An expression is shown below: 3pf2 − 21p2f + 6pf − 42p2 Part A: Rewrite the expression by factoring out the greatest common factor. (4 points) Part B: Factor the entire expression completely. Show the steps of your work. (6 points)

If an item has a listed price of $65 and the sales tax rate is 5%, the sales tax will be $3.25 and the total cost will be $68.25.

If the item has a listed price of $65, the sales tax rate is 5%, we can first calculate the amount of sales tax as follows:

Sales Tax = Listed Price x Sales Tax Rate

Sales Tax = $65 x 0.05

Sales Tax = $3.25

So the sales tax on the item is $3.25.

To calculate the total cost, we simply add the sales tax to the listed price:

Total Cost = Listed Price + Sales Tax

Total Cost = $65 + $3.25

Total Cost = $68.25

So the total cost of the item with sales tax is $68.25.

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Answer:

173.83

Step-by-step explanation:

An octagon has 8 sides (octa-, right, like octopus)

"regular" means all the sides are the same length.

The apothem is given, 7.243, it goes from the center to the middle of a side, at a right angle. See image. Make a triangle. Use area formula for triangle.

A = 1/2bh

Multiply by 8, bc there are 8 of these identical triangles.

OR, use the area formula for an n-sided regular polygon:

A = 1/2asn

see image.

The evaluated probability that a randomly choosing a point within the circle falls in the red-shaded triangle is 0.30,  

To evaluate this probability, we have to calculate the ratio of the area of the red-shaded triangle to the area of the circle.

The area of the circle is derived as πr²

Here,

r = radius of the circle.

In this case, r = 5.

Staging the values

π(5)² = 25π.

So, the area of the circle is 25π.

The area of the red-shaded triangle can be evaluated by using the formula for the area of a triangle which is

[tex]1/2 * base * height[/tex]

In this case, the base is 6 and the height is 8. So, the area of the red-shaded triangle is

1/2 x 6 x 8

= 24.

Hence, the probability that a randomly selected point within the circle falls in the red-shaded triangle is

P = (Area of red-shaded triangle) / (Area of circle)

= 24 / (25π)

= 24 / 25 x 3.14

= 24 / 78.5

≈ 0.30

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The area of the composite figure is 336 centimeters²

Since, A right triangle or right angled triangle, or more formally an orthogonal triangle, formerly referred as rectangle triangle, is a triangle in which one angle is a right angle, that is in which two sides are perpendicular to each other. It forms the basis of trigonometry on the basis of its sides and angles.

Now, The area of a figure is the number of unit squares that cover the surface of a closed figure. Area is measured in square units like cm2 and m2. Area of a shape is a two dimensional quantity.

Divide the composite figure into smaller shapes.

A is a rectangle with width 6cm and length 12cm

Hence, Area of A= 12 x 6 = 72cm²

B is a triangle with base 6cm and height 8cm

Therefore area of triangle=1/2 x base x height

= 1/2 x 6x 8 = 24cm²

C is a rectangle with breadth 30cm and length 8cm

Area of figure C =30  x 8 = 240cm²

So, total area of the figure is =(72+24+240)cm²

=336cm²

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Complete question is,

What is the area of this composite figure?

A figure is comprised of 1 rectangle and 2 triangles. The rectangle has a length of 16 centimeters and width of 10 centimeters. Both triangles have a base of 16 centimeters and height of 6 centimeters.

96 centimeters squared

160 centimeters squared

208 centimeters squared

256 centimeters squared

Answer:

Step-by-step explanation:

First, we need to determine the slope of the line we want to find since it is perpendicular to the given line. The slope of a line perpendicular to another line is the negative reciprocal of the slope of the given line.

The given line has a slope of -1/3, so the slope of the line we want to find is 3 (the negative reciprocal of -1/3).

Now we can use the point-slope form of the equation of a line to find the equation of the line passing through (1,-10) with a slope of 3:

y - y1 = m(x - x1)

where m is the slope and (x1, y1) is the point. Substituting the values we have:

y - (-10) = 3(x - 1)

Simplifying:

y + 10 = 3x - 3

Subtracting 10 from both sides:

y = 3x - 13

Therefore, the equation of the line perpendicular to y=-1/3x+5 and passing through the point (1,-10) is y = 3x - 13.

The ratio is not approximately 3.1.

Option B is the correct answer.

We have,

Let's use the formula for the ratio of the circumference to the diameter of a circle:

C/d = pi (where pi is approximately 3.14).

Therefore, the ratio should be approximately 3.14, not 3.1.

Using the given answer choices:

A. C/d = 40.3/13 = 3.1... approximately correct

B. C/d = 29.7/9 = 3.3... not correct

C. C/d = 21.7/7 = 3.1... approximately correct

D. C/d = 37.2/12 = 3.1... approximately correct

Therefore,

The ratio is not approximately 3.1.

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The results for each composite function at each x-value are listed below:

Case 13: (f ° g) (1) = 26 (Right choice: D)

Case 14: (f + g) (3) = 20 (Reight choice: E)

In this problem we find two cases of composite functions that must be evaluated at given x-value. The procedure is described below:

Now we proceed to solve for each case:

Case 1: f(x) = x² + x - 4, g(x) = 3 · x + 2

(f ° g) (x) = [(3 · x + 2)² + (3 · x + 2) - 4]

(f ° g) (1) = [(3 · 1 + 2)² + (3 · 1 + 2) - 4]

(f ° g) (1) = (5² + 5 - 4)

(f ° g) (1) = 26

Case 2: f(x) = x² + x, g(x) = x² - 1

(f + g) (x) = (x² + x) + (x² - 1)

(f + g) (x) = 2 · x² + x - 1

(f + g) (3) = 2 · 3² + 3 - 1

(f + g) (3) = 18 + 3 - 1

(f + g) (3) = 20

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The value of x is 3 when the two angles are vertical.

Vertical angles are formed when two lines meet each other at a point.

They are always equal to each other.

The first angle measures (2x + 15)°, and the second measures (4x + 9)°.

2x+15 = 4x+9

We have to find the value of x

Take the variable terms on one side and constants on other side

15-9=4x-2x

6=2x

Divide both sides by 2

x=3

Hence, the value of x is 3 when the two angles are vertical.

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The the distance between these points (6,0) and (26,0) is 20 units.

We have,

The distance between two points is the length of the line joining the two points. Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria.

Formula: distance= √(x_2-x_1)²+(y_2-y_1)²

Given that,

endpoints are  (6,0) and (26,0) of the line segment.

The distance= √(26-6)²-(0-0)²

                 =20 units

Hence, The the distance between these points (6,0) and (26,0) is

20 units.

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Computing the total number of counters in the class as 1,108, the mean number of counters that the 46 children have is 24.

The mean refers to the average value.

The average is the quotient of the total value divided by the number of items in the data set.

The number of boys in the class = 26

The number of girls in the class = 20

The total number of boys and girls in the class = 46

The mean number of counters that the boys have = 28

The total number of counters that the boys have = 728 (28 x 26)

The mean number of counters that the girls have =19

The total number of counters that the girls have = 380 (19 x 20)

The total number of counters that the class has = 1,108 (728 + 380)

The average or mean number of counters in the class = 24 (1,108 ÷ 46)

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Answer:

B. 10/2, E. 6.3

Step-by-step explanation:

First, you need to solve for x by isolating it on one side of the inequality. You can do this by adding 3 to both sides and then dividing by 2:

2x - 3 > 1

2x - 3 + 3 > 1 + 3

2x > 4

2x / 2 > 4 / 2

x > 2

Next, you need to find two numbers that are greater than 2 and plug them into the inequality to check if they make it true. You can choose any numbers that are larger than 2, but it is easier to pick from the given options. For example, let’s try A. 3/2 and B. 10/2:

A. x = 3/2

2(3/2) - 3 > 1

3 - 3 > 1

0 > 1, False

B. x = 10/2

2(10/2) - 3 > 1

10 - 3 > 1
7 > 1, True

Then, you need to repeat the same process for the remaining options until you find another number that makes the inequality true. For example, let’s try C. 1, D. 2, and E. 6.3:

C. x = 1

2(1) - 3 > 1

2 - 3 > 1

-1 > 1 False

D. x = 2

2(2) - 3 > 1

4 - 3 > 1

1 > 1, False

E. x = 6.3

2(6.3) - 3 > 1

12.6 - 3 > 1

9.6 > 1, True

Finally, you need to select the two numbers that make the inequality true from the options. The answer is B. and E. B. 10/2 and E. 6.3 are two numbers that make the inequality true.

Answer:

your selection is correct. The slope is positive. As x increases y increases.

Answer: e, an=6n-7.

Step-by-step explanation: Any pair of consecutive terms can be used to determine the arithmetic sequence's common difference (d), which can then be used in conjunction with the given term to determine the nth term using the following formula:

an = a1 + (n - 1) d.

Let's use the pair a11=53 and a10=47 to find d:

d = a11 - a10 = 53 - 47 = 6

Now we can use the formula to find any term of the sequence. Let's use the given value of a13=65 to find a13:

a13 = a1 + (13 - 1)d

65 = a1 + 12(6)

65 = a1 + 72

a1 = -7

Therefore, the formula that can be used to find the nth term of the arithmetic sequence is:

an = -7 + (n - 1)6

Simplifying this expression, we get:

an = 6n - 7.

Answer:

the answer is B

Step-by-step explanation:

all trapezoids are quadrilaterals

Answer:

A soup can with a height of 4cm and a radius of 3cm holds approximately 113.1 cubic centimeters of soup.

Explanation:

The volume of a cylindrical can is calculated using the formula V = πr^2h, where V is the volume, r is the radius, and h is the height of the can.

In this case, the radius is 3cm and the height is 4cm. Plugging these values into the formula, we get:

V = π × 3^2 × 4

V = 113.1 cubic centimeters

The equilibrium price and quantity if the government imposes a fixed tax of $36 on each good is $26 and  the corresponding value of the government tax revenue is $648

To find the equilibrium price and quantity, we need to set Qs = Qd and solve for the price.

Qs = Qd

=> p = -3Qd + 80 = Qs + 8

=> -3Qd + 80 = Qd + 8 (substituting Qs = Qd)

=> 4Qd = 72

=> Qd = 18

=> Qs = 18

Therefore, the equilibrium price is:

p = Qs + 8 = 18 + 8 = 26

Now, if the government imposes a fixed tax of $36 on each good, the new supply function becomes:

p = Qs + 8 - 36

=> Qs = p - 28

And the demand function remains the same:

p = -3Qd + 80

Setting Qs = Qd and substituting Qs and Qd in terms of p, we get:

p - 28 = -3Qd + 80

=> -3Qd = p - 52

=> Qd = (52 - p) / 3

The government tax revenue is the product of the tax and the quantity sold, which is:

Tax revenue = tax per unit × quantity sold

=> Tax revenue = 36 ×Qs

Substituting Qs = 18, we get:

Tax revenue = 36 × 18 = $648

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The slope of the line shown is equal to: A. 1/4.

In Mathematics and Geometry, the slope of any straight line can be determined by using this mathematical equation;

Rate of change (slope) = (Change in y-axis, Δy)/(Change in x-axis, Δx)

Rate of change (slope) = rise/run

Rate of change (slope) = (y₂ - y₁)/(x₂ - x₁)

By substituting the given data points into the formula for the slope of a line, we have the following;

Rate of change (slope) = (y₂ - y₁)/(x₂ - x₁)

Rate of change (slope) = (1 + 1)/(5 + 3)

Rate of change (slope) = 2/8

Rate of change (slope) = 1/4.

Based on the graph, the slope is the change in y-axis with respect to the x-axis and it is equal to 1/4.

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The measure of exterior angle 1 is 120 degrees. (option a)

In this case, we are given two interior angles of the polygon, which are 60 degrees and 25 degrees. The sum of the measures of the interior angles of a polygon with n sides is given by the formula (n-2) x 180 degrees. Therefore, for this polygon, the sum of its interior angles can be calculated as

=> (n-2) x 180 = (3-2) x 180 = 180 degrees.

Next, we can use the fact that the sum of an exterior angle and its adjacent interior angle is always 180 degrees. Therefore, we can calculate the measure of the exterior angle 1 as follows:

Exterior angle 1 + Interior angle 1 = 180 degrees

Exterior angle 1 + 60 degrees = 180 degrees (since we know that Interior angle 1 is 60 degrees)

Exterior angle 1 = 180 degrees - 60 degrees

Exterior angle 1 = 120 degrees

Hence the correct option is (a).

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The points (-4, 3), (-1, 1), and (1, 3) cannot form the vertices of a right triangle.

To determine if the points (-4, 3), (-1, 1), and (1, 3) can form the vertices of a right triangle

we need to check if the square of the length of one side of the triangle is equal to the sum of the squares of the lengths of the other two sides.

We calculated the distances between the three points using the distance formula, and checked if any of the three sides of the triangle satisfied the Pythagorean theorem, which relates the sides of a right triangle.

Since none of the three sides satisfied the Pythagorean theorem, the given points cannot form the vertices of a right triangle.

Therefore, the points (-4, 3), (-1, 1), and (1, 3) cannot form the vertices of a right triangle.

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A formula relating y to t is y = 6.55[tex]e^{k(22)}[/tex]ln(26/6.5)t)

Let's consider a sample culture of bacteria with an initial area of 6.5. We observe that the population of bacteria doubles in size every 22 minutes. We want to find a formula that relates the area of the sample culture, denoted as y, to the time passed, denoted as t.

Let's call this proportionality constant k. We can then write the differential equation that describes the growth of the population as:

dy/dt = ky

where dy/dt is the rate of change of the area with respect to time.

We can solve this differential equation by separation of variables. We first separate the variables and then integrate both sides:

dy/y = k dt

∫(dy/y) = ∫k dt

ln|y| = kt + C

where C is a constant of integration. To solve for C, we use the initial condition that the area of the sample culture is 6.5 when t = 0. This gives:

ln|6.5| = k(0) + C

C = ln|6.5|

Substituting this value of C back into the solution, we get:

ln|y| = kt + ln|6.5|

ln|y/6.5| = kt

Finally, we can solve for y to get the formula we were looking for:

y = 6.5[tex]e^{kt}[/tex]

where k is the proportionality constant that depends on the doubling time of the bacteria. We can use the fact that the population doubles in size every 22 minutes to find k. If the population doubles, then the area of the population quadruples. This means that after 22 minutes, the area is 6.5 x 4 = 26. We can use this to find k:

26 = 6.5[tex]e^{k(22)}[/tex]

[tex]e^{k(22)}[/tex] = 26/6.5

k(22) = ln(26/6.5)

k = (1/22)ln(26/6.5)

Substituting this value of k back into the formula we derived earlier, we get the exact expression relating y to t:

y = 6.55[tex]e^{k(22)}[/tex]ln(26/6.5)t)

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Logan will pay $18.02 more if he buys the wooden stacking toy instead of the plastic stacking toy.

The cost of the plastic stacking toy is 1/3 the cost of the wooden stacking toy:

Cost of plastic toy = (1/3) x $25.50 = $8.50

If Logan buys the wooden toy, he will pay:

Cost of wooden toy = $25.50

To find the amount of sales tax on either toy, we can multiply the cost of the toy by the tax rate of 6%:

Sales tax = 6% x cost of toy

Sales tax on the wooden toy = 6% x $25.50 = $1.53

Sales tax on the plastic toy = 6% x $8.50 = $0.51

If Logan buys the wooden toy, he will pay an extra:

Total cost difference = Cost of wooden toy - Cost of plastic toy + Sales tax on wooden toy - Sales tax on plastic toy

Total cost difference = ($25.50 - $8.50) + ($1.53 - $0.51)

Total cost difference = $17 + $1.02

Total cost difference = $18.02

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So expressing the given complex solution x = (- 6 ± √(-432))/6 as x = a ± bi we can write that x = -1 ± 2√3i and the values of a and b are -1 and 2√3 respectively.

Given that the standard form of writing a complex solution is,

x = a ± bi, where a and b are the real numbers and i = √(-1)

Given that one of this type of solution is,

x = (- 6 ± √(-432))/6

x = -6/6 ± √(-432)/6, differentiate denominators with each numerator

x = -1 ± √((-1)*144*3)/6

x = -1 ± 12√3i/6

x = -1 ± 2√3i

Comparing with x = a ± bi we get, a = -1 and b = 2√3.

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Answer:1

Step-by-step explanation:

The type of arrangement used in the picture to pack cans is a linear arrangement.

If the cans are arranged in a straight line, this would be called a linear arrangement. In a linear arrangement, the objects are arranged in a single line or row.

This type of arrangement is commonly used for packing and displaying items that need to be easily accessible or for creating a uniform and organized appearance. It can be used for packing cans, bottles, or other similarly shaped objects.

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Identify the type of arrangement used in the picture to pack cans

​

Based on an equation, the salaries of the two employees whose average weekly salary is $5,500 are as follows:

X = $5,25

S = $5,725.

An equation is a mathematical statement that shows that two or more algebraic expressions are equal or equivalent.

While equations use the equal symbol, mathematical or algebraic expressions combine variables with numbers, constants, and values using mathematical operands.

The average weekly salary of two employees = $5,500

The number of employees = 2

Th,e total weekly salaries of the two employees = $11,000 ($5,500 x 2)

Let the salary of Employee One = x

Let the salary of Employee Two = s

x = s - 450

11,000 = s - 450 + s

11,450 = 2s

s = $5,725

x = $5,275 ($5,725 - $450)

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Using the Pythagorean theorem: a^2 + b^2 = c^2, where a=36, b=77, and c is the hypotenuse.

36^2 + 77^2 = c^2

1296 + 5929 = c^2

7225 = c^2

c = √7225

c = 85

The hypotenuse is 85 units

I am also in confused

Using the exponential function we can rewrite the expression as:

[tex]H = 10^{-4}[/tex]

Here we start with the equation:

log₁₀(H) = -4

We can rewrite the logarithm part as follows:

log₁₀(H) = ln(H)/ln(10)

Then we can rewrite:

ln(H)/ln(10) = -4

ln(H) = -4*ln(10)

Now we can move the coefficeint -4 as a exponent:

[tex]ln(H) = ln(10^{-4})[/tex]

Now apply the exponential equation to both sides:

[tex]exp(ln(H)) = exp(ln(10^{-4}))[/tex]

[tex]H = 10^{-4}[/tex]

That is what we wanted.

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Answer :

In the first figure,

△BCA

a = Perpendicular

b = base

c = hypotenuse

In the second figure,

△ABC

a = base

b = perpendicular

c = base

tan B = Perpendicular/Base = b/a

cos B = Base/Hypotenuse = a/c

cos A = Base/Hypotenuse = a/c

sin B = Perpendicular/Hypotenuse = b/c

tan A = Perpendicular/Base = a/b

sin A = Perpendicular/Hypotenuse = a/c