One day, a king left his castle with a bag of silver coins to wander his kingdom. to the first peasant he met, he gave one-half of the coins plus two more. A little later, he met another peasant to whom he also gave one-half of his coins plus two more. Walking on, he met a third peasant and again gave half his coins plus two more. FInally, the king went home with two coins left in his bag. How many coins did he have to begin with? Create an algebraic equation. Please explain why.

Volume is a measurement that quantifies the amount of three-dimensional space occupied by an object, substance, or region.

Volume is a fundamental physical quantity that describes the extent or size of an object in the context of its three dimensions: length, width, and height.

Volume is commonly expressed in cubic units, such as cubic meters (m³), cubic centimeters (cm³), cubic inches (in³), or cubic feet (ft³), depending on the system of measurement used. It can apply to various entities, including solid objects, liquids, gases, or even abstract concepts like the volume of data.

For solid objects, the volume represents the total amount of space enclosed within its boundaries. It is typically determined by measuring or calculating the product of the object's length, width, and height or through more complex mathematical formulas for irregular shapes.

In the case of liquids or gases, the volume refers to the amount of space occupied by the substance within a container. It can be measured directly using tools such as graduated cylinders or indirectly through displacement methods.

Understanding volume is essential in fields like physics, engineering, architecture, chemistry, and many others where knowledge of spatial quantities is required for designing, analyzing, or describing various objects or systems.

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The scientific notation of 320 million is 3.2 x [tex]10^8[/tex].

It is the way of writing a large number between 1 to 10 with a power of 10.

Example:

2.9 x 10² is in scientific notation

3.4 x 10³ is in scientific notation

4.4 x 10³ is in scientific notation

We have,

320 million

This can be written as,

1 million = 1000000

So,

= 320 x [tex]10^6[/tex]

= 32 x [tex]10^7[/tex]

Multiply 10/10 to make a number between 1 and 10.

= 32/10 x 10 x [tex]10^7[/tex]

Make an exponent of base 10.

= 3.2 x [tex]10^8[/tex]

Thus,

The scientific notation of 320 million is 3.2 x [tex]10^8[/tex].

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An equation in slope-intercept form models given situation is y=0.6x+20 and the length of the sunflower after two weeks is 21.2 meters.

Given that, a sunflower in Julia Rosario's garden was 12 centimetres tall when it was first planted. Since then, it has grown approximately 0.6 meters per day.

We need to write an equation in slope-intercept form that models this situation.

The slope-intercept form of a straight line is one of the most common forms used to represent the equation of a line. The slope intercept formula can be used to find the equation of a line when given the slope of the straight line and the y-intercept( the y-coordinate of the point where the line intersects the y-axis). The equation of the line is the equation that is satisfied by each point that lies on that line.

The slope intercept form equation for a straight line with a slope, 'm', and 'b' as the y-intercept can be given as y = mx + b.

Let y be the length of a sunflower plant and the number of days be x.

Now, y=0.6x+20

The length of the sunflower after two weeks is y=0.6×2+20

⇒y=1.2+20=21.2 meters

Therefore, an equation in slope-intercept form models given situation is y=0.6x+20 and the length of the sunflower after two weeks is 21.2 meters.

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Answer:

0.1111.... = [tex]\frac{1}{9}[/tex].

Step-by-step explanation:

Given  : 0.1 repeating

To find : Which rational number equals 0.1111......

Solution : We have given 0. 11111.........

We can see there is only one number is repeating , we will multiply it by 10

Let  x = 0.1111.......

Multiply both sides by 10

10x = 10 * 0.1111.......

10x = 1.1111.......

We can write it in terms of x

10x = 1. 000 + 0.11111....

10 x = 1 + x

On subtracting both sides by x

10 x -x = 1

9 x = 1

On dividing both sides by 9

x = [tex]\frac{1}{9}[/tex].

0.1111.... = [tex]\frac{1}{9}[/tex].

Therefore, 0.1111.... = [tex]\frac{1}{9}[/tex].

The rational number that equals 0.1 repeating is 1/9.

The rational number that equals 0.1 repeating can be represented as follows:

Let x = 0.1 repeating

To convert the repeating decimal to a fraction, we can use the following method:

Let's multiply both sides of the equation by 10 to shift the decimal point one place to the right:

10x = 1.1 repeating

Now, subtract the original equation from this new equation to eliminate the repeating part:

10x - x = 1.1 repeating - 0.1 repeating

9x = 1

Divide both sides of the equation by 9 to solve for x:

x = 1/9

Therefore, the rational number that equals 0.1 repeating is 1/9.

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The number 5.38516 is greater than the square root of 29 or the square root of the number 29 is less than the number 5.7145.

A number is a mathematical entity that can be used to count, measure, or name things. For example, 1, 2, 56, etc. are the numbers.

It is given that:

The two numbers are:

5.7145 and √29

It is required to compare the numbers

The square root of the number 29 can be found using the division method.

The square root of the number 29 is 5.38516

As we know, the number line can be defined as the representation of the numbers on a straight line that goes infinitely on both sides.

Plotting the numbers 5.7145  and 5.38516 on a number line.

We will see the:

5.38516 < 5.7145  OR

5.7145  > 5.38516

Thus, the number 5.38516 is greater than the square root of 29 or the square root of the number 29 is less than the number 5.7145.

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The approximation of 89972 to three significant digits is 900

The number is given as:

89972

To round to three signficant digits, we simply count three digits from the left.

i.e. 899

Next, we start the approximation from the next digit

i.e. 7

Hence, the approximation of 89972 to three significant digits is 900

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Final answer:

To calculate the total distance Gabriella flew in four shows at the circus, add up the distances she covered in each performance.

Explanation:

The total distance Gabriella flew on Saturday can be calculated by adding up the distances she covered in each show:

Distance 1: 8.05 meters

Distance 2: 17.71 meters

Distance 3: 14.38 meters

Distance 4: 19.35 meters

By adding these distances together, you can find the total distance Gabriella flew that day i.e. 8.05+17.71+14.38+19.35 =  59.49 m

The side of the four-square court is 8.1 ft.

The area of a square is given as the square of its side. the are of a square whose side is a is given as,

Area of Square = a²

the area of a four-square court = 66 ft²

Let's assume that the side of the four-square court is x.

the area of a four-square court = area of square

[tex]66 = x^2\\x^2 = 66\\x = \sqrt{66}\\x=8.124[/tex]

Thus, the side of the four-square court is 8.124 ft. but we want the value to be the nearest tenth of a foot,

[tex]\rm{8.124\ ft \approx 8.1\ ft[/tex]

Hence, the side of the four-square court is 8.1 ft.

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multiply the 7 (the denominator), and the 6 (the whole number). To this product, add the 4 (the numerator)
giving 46, to form the new numerator, and use the 7 as the new denominator.

So, 6 4/7 as an improper fraction is 46/7

The improper fraction equivalent to [tex]6\frac{4}{7}[/tex] is [tex]\frac{46}{7}[/tex].

An improper fraction is a fraction where the numerator is greater than the denominator.

A mixed fraction is defined as a fraction formed by combining a whole number and a fraction.

Since the given fraction is in mixed fraction so on solving it we get,

[tex]\dfrac{46}{7}[/tex].

Hence the required improper fraction will be [tex]\dfrac{46}{7}[/tex].

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Answer:

3 9/20

Step-by-step explanation:

3.45 is 69/20

60 goes into 20 3 times

which leaves you with 3 9/20

Answer:  the correct option is

(D) [tex]c=\dfrac{d+ab}{a}.[/tex]

Step-by-step explanation:  We are given to solve the following equation for the value of c :

[tex]a(c-b)=d~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]

To solve the given equation for c, we need to take c on one side and all other values on the other side of the equation.

The solution for c of equation (i) is as follows :

[tex]a(c-b)=d\\\\\Rightarrow ac-ab=d\\\\\Rightarrow ac=d+ab\\\\\Rightarrow c=\dfrac{d+ab}{a}.[/tex]

Thus, the required value of c is [tex]c=\dfrac{d+ab}{a}.[/tex]

Option (D) is CORRECT.

Using Pythagoras theorem,  the ladder must be 19.7ft long.

Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides“.

Perpendicular = height of the window from the ground = 17ft

Base = distance of the foot of the ladder from the base of the building = 10ft

Hypotenuse = [tex]\sqrt{perpendicular^{2} +base^{2} }[/tex]

H = [tex]\sqrt{17^{2}+10^{2} }[/tex] = 19.4ft

Length of ladder  = 19.4ft

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To obtain a 50% mixture, 2.4 quarts of water must be added to 4 quarts of an 80% mixture.

To solve this problem, we'll use the concept of mixtures.

Let's denote:

x = the number of quarts of water to be added.

We have 4 quarts of an 80% mixture. This means that in these 4 quarts, 80% is the actual substance (let's call it A) and 20% is water.

So, the amount of substance A in the 4 quarts is [tex]\(0.80 \times 4 = 3.2\)[/tex]quarts, and the amount of water is [tex]\(0.20 \times 4 = 0.8\)[/tex] quarts.

When we add \(x\) quarts of water, the total amount of water becomes 0.8 + x, and the total volume of the mixture becomes 4 + x quarts.

For the mixture to be 50%, the amount of substance A in the mixture should be equal to the amount of water. So, we set up the equation:

[tex]\[3.2 = 0.5 \times (4 + x)\][/tex]

Solving for x:

[tex]\[3.2 = 2 + 0.5x\]\[0.5x = 3.2 - 2\]\[0.5x = 1.2\]\[x = \frac{1.2}{0.5}\]\[x = 2.4\][/tex]

So, 2.4 quarts of water must be added to 4 quarts of an 80% mixture to obtain a 50% mixture.