Commutative property

Answers

Answer 1

Answer:

Commutative property of addition: Changing the order of addends does not change the sum. For example, 4 + 2 = 2 + 4 4 + 2 = 2 + 4 4+2=2+44, plus, 2, equals, 2, plus, 4. Associative property of addition: Changing the grouping of addends does not change the sum


Related Questions

A triangle is shown in the image 28 inches 32 inches 8 inches what is the area of the triangle

Answers

The area of the triangle is 128 square inches.

How to find the area of the triangle

To find the area of a triangle, you can use the formula:

Area = (1/2) * base * height

In this case, the base of the triangle is 32 inches and the height is 8 inches.

Plugging these values into the formula, we get:

Area = (1/2) * 32 * 8

Area = 16 * 8

Area = 128 in²

So, the area of the triangle is 128 square inches.

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A triangle is shown in the image. A triangle with a height of 8 inches. The height is perpendicular to the base labeled 32 inches. The side from the top of the perpendicular side to the base is labeled 28 inches. What is the area of the triangle? 256 in2 224 in2 128 in2 112 in2

What is the meaning of "Finally [tex]s _{i}\circ s_{i}=s _{j}\circ s_{i}=1[/tex] hence [tex]s _{j}=s_{i}\circ r_{i-j}[/tex] and [tex]s _{i}=r_{i-j}\circ s_{j}[/tex]?

Answers

This statement describes the relationship between symmetries and rotations

Explaining the meaning of the statement as stated

This statement describes a relation between two symmetry operations, denoted by [tex]s_i[/tex] and [tex]s_j[/tex], and a rotation operation denoted by [tex]r_{i-j}[/tex]. The notation [tex]"s_i \circ s_i"[/tex] and [tex]"s_j \circ s_i"[/tex] denotes the composition of these operations, which means applying one operation after the other.

The statement says that [tex]s_i[/tex] and [tex]s_j[/tex] are both symmetries, and that they have a specific relationship with each other: when [tex]s_i[/tex] is composed with itself, it gives the identity operation (denoted by 1), and when [tex]s_j[/tex] is composed with [tex]s_i[/tex], it also gives the identity operation.

Then the statement concludes that this relationship implies that [tex]s_j[/tex] can be obtained by first applying [tex]s_i[/tex] and then applying a rotation operation [tex]r_{i-j}[/tex], and that [tex]s_i[/tex] can be obtained by first applying a rotation operation [tex]r_{i-j}[/tex] and then applying [tex]s_j[/tex].

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Which of the following statements about the number line
is true?

A. Number lines can't help you compare numerals.

B. Numbers get larger as you move to the left on the
number line.

C. Numbers get smaller as you move to the left on the
number line.

D. Number lines end with the number 10.

Answers

Hi the answer is C hope this helps have a great day

Find the area of this triangular prism. Be sure to include the correct unit in your answer

Answers

The surface area of the prism is 360in²

What is surface area of prism?

A prism is a solid shape that is bound on all its sides by plane faces.

The surface area of a prism is expressed as;

SA = 2B + ph

where h is the height of the prism and B is the base area , p is the perimeter of the base.

Base area = 1/2 × 5 × 12

= 5 × 6 = 30 in²

Perimeter of the base = 5+12+13

= 30 in²

height = 10 in

SA = 2 × 30+30× 10

= 60 + 300

= 360 in²

therefore the surface area of the prism is 360 in²

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A woman earned wages of 33,900 receive 2600 and interest from savings account and contributed 3600 to tax deferred retirement plan she was entitled to a personal exemption of 3500 and her deductions totaling 5440 find her gross income adjusted income and taxable income

Answers

The woman's gross income, adjusted income and taxable income are:

gross income - $ 36, 500adjusted income - $ 32, 900taxable income -$ 23, 960

How to find the types of income ?

The gross income to the woman is:

= Wages + interest

= 33, 900 + 2, 600

= $ 36, 500

The adjusted gross income :

= Gross income - Retirement plan contribution

= 36, 500 - 3, 600

= $ 32, 900

The taxable income ;

= AGI - Personal exemption - Deductions

= 32, 900 - 3, 500 - 5, 440

= $ 23, 960

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A subscription to a live streaming program normally costs $29 but is on sale for 25% off. An 8% tax is added at checkout. What is the total cost?

$7.25
$21.75
$23.49
$39.15

Answers

$23.49, $7.25 is how much money you get off, the sale price is $21.75 before tax, after tax it’s 23.49.

A subscription to a live streaming program normally costs $29 but is on sale for 25% off. An 8% tax is added at checkout then the total cost is $23.49.

To calculate the total cost, first, we find the discounted price of the subscription. A 25% discount on $29 is calculated by multiplying $29 by 0.25 (25% as a decimal), which gives us a discount of $7.25. Subtracting the discount from the original price, we get the discounted price of $29 - $7.25 = $21.75.

Next, we need to add the 8% tax to the discounted price. To calculate the tax amount, we multiply $21.75 by 0.08 (8% as a decimal), which gives us $1.74. Adding this tax amount to the discounted price, we get the total cost as $21.75 + $1.74 = $23.49.

Step 1: Calculate the discount amount:

Discount = 25% of $29

Discount = 0.25 * $29

Discount = $7.25

Step 2: Find the discounted price:

Discounted Price = Original Price - Discount

Discounted Price = $29 - $7.25

Discounted Price = $21.75

Step 3: Calculate the tax amount:

Tax Amount = 8% of Discounted Price

Tax Amount = 0.08 * $21.75

Tax Amount = $1.74

Step 4: Calculate the total cost:

Total Cost = Discounted Price + Tax Amount

Total Cost = $21.75 + $1.74

Total Cost = $23.49

So, the total cost after applying the 25% discount and adding an 8% tax at checkout is $23.49.

- Discount = 0.25 * $29 = $7.25

- Discounted Price = $29 - $7.25 = $21.75

- Tax Amount = 0.08 * $21.75 = $1.74

- Total Cost = $21.75 + $1.74 = $23.49

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what is the interquartile range of this data set? enter answer in box

Answers

Answer:

do it by yourself what you do while teacher is teaching

A scale drawing for an apartment is shown below. In the drawing, 3cm represents 4cm. Assuming the bedroom is rectangular, find the area of the real room

Answers

The calculated value of the area of the real room is 10.67 cm²

Finding the area of the real room

From the question, we have the following parameters that can be used in our computation:

In the drawing, 3cm represents 4cm

This means that

Scale factor = 4/3

Also, we have the dimension of the scale drawing to be

Dimension = 3cm by 2 cm

So, the area of the bedroom in the scale is

Area = 3cm * 2cm

For the actual room, we have

Area = 3cm * 2cm * (4/3)²

Evaluate the exponent

Area = 3cm * 2cm * (16/9)

Evaluate the products

Area = 10.67 cm²

This means that the real area is 10.67 cm²

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The volume of a juice box is about 24 cubic inches.

Answers

1. Length = 4 cm, width = 3 cm, height = 2 cm
The volume of this juice box is 4 x 3 x 2 = 24 cubic cm.

Total score:
of 15 points
(Score for Question 1: of 5 points)
1. The diagram shown is two intersecting lines. The measure of 22 is 29
2
(a) What is the measure of 24? How do you know? Explain your answer in complete sentences.
(b) Suppose the measure of 3 can be represented by (3x - 8)*. What equation can be written to solve for the value of x?
(c) What is the value of x? Show all work.
Answer:
•(Score for Question 2: _ of 5 points)
2. Draw a right triangle with side lengths of 6, 8, and 10 units. For full credit, be sure to include appropriate labels and symbols where necessary. © Stride, Inc. All rights reserved. No reproduction without written consent of Stride, Inc.
Answer:
•Score for Question 3:
_ of 5 points)
3. At a park, a statue has a ring-shaped brick path around it. The outer edge of the brick path is a circle with a radius of 13 m. The inner edge of the brick path is a circle with a radius of 8 m.
Brick path
13 m
8 m
Statue
(a) Write and simplify an expression for the exact area of the brick path. Show all work.
(b) Find the approximate area of the brick path. Use 3.14 to approximate r. Show all work.

Answers

The measure of angle ∠4 is 29°, the equation is 3x = 159 and the value of x is 53

The right angle using the given lengths is attachedThe exact area is 105π square meters and the approximate area is 330 square meters

The two intersecting lines

Given that, we have

∠2 = 29°

(a)

Angles 2 and 4 are vertical angles

So, we have

∠4 = 29°

(b) and (c)

We have

∠3 = (3x - 8)°

Angles 3 and 4 are angles on a straight line

So, we have the equation to be

3x - 8 + 29 = 180

Solving for x, we have

3x = 159

Divide by 3

x = 53

This means that the equation is 3x = 159 and the value of x is 53

The right angle with the given lengths

In this section, we have the following side lengths

Lengths = 6, 8 and 10

This means that

Side lengths = 6 and 8

Hypotenuse = 10

Next, we create the right angle using the given lengths of the legs and the hypotenuse

Calculating the area of the brick path

Here, we have

R = 13

r = 8

The exact area is calculated as

A = π(R² - r²)

So, we have

A = π(13² - 8²)

Evaluate

A = 105π

For the approximate area, we have

A = 105 * 22/7

A = 330

This means that the exact area of the brick path is 105π square meters and the approximate area is 330 square meters

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Use the unit circle to find exact value of the trig function
sin(135°)

Answers

The exact value of sin is √2/2

Calculate standard deviation for the following by direct and shortcut method.assume mean as 32
Serial no.of workers
1
2
3

Answers

The standard deviation for the given distribution is 8.739 using either the direct method or the shortcut method.

How to find the standard deviation?

Standard deviation is defined as a measure of how spread out numbers are. It is also referred to as the square root of the Variance, and then the Variance is the average of the squared differences from the Mean.

To calculate the standard deviation for the given distribution, we will utilize either the direct method or the shortcut method.

Direct method:

Calculate the mean wage (F) for the distribution:

Mean = (20 + 22 + 27 + 30 + 31 + 32 + 35 + 45 + 40 + 48)/10

Mean = 34

Calculate the differences between each wage and the mean wage:

20 - 34 = -14

22 - 34 = -12

27 - 34 = -7

30 - 34 = -4

31 - 34 = -3

32 - 34 = -2

35 - 34 = 1

45 - 34 = 11

40 - 34 = 6

48 - 34 = 14

Square each of the differences:

(-14)² = 196

(-12)² = 144

(-7)² = 49

(-4)² = 16

(-3)² = 9

(-2)² = 4

1² = 1

11² = 121

6² = 36

14² = 196

Add up the squared differences: 196 + 144 + 49 + 16 + 9 + 4 + 1 + 121 + 36 + 196 = 756

Divide the sum of the squared differences by the number of workers: 756/10 = 75.6

Take the square root of the result to get the standard deviation:

√75.6 = 8.739

Shortcut method:

The mean wage (F) for the distribution:

(20 + 22 + 27 + 30 + 31 + 32 + 35 + 45 + 40 + 48)/10 = 34

The sum of the squares of the differences between each wage and the mean wage:

(20 - 34)² + (22 - 34)² + (27 - 34)² + (30 - 34)² + (31 - 34)² + (32 - 34)² + (35 - 34)² + (45 - 34)² + (40 - 34)² + (48 - 34)² = 756

Divide the sum by the number of workers: 756/10 = 75.6

Take the square root of the result to get the standard deviation:

Standard deviation = √75.6 = 8.739

Therefore,

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Complete question is:

Calculate standard deviation for the following

distribution by direct and shortcut method.

Assume mean as 32.

Serial No. of Workers

1

2

3

4

5

6

7

8

9

10

Wage (F)

20

22

27

30

31

32

35

45

40

48

Screenshot below i have no idea how to solve this

Answers

Answer:

x^1/4 + 8x^1/4 + 12

Step-by-step explanation:

Using the FOIL method, we get x^1/4 + 8x^1/4 + 12

A waitress sold 11 ribeye steak dinners and 14 grilled salmon dinners, totaling $551.11 on a particular day. Another day she sold 15 ribeye steak dinners and 7 grilled salmon dinners, totaling $581.47. How much did each type of dinner cost?

Answers

Answer:

A ribeye steak dinner costs $32.23 and a grilled salmon dinner costs $14.04.

Step-by-step explanation:

Let's use variables to represent the cost of each dinner. Let's say that the cost of a ribeye steak dinner is "r" and the cost of a grilled salmon dinner is "s".

From the first day's sales, we know:

11r + 14s = 551.11

From the second day's sales, we know:

15r + 7s = 581.47

We now have two equations with two variables, which we can solve using elimination or substitution.

Let's use elimination. If we multiply the first equation by 15 and the second equation by 11, we can eliminate the "r" variable:

165r + 210s = 8266.65165r + 77s = 6396.17

Subtracting the second equation from the first, we get:

133s = 1870.48

Dividing both sides by 133, we get:

s = 14.04

So a grilled salmon dinner costs $14.04.

We can substitute this value back into one of the original equations to solve for "r". Let's use the first equation:

11r + 14(14.04) = 551.1111r + 196.56 = 551.1111r = 354.55r = 32.23

So a ribeye steak dinner costs $32.23.

Therefore, a ribeye steak dinner costs $32.23 and a grilled salmon dinner costs $14.04.

Find the approximate area of the shaded region below, consisting of a right triangle with a circle cut out of it. Use 3.14 as an
approximation for pi.

a. 1.254 square meters
b. 312 square meters
c. 2, 822 square meters
d. 314 square meters

Answers

Answer:

(a) 1.254 square meters

Step-by-step explanation:

We are given a figure, consisting of a right triangle with a circle cut out of it.

To Calculate : area of the shaded region below,

First Let's find the Area of the right triangle:

Area(triangle) = 1/2 × Base × Height

= 1/2 × 56 × 56

= 1/2 × 3136

= 3136/2

= 1568 m².

Now, We are diameter of the circle = 20 m

Radius of the circle = 20/2 = 10 m

Area of circle = πr²

Acc. to question we have to use 3.14 as an

approximation for pi.

Area (circle) = 3.14 × 10 × 10

Area (circle) = 3.14 × 100

Area (circle) = 314 m²

Area of shaded region = Area of triangle - Area of circle =

= 1568 m² - 314 m²

= 1.254 square meters

Therefore, The approximate area of the shaded region is 1.254 square meters.

Option (a) is the required answer.

The questions I need answered are in the picture… help plsss

Answers

The graph of the linear equation y = (1/2)*x + 3 can be seen in the image at the end.

How to graph the linear equation?

Here we want to graph the linear equation:

y = (1/2)*x + 3

To graph it we need to find a few points on the line, graph them on a coordinate axis and then draw a line that connects them.

The first point that we can use is the y-intercept, it is what we get when we evaluate in x = 0.

y = (1/2)*0 + 3

The point is (0, 3)

Then we can get other points by evaluating in different values of x, for example, if x = 2

y = (1/2)*2 + 3 = 1+ 3 =4

If x = 4

y = (1/2)*4 + 3 = 2 + 3 = 5

So we have the points (2, 4) and (4, 5)

Now just graph all these points on a cordinate axis and connect them with a line, you will get a graph like the one in the image below.

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Write an equation of a circle with the given center and radius. center (3,4) and radius 6

(x-3)^2+(y-4)^2=36
(x-3)^2+(y-4)^2=6
(x+3)^2+(y-4)^2=36
(x+3)^2+(y-4)^2=6

Answers

Answer:

[tex] {(x - 3)}^{2} + {(y - 4)}^{2} = 36[/tex]

Helppppppp anyoneeee?

Answers

Answer:

( -6, 6)

Step-by-step explanation:

help please

the triangle above has the following measures. q = 7cm m

Answers

Answer:

Put your calculator in degree mode.

sin(35°) = 7/r

r sin(35°) = 7

r = 7/sin(35°) = 12.2 cm

pls pls help me with answer(correct it)

Answers

I think the answer should be 450 cm squared. Because to find the area of the triangle, you need to do this: 0.5*one leg of triangle*other leg of triangle

Explain by using an example why you need to multiply when converting from a larger unit to a smaller unit

Answers

Multiplying is necessary because each larger unit contains a certain number of the smaller units and we need to account for that conversion factor.

Why is multiplication necessary with examples?

For example, when converting from meters to centimeters where we know that 1 meter is equal to 100 centimeters.

To convert 3 meters to centimeters, we need to multiply by the conversion factor of 100 which gives us:

= 3 meters * 100 centimeters/meter

= 300 centimeters

So, without the multiplication, we would not account for the fact that each meter contains 100 centimeters. This principle applies to all conversions from larger units to smaller units and it is important to remember to ensure accurate conversions.

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can someone help me please

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The matrix formed by performing the row operation 4R₁ + R₂ — R₂ on M will have R₁ = [1 0 3] and R₂ = [ -1 2 10]

What is a matrix row operation

Row operations are a set of operations that can be performed on the rows of a matrix in order to transform it into a row equivalent matrix, which has the same solution set as the original matrix.

performing the row operation 4R₁ + R₂ — R₂ on M, we have;

4(1) + (-5) = -1 {row 2 column 1}

4(0) + 2 = 2 {row 2 column 2}

4(3) + (-2) = 10 {row 2 column 3}

Therefore, the matrix formed by performing the row operation 4R₁ + R₂ — R₂ on M will have R₁ = [1 0 3] and R₂ = [ -1 2 10]

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3. Write the following using exponential notation:
(5.
5.5.5.5.5.5.5.5.5) (7.7.7.7.7.7.7.7.7.7.7.7.7.7)

Answers

Answer:

5.5.5.5.5.5.5.5.5 in exponential notation is 5^9

7.7.7.7.7.7.7.7.7.7.7.7.7.7 in exponential notation is 7^13

Therefore, (5.5.5.5.5.5.5.5.5) (7.7.7.7.7.7.7.7.7.7.7.7.7.7) in exponential notation is:

5^9 * 7^13

is 1 over 16 the same as 16 A: Yes B.NO ​

Answers

Answer:

1 over 16 is not the same as 16.

1 over 16 is a fraction, which means that it represents a part of a whole. It can be written as 1/16, which means one part out of a total of sixteen equal parts.

On the other hand, 16 is a whole number, which represents a quantity of sixteen units.

So, 1/16 is not equal to 16.

Step-by-step explanation:

1/16 is definitely not the same as 16. 1/16 is seen as 1 part of 16 when 16 is seen as a whole

Jeannie designs jewelry. The figure shows a
pendant on a necklace Jeannie is making. It is
composed of three individual beads.
The larger triangular bead cost $1.52. The
rectangular bead costs $0.42, and the smaller
triangular bead costs $0.92.
What is the cost per square millimeter of the
pendant? Round your answer to the nearest
hundredth.

Answers

Answer: ≅ $0.02

Step-by-step explanation:

First, lets find the area of the large triangle

0.5 x 14 x 11 = 77 mm^2

Next lets find the area of the middle rectangle:

4 x 14 = 56 mm^2

Last lets find the area of the small triange:

0.5 x 14 x 3 = 21 mm^2

So the total area is: 77 + 56 + 21 = 154 mm^2

the total cost is given to be: 1.52 + 0.42 + 0.92 = $2.86

dividing the cost by the area to find the cost/mm:

2.86/154 ≅ $0.02

A cube has shaded shapes on three of its faces. Here is a net of the cube. Draw in the two missing shaded shapes.​

Answers

The appropriate diagram to illustrate the cube is given.

What is a cube?

A cube is a three-dimensional shape featureing six equal square faces, with its edges and vertices being of congruent length. It counts as one of the five platonic solids distinguished by all of its features having matching sizes.

The cube can be perceived as a special instance of a rectangular parallelepiped where all six are perfectly square in shape. In order to calculate the volume of such an ordinary cube, the number of its edge needs to be multiplied by itself thrice (V = a³).

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Find the average rate of change of g(x)= 4x^2+3 on the interval [-4,1]

Answers

The rate of change is 4/1

The mass of 12 packets of flour and 8 packets of salt is 19 kg. When 7 packets of flour are removed, the mass became 9.9 kg. Find the mass of each packet of salt in kilogram.​

Answers

Answer:

1.235 kg

Step-by-step explanation:

Let's assume that each packet of flour and salt weighs the same.

When we remove 7 packets of flour, we are left with 5 packets of flour and 8 packets of salt. The total number of packets is 13 and their total mass is 9.9 kg.

Therefore, the average mass of each packet is 9.9 kg ÷ 13 = 0.76 kg.

Since we know that 12 packets of flour and 8 packets of salt have a total mass of 19 kg, we can subtract the mass of the 12 packets of flour (12 × 0.76 kg = 9.12 kg) from the total mass to find the mass of the 8 packets of salt:

19 kg - 9.12 kg = 9.88 kg

Finally, we can divide the mass of the 8 packets of salt by the number of packets to find the mass of each packet:

9.88 kg ÷ 8 = 1.235 kg

Therefore, each packet of salt weighs approximately 1.235 kg.

Please help Thanks
there is a FRACTION symbol two variable symbols y and x


I need to write an equation

Answers

The required equation of line is y = 2x

What is equation of line?

A straight line's equation is y=mx+c, where m denotes the gradient and c denotes the height at which the line crosses the y-intercept, also known as the y-axis.

According to question:

Given points:

(1,2) and (2,4)

then

[tex]Slope of line = \frac{4-2}{2-1}= 2[/tex]

then

y-2 = 2(x-1)

y = 2x - 2 + 2

y = 2x

Thus, required equation of line is y = 2x

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Consider the line 6x-7y = 4. What is the slope of a line parallel to this line? What is the slope of a line perpendicular to this line?

Answers

The slope of a line parallel to this line is 6/7.

The slope of a line perpendicular to this line is -7/6.

What are parallel lines?

In Mathematics and Geometry, parallel lines can be defined as two (2) lines that are always the same (equal) distance apart and never meet.

In Mathematics and Geometry, two (2) lines are parallel under the following conditions:

Slope, m₁ = Slope, m₂

Based on the information provided about this line, we have the following equation in standard form;

6x - 7y = 4

By making y the subject of formula, we have:

7y = 6x - 4

y = 6x/7 - 4

In Mathematics and Geometry, a condition that is true for two lines to be perpendicular is given by:

m₁ × m₂ = -1

6/7 × m₂ = -1

6m₂ = -7

Slope, m₂ = -7/6

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Who is referred to as the grey in the passage of the wealth mare In chapter 6 of the Great Gatsby, when did James slowly piece together the first conceptions of his new identity?No cheating According to Bratich and Banet-Weiser, what are commonalities between PUA and incels (consider gender, sexuality, race, beliefs, goals, etc)? A bullet list here is fine. In addition, according to the authors, how did a community of failed PUAs become incels? Write 3-4 sentences. Which hormone is absolutely necessary for ovulation to occur?. social skill, one of the components of emotional intelligence (ei), refers to personal proficiency in managing relationships and building networks.group startstrue or false What is the status if the Baoule or Buginese language? A 30g piece of metal absorbs 1,200 joules of heat energy, and itstemperature changes from 25C to 175C. Calculate the specific capacity ofthe metal. What is the likely metal? Analyze the three important events in Jamestown in 1619. Why were they important collectively? How would one of these events spell trouble later? What do u think happened aboard the mary celeste? Using your ability to infer, plan and write a first person description in the role of surviving crew member, of the moment when you realised you would have to abandon ship. Include what you can see and hear, as well as your thoughts and feeling at the time. Read your piece,of about 200 words, to the class, who will vote on the most convincing account. which of the following is true concerning the materials price variance? group of answer choices none of these. it is the difference between the actual and standard unit price of an input multiplied by the number of inputs used. it is the difference between the actual and standard unit price of an output multiplied by the number of inputs purchased. it is the difference between the actual and standard unit price of an output multiplied by the number of inputs used. it is the difference between the actual and standard unit price of an input multiplied by the number of inputs purchased. This echidna, this capuchin and this bottlenose dolphin similarities and differences in their body structures. what does the information about these structures tell you about the ancestors of these species? What is phloem?What does it phloem carry?Explain the function of sieve tubes and their association with companion cells.What is translocation? Explain how pressure and water potential aid translocation.How can aphid stylets be used to measure phloem transport rates? How can radioactively-labeled CO2 be used to measure phloem transport rates? Haw many valance electrons in the following atoms.O Na Sr After a recent collapse of a structure in which many people died, an Engineer came to know about a bridge which is marginally safe. He informed his superior who asked him to stay calm and not to discuss with anyone, while waiting for the next year budget sessions to get some financial help for the repair required. What should the engineer do? can someone explain this to me. traveling 60 mhp in the car after 3 hours how long would that be The graph below shows the relationship between temperature in degrees Fahrenheit and degrees Celsius. What is the meaning of the y-intercept? A the change in degrees Fahrenheit for each change of one degree Celsius Bel change in degrees Celsius for each change of one degree Fahrenheit the temperature in degrees Fahrenheit when the temperature is zero degrees Celsius D the temperature in degrees Celsius when the temperature is zero degrees Fahrenheit What do historians mean by "universalizing empires"? discuss two ways either imperial rome or han dynasty china fit this definition. add a comma or semicolon if needed. Otherwise, submit the text without any additional punctuation.Mother yaks climb to higher altitudes than do males or females without calves researchers believe the steepmountain slopes may help protect calves from predators and provide a richer source of food. fill in the blanks with suitable idiom, I need help please