find y. Let's use equation (1):

8x - 5y = 11

8(2) - 5y = 11

16 - 5y = 11

Subtracting 16 from both sides, we get:

-5y = -5

Dividing both sides by -5, we get:

y = 1

Therefore, the solution to the system of equations is (x, y) = (2, 1).

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Complete question:

Given the system of equation:

8x - 5y = 11

4x - 3y = 5

Find the value of x and y.

The correct interpretations of the given quantity are C and E.

C. Your daily viewing time is greater than 90% of all Netflix users.

E. 90% of all Netflix users viewing times per day are less than your daily viewing time.

To understand the correct interpretation, we need to first understand what the 90th percentile means. It refers to the value below which 90% of the data falls.

If your daily viewing time is in the 90th percentile of all Netflix users, it means that your viewing time is greater than 90% of all users. In other words, only 10% of users watch more Netflix than you, and your viewing time is greater than the viewing time of 90% of users.

Therefore, options C and E are the correct interpretations.

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Complete Question:

Suppose the amount of time (in minutes) per day that you spend watching Netflix is in the 90th percentile of all account users. Which interpretation(s) of this quantity are correct? Select ALL that apply.

A. You spend approximately 90 minutes per day watching Netflix.

B. 90% of all viewers spend more time on Netflix per day than you.

C. Your daily viewing time is greater than 90% of all Netflix users.

D. There is a 90% probability that a randomly selected viewer spends less time than you watching Netflix per day.

E. 90% of all Netflix user viewing times per day are less than your daily viewing time.

F. 90% of all Netflix user viewing times per day are less than or equal to your daily viewing time.

2 and 2/3 divided by 6 equals 4/9.

So, 2 and 2/3 divided by 6,

we can simplify it step by step as follows:

First, we need to convert the mixed numbers 2 and 2/3 to an improper fraction:

2 and 2/3 = (2 × 3 + 2) / 3 = 8/3

Therefore, the expression becomes:

8/3 ÷ 6

To divide fractions, we need to invert the second fraction and multiply:

8/3 ÷ 6 = 8/3 × 1/6

Simplifying the multiplication of fractions:

8/3 × 1/6 = (8 × 1) / (3 × 6) = 8/18

Now, we can simplify the fraction 8/18 by dividing both the numerator and denominator by their greatest common factor, which is 2:

8/18 = (8 ÷ 2) / (18 ÷ 2) = 4/9

Therefore, 2 and 2/3 divided by 6 equals 4/9.

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10-hole harmonicas, which can produce a wider range of notes and are more commonly used in professional music settings.

An equation is a statement that expresses the equality of two mathematical expressions using mathematical symbols such as variables, numbers, and mathematical operations. The equality is represented by an equal sign "=" between the two expressions. Equations are used to represent mathematical relationships and solve problems in various fields such as physics, chemistry, engineering, and economics.

Given by the question.

The terms "6 holes and 1/2" and "10 holes" are often used to refer to harmonicas, which are small wind instruments that produce sounds when air is blown into or drawn out of them.

The main difference between a 6-hole harmonica and a 10-hole harmonica is the number of holes on the instrument. As the name suggests, a 6-hole harmonica has 6 holes, while a 10-hole harmonica has 10 holes.

In addition to the number of holes, the two types of harmonicas may also differ in their size, range, and the specific notes that they can produce. 6-hole harmonicas are typically smaller and produce a more limited range of notes compared

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The ratio of QR to ST in the triangle is 5/7

Three angles and the three sides of a triangle are equal to the corresponding angles and the corresponding sides of another triangle, then both the triangles defined as congruent.

To establish that two triangles are congruent, not all six matching elements of either triangle must be located. There are five condition for two triangles to be congruence,  according to  the trials. The congruence qualities are SSS, SAS, ASA, AAS, and RHS.

PQR and PST are congruent

So PQ/PS=QR/ST

We know that

PQ=10

PS=14

QR/ST=10/14

QR/ST=5/7

hence, the ratio of QR to ST in the triangle is 5/7.

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During bracket off, when 6x⁴ * 6x⁴= 36x⁸ instead of 36x¹⁶ using Perfect square method.

When 4y² * 4y² = 16y^4 instead of 25y⁴⁴ gotten in the question. To correct the error, using the perfect square method, it can be corrected.

The answer will be (6x⁸ - 5y²)(6x⁸ + 5y²)The last error found is sign rule-4y^2  x -4y^2 = +16y⁴ instead of -4y² * +4y² = -16y⁴

Perfect square is  is a number that is generated by multiplying two equal integers by each other.

Given 36x¹⁶ - 25y⁴

The rule = A difference of two perfect squares, A² - B²²  can be factored into (A+B) x (A-B)36x¹⁶ is a perfect square = 6₂ * x⁸2)= (6x⁸)^225y⁴ is also a perfect square= 5² * x²(2)= (5x²)²

Using the rule for Perfect square,

When A² - B² = (A + B) x (A - B)

So,(6x⁸)₂ - (5x²)²

=(6x⁸ + 5x²) * (6x⁸ - 5x²)

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The length for which the volume of Box A equals the volume of Box B, obtained by resolving the cubic equation is 4 inches. The correct option is therefore;

A cubic equation is an equation of the form; a·x³ + b·x² + c·x + d, where a is a non zero number.

The x represent the width of Box A, we get;

The length of Box A + 4 = x

The length of Box A = x - 4

Height of Box A = x - 4 - 1 = x - 5

Volume of Box A = x·(x - 4)·(x - 5) = x³ - 9·x² + 20·x

Length of Box B = 2 × (x - 4)

Width of Box B = (2 × (x - 4))/2 - 2 = x - 4 - 2 = x - 6

Height of Box B = (2 × (x - 4))/2 + 2 = x - 4 + 2 = x - 4

The volume of Box B = 2 × (x - 4) × (x - 6) × (x - 4) = 2·x³ - 28·x² + 128·x - 192

The volume of Box A is equal to the volume of Box B when we get the following cubic equation;

x³ - 9·x² + 20·x = 2·x³ - 28·x² + 128·x - 192

The above cubic equation can be factored using an online tool to get;

(x - 4)·(x² - 15·x + 48) = 0

Therefore; (x - 4) = 0, and x² - 15·x + 48 = 0

x = 4, x = (15 + √(33))/2, and x = (15 - √(33))/2

Therefore, one of the lengths for which the volume of Box A is the same as Box B ; x = 4 inches. The correct option is option A.

A. x = 4 inches

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a. The equation to represent the situation is y = -12x + 120, where x is the number of movies and y is the amount of money remaining on the gift card.

Money is a medium of exchange that is widely accepted as a way to pay for goods and services or to settle debts. Money also serves as a store of value, providing a way for people to save for the future. Money is generally created through government-backed fiat currencies, such as the U.S. dollar, which are issued and regulated by central banks. Money can also be created in the form of crypto-currencies, such as Bitcoin, which are not issued by any single government or central bank. Money is essential for economic growth and stability, as it allows for efficient exchanges of goods and services. Money can also be a source of financial security, providing people with a way to manage their finances and plan for the future.

b. The slope of -12 indicates that for every movie that Elyse sees, she will spend $12 from her gift card. The y-intercept of 120 indicates that if Elyse has not seen any movies, she will have $120 remaining on her gift card.

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The situation is,

a) The line equation is [tex]y = -3x - 6[/tex]

b) The line's y-intercept is -6, which indicates that when the amount of movies x = 0 , the amount on gift y = -6

c) The slope of the line is -3, indicating that as the number of movies x increases, the rate of change of the amount on the gift is declining.

a). The equation provides the line's slope.

Slope,

[tex]m=\frac{(y_2-y_1)}{(x_2-x_1)}[/tex]

Changing the numbers indicated in the slope equation,

Slope,

[tex]m=\frac{(6-12)}{(4-2)}[/tex]

Slope m = -6/2

Slope m = -3

The slope is -3

The equation of the line is,

[tex]y - y_1 = m ( x - x_1 )[/tex]

Substitute the given values in the equation,

[tex]y - 12 = -3 ( x - 2 )[/tex]

Simplify the equation,

[tex]y - 12 = -3x + 6[/tex]

Adding 12 on both sides

[tex]y = -3x - 6[/tex]

The equation of line is [tex]y = -3x - 6[/tex]

b). The y-intercept of the equation of line [tex]y = -3x - 6[/tex] is [tex]-6[/tex], when [tex]x=0[/tex]

c). The slope of the line [tex]y = -3x - 6[/tex] is [tex]m = -3[/tex] and the value is decreasing

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The completr question and graph attached below,

c. What is the slope, and what does it mean in the context of the situation?

Answer:

3s+7.99=71.83

Step-by-step explanation:

Answer:

To find the explicit formula for the sequence, we need to determine the common ratio between each term.

The common ratio is found by dividing any term by the previous term. For example:

The common ratio between the second and first terms is 150/30 = 5.

The common ratio between the third and second terms is 750/150 = 5.

Since the common ratio is the same for all terms, we can use it to find any term in the sequence.

Let's call the first term a₁, and let r be the common ratio. Then we have:

a₁ = 30

a₂ = a₁ * r = 30 * 5 = 150

a₃ = a₂ * r = 150 * 5 = 750

a₄ = a₃ * r = 750 * 5 = 3750

a₅ = a₄ * r = 3750 * 5 = 18750

We can see that the explicit formula for the sequence is:

aₙ = a₁ * r^(n-1) = 30 * 5^(n-1)

Therefore, the explicit formula for the sequence is aₙ = 30 * 5^(n-1).