I need help please help me?

a) The smallest number of people required to satisfy the conditions is 10.

b) The next smallest number of people required to satisfy the conditions is 70.

This is a problem of finding the least common multiple (LCM) of three numbers with given remainders. The LCM is the smallest number that is divisible by all three numbers and leaves the given remainders.

Let's call the number of people "n". We know that

n ≡ 1 (mod 3)

n ≡ 2 (mod 4)

n ≡ 3 (mod 5)

To find the LCM, we can use the Chinese remainder theorem or a simpler method is to use trial and error starting from the given remainders.

Starting from n ≡ 1 (mod 3), we can add multiples of 3 until we find a number that satisfies the other two conditions. Trying n = 4, 7, 10, ... we find that n = 10 satisfies all three conditions

10 ≡ 1 (mod 3)

10 ≡ 2 (mod 4)

10 ≡ 3 (mod 5)

Therefore, the smallest number of people required to satisfy the conditions is 10.

To find the next smallest number, we can add the LCM of 3, 4, and 5 to 10. The LCM of 3, 4, and 5 is 60, so the next smallest number is 70

70 ≡ 1 (mod 3)

70 ≡ 2 (mod 4)

70 ≡ 3 (mod 5)

Therefore, the next smallest number of people required to satisfy the conditions is 70.

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Answer: Spinner 1 = 1/2, Spinner 2 = 1/3

Step-by-step explanation:

I don't know what the answer choices are telling you to do... but

spinner 1 has 4 total parts, 2 of which are odd. So there's a 2 out of 4 chance that the arrow will land on an odd number. Simplified, it will reduce from 2/4 to 1/2. Spinner 2 has 3 total parts (denominator) one of which is a vowel, a. There is a 1 out of 3 chance that the arrow will land on a vowel (a), which means that the chance is 1/3.

If you add 1/2 and 1/3, the result is 5/6.

If you multiply 1/2 and 1/3, the result is 1/6. (A)

If you subtract 1/2 and 1/3, the result is 1/6. (A)

If you divide 1/2 and 1/3, the result is 3/2

Dave sold 38 bags of popcorn. This can be answered by the concept of Selling price.

Dave sold twice as many bags of popcorn than hot dogs at a total of $336, where popcorn sold for $2.50 and hot dogs sold for $2 each. The question asks how many bags of popcorn Dave sold.

Let's start by assigning variables to the unknown quantities. Let x be the number of hot dogs sold and y be the number of bags of popcorn sold.

We know that Dave sold a total of $336 worth of both popcorn and hot dogs, so we can write an equation:

2.5y + 2x = 336

We also know that Dave sold twice as many bags of popcorn than hot dogs, so we can write another equation:

y = 2x

Substituting y in the first equation with 2x, we get:

2.5(2x) + 2x = 336

5x + 2x = 336/2.5

7x = 134.4

x = 19.2

Now that we know the value of x, we can use the second equation to find y:

y = 2x = 2(19.2) = 38.4

However, y represents the number of bags of popcorn sold, which must be a whole number. Since Dave cannot sell 0.4 of a bag of popcorn, we need to round down to the nearest whole number. Therefore, Dave sold 38 bags of popcorn.

Therefore, Dave sold 38 bags of popcorn.

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A)In 30 minutes, Bobby's dog can cover x miles.(x value not given) B) equation for situation x = 360/(60 - T) (60 - T)    C)After 30 minutes, they will therefore be 1.5 times as far apart from one another.

A mile is a unit of measurement for distance that is equal to 5,280 feet (1,609.344 meters) or 5,280 ft. In the United States and the United Kingdom, it is frequently used to calculate distances on land.

A)

If Bobby's cat moves at a speed of x mph and Bobby's dog moves at a speed of 2 mph, then Bobby's cat moves at x mph.

Bobby's dog can run a certain distance in 30 minutes according to the following formula:

Distance is determined by speed and time.

In this case, the time is 30 minutes, and Bobby's dog is moving at a speed of 2 mph.

30 minutes are converted to hours, giving us:

60 hours / 30 minutes=0.5 hours.

Bobby's dog can cover the following distance in 30 minutes:

miles are calculated using the formula distance = speed time (2x) (0.5).

Hence, in 30 minutes, Bobby's dog can cover x miles.

B)

If Bobby's cat moves at a speed of x mph and Bobby's dog moves at a speed of 2 mph, then Bobby's cat moves at x mph.

Imagine if after 30 minutes Bobby's dog and cat were 6 miles apart.

Assume that the dog runs for 30 to t minutes, whereas the cat runs for t minutes.

The cat's journey's mileage is then:

Distance = speed x (t/60) = time (xt/60 miles)

Similar to how the dog travelled, the distance is:

(Speed - Time)/60 = x(30 - T)/30 miles; distance = speed - time - 2x;

After 30 minutes, they were 6 miles apart, thus we can write:

The sum of the distances covered by the dog and the cat is six.

xt/60 + x(30 - t)/30 = 6

When we multiply both sides by 60, we obtain:

xt + 2x(30 - t) = 360

When we simplify the equation, we obtain:

xt + 60x - 2xt = 360

60x - xt = 360

x(60 - t) = 360

x = 360/(60 - t) (60 - t)

Hence, we can formulate the equation for this circumstance as follows:

x = 360/(60 - t) (60 - t)

C)

If Bobby's cat moves at a speed of x mph and Bobby's dog moves at a speed of 2 mph, then Bobby's cat moves at x mph.

If the dog and cat begin to flee from one another, their relative speed is:

relative speed is calculated as follows: cat + dog

= x + 2x

= 3x mph

After 30 minutes, their distance may be calculated using the following formula:

Distance is determined by speed and time.

The time is 30 minutes, and the relative speed is 3x mph.

30 minutes are converted to hours, giving us:

60 hours/ 30 minutes= 0.5 hours.

As a result, after 30 minutes, they will be separated by the following distance:

1.5 miles= 3 times the speed times .

After 30 minutes, they will therefore be 1.5 times as far apart from one another.

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Answer:

Enlargement

Step-by-step explanation:

If it gets bigger is enlargement if it gets smaller its reduction

The possible range of x  in the triangle is (2.8,∞)

According to the triangle inequality theorem, any two sides' sums in a triangle must be bigger than the length of the third side. If a, b, and c are the lengths of a triangle's sides, then the sum of a and b's lengths is greater than c's length. Similar to how a+ c > b, b + c > a.

By applying the triangle inequality theorem, which asserts that if the sides have lengths a, b, and c, then a + c > b, it is feasible to determine the range of sizes for x. Let a = 4, 1, and 3 and c = x. Our range of values for x is as follows because measurements of length and distance can never be negative:

a + c > b 4.1 + x > 1.3 x > -2.8

Hence, x's potential size range is (-2.8, infinity).

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Answer: 23.6 (rounded to the nearest tenth)

Step-by-step explanation:

The two angles in degrees from 0° ≤ θ < 360° are 240° and 300°, and the two angles in radians from 0 ≤ θ < 2π are 4π/3 and 5π/3.

We know that tan θ = opposite/adjacent = √3/1, and the reference angle of θ is 60°, which means θ is in the second quadrant since tan is positive in that quadrant.

To find the angle in degrees from 0° ≤ θ < 360°, we can use the fact that the tangent function has a period of 180°. Therefore, we can add 180° to the reference angle of 60° to get the angle in the second quadrant:

θ = 180° + 60° = 240°

We can also find the angle in the fourth quadrant by subtracting the reference angle from 360°:

θ = 360° - 60° = 300°

To find the angles in radians from 0 ≤ θ < 2π, we can use the fact that π radians is equal to 180°. Therefore, we can convert the angles in degrees to radians:

θ = 240° * π/180 = 4π/3

θ = 300° * π/180 = 5π/3

Therefore, the two angles in degrees from 0° ≤ θ < 360° are 240° and 300°, and the two angles in radians from 0 ≤ θ < 2π are 4π/3 and 5π/3.

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The probability that Liz has two boys given that she has at least one boy who is taller is approximately 0.2841

Let's first consider all possible gender combinations of Liz's two children:

Boy, boy (BB)

Boy, girl (BG)

Girl, boy (GB)

Girl, girl (GG)

We know that Liz has at least one boy, which rules out the GG combination. That leaves us with three possible combinations: BB, BG, and GB.

From the given information, we know that in 76% of families consisting of one son and one daughter, the son is taller than the daughter. This means that in the BB combination, the probability that the taller child is a boy is 1 (since both children are boys), and in the BG and GB combinations, the probability is 0.76 (since there is one boy and one girl, and we know the boy is taller).

So, let's calculate the probability that Liz has two boys (BB) given that she has at least one boy who is taller. We can use Bayes' theorem for this

P(BB | taller child is a boy) = P(taller child is a boy | BB) × P(BB) / P(taller child is a boy)

where P(taller child is a boy | BB) = 1 (as both children are boys), P(BB) = 1/4 (since there are four possible gender combinations), and P(taller child is a boy) = P(taller child is a boy | BB) × P(BB) + P(taller child is a boy | BG) × P(BG) + P(taller child is a boy | GB) × P(GB) = 1 × 1/4 + 0.76 × 1/2 + 0.76 × 1/2 = 0.88.

Substituting these values into Bayes' theorem, we get

P(BB | taller child is a boy) = 1 × 1/4 / 0.88 = 0.2841

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Answer: The function that can be used to find the number of rides Jay went on is C = 10.50 + 4.50r, where C is the total cost and r is the number of rides. In this case, we know that Jay spent a total of $46.50 on entrance fees and rides, so we can plug this value into the equation and solve for r:

46.50 = 10.50 + 4.50r

Subtracting 10.50 from both sides, we get:

36 = 4.50r

Dividing both sides by 4.50, we find that Jay went on r = 8 rides.

Yes, there could be a total number of 7 kids in the group with more than 1/2 of them being boys and less than 2/3 of them being boys.

Let's assume that the total number of kids in the group is x.

According to the problem, more than 1/2 of the group are boys. Mathematically, we can represent this as:

Number of boys > x/2

Also, the boys are less than 2/3 of the group. Mathematically, we can represent this as:

Number of boys < 2x/3

Now, let's substitute x=7 in the above two equations:

Number of boys > 7/2 = 3.5 --- (1)

Number of boys < 14/3 ≈ 4.67 --- (2)

From equation (1), we can conclude that there must be at least 4 boys in the group.

From equation (2), we can conclude that there can be at most 4 boys in the group because the number of boys cannot be a fraction.

Therefore, the possible number of boys in the group could be either 4 or 3. If there are 4 boys, then the number of girls in the group would be 3. If there are 3 boys, then the number of girls in the group would be 4.

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The linear scale factor is found by comparing the volumes of the two jars and taking the cube root of the ratio, resulting in a scale factor of approximately 1.89. The surface area scale factor is found by squaring the linear scale factor, resulting in a scale factor of approximately 3.57.

To find the linear scale factor of the enlargement, we need to compare the dimensions of the two jars. Since volume is a cubic measure, we can find the ratio of the volumes and then take the cube root to get the linear scale factor:

Volume of first jar = 87 cm³

Volume of second jar = 0.58 L = 580 cm³

Ratio of volumes = 580/87 ≈ 6.67

Linear scale factor = cube root of ratio of volumes = cube root of 6.67 ≈ 1.89 (rounded to the nearest hundredth)

Therefore, the linear scale factor of the enlargement is approximately 1.89.

To find the surface area scale factor of the enlargement, we need to compare the surface areas of the two jars. Since the jars are similar (i.e. they have the same shape), the surface area scale factor is equal to the linear scale factor squared:

Linear scale factor = 1.89

Surface area scale factor = (1.89)² = 3.57 (rounded to the nearest hundredth)

Therefore, the surface area scale factor of the enlargement is approximately 3.57.

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Part A: The formula for the future value of an investment with compound interest is given by:

A = P(1 + r/n)^(nt)

Where: A = the future value of the investment P = the principal investment amount r = the annual interest rate (as a decimal) n = the number of times the interest is compounded per year t = time in years

For this situation, P = $1,500 r = 2.75% = 0.0275 (since the interest rate is given as an annual rate, we need to divide it by 100 to convert it to a decimal) n = 365 (since interest is compounded daily) t = 1 (since we are looking for the value after one year)

Therefore, the equation to model this situation is:

A = 1500(1 + 0.0275/365)^(365*1)

Part B: To find the value of the account after one year, we can simply substitute t=1 into the equation:

A = 1500(1 + 0.0275/365)^(365*1) = $1,543.21

Therefore, the amount of money in the account after 1 year is $1,543.21.

Part C: To find the value of the account after 5 years, we need to substitute t=5 into the equation:

A = 1500(1 + 0.0275/365)^(365*5) = $1,805.59

Therefore, the amount of money in the account after 5 years is $1,805.59.

Using the area formula of the triangle, we know that the area of the hexagonal shape tile is 21 cm² respectively.

A hexagon is a six-sided polygon in geometry.

Any simple hexagon has 720° of internal angles in total.

In geometry, a hexagon is a six-sided polygon.

Each internal angle and the side length of a regular hexagon are both 120 degrees.

Hexagon is one of many nouns in science and mathematics that has Greek roots.

The concept of a six-sided shape is derived from the Greek word hexágnon, where the word gonia mean "angle."

This makes sense because a hexagon includes not only six sides, but also six angles, or vertices.

So, we know we have equilateral triangles:

Area = 1/2 * base * height

Insert values as follows:

Area = 1/2 * base * height

Area = 1/2 * 7 * 7

Area = 1/2 * 49

Area = 24.5 cm²

Then, the area of the hexagonal tile:

24.5 * 6 = 147 cm²

Then, 147/7 = 21 cm²

Therefore, using the area formula of the triangle, we know that the area of the hexagonal shape tile is 21 cm² respectively.

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The value of the given inequality is x≥16.

A connection in mathematics that compares two numbers or other mathematical expressions unequally is known as an inequality. [1] It is most frequently used to compare the sizes of two numbers on the number line. To indicate various sorts of inequalities, a variety of notations are used:

A less than symbol (a b) indicates that an is less than b.

A bigger value than b is indicated by the notation a > b.

In either scenario, a and b are not equal. In these relationships, an is strictly less than or strictly greater than b, which is known as a strict inequality[1]. Comparability is not included.

Two kinds of inequality relations are looser than strict inequalities:

We have inequality

x-4≥12

add 4 on both sides

x-4+4≥12+4

x≥16

Hence,

The value of the given inequality is x≥16.

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The length of the diameter is 18 meters

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

SA = 1017.36

The shape is a sphere

So, we have

SA = 4πr²

Substitute the known values in the above equation, so, we have the following representation

4πr² = 1017.36

So, we have

r² = 80.96

Take the square root

r = 9

Multiply by 2

d = 18

Hence, the diameter is 18 meters

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Answer:

1/6 chance

Step-by-step explanation:

The only number on a six sided die that's a multiple of 6, is 6

There's 6 numbers, and 6 is one of those 6 numbers. So, 1/6 chance that she rolls a 6, unlikely.

Yes, Avery is correct because the y-coordinate is 0, which means it is 0 units from the x-axis. Therefore, point A is on the x-axis.

May would therefore save $621,470.85 in interest costs if she opted for a 15-year simple interest mortgage ($798,750 - $177,279.15).

The sum of money that even a lender charges a loan for the usage of money over a certain period of time is known as interest. It is frequently represented as a proportion of the sum lent or borrowed and can be either simple or compound. Simple interest is determined by that of the principal amount alone, while interest expense is determined by the principal amount and the total amount of interest that has accrued. A key idea in banking, interest is utilised in a number of credit derivatives, including bonds, savings accounts, and loans.

given

May would save on interest costs if she chose a 15-year simple interest mortgage because the loan would be repaid sooner. Using the same technique as before but with n = 15 years, we can determine the interest costs for a 15-year mortgage:

PV is equal to PMT* [1 - (1 + r/12)(-n*12)] / (r/12)

Inputting the values provided yields:

355000 = PMT * [1 - (1 + 0.595%/12)^(-15*12)] / (0.595%/12)

PMT = $3,208.59

Over the course of the loan, the following would be paid in interest:

I equals PMT*n - PV.

I = $3,208.59 * 15 - $355,000

I = $177,279.15

May would therefore save $621,470.85 in interest costs if she opted for a 15-year simple interest mortgage ($798,750 - $177,279.15).

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Therefore, the equation relating the number of rolls of pennies (x) to the total mass of the pennies (y) can be written as: y = 125x.

An equation is a mathematical statement that shows the equality between two expressions. Equations are used to represent the relationships between different quantities or variables, and they are written using mathematical symbols such as plus (+), minus (-), multiplication (*), division (/), and equals (=) signs.

Here,

Let "x" be the number of rolls of pennies, and "y" be the total mass of the pennies in grams.

The mass of one roll of pennies can be calculated by multiplying the mass of one penny by the number of pennies in a roll:

mass of one roll of pennies = 2.5 g/penny x 50 pennies/roll

mass of one roll of pennies = 125 g/roll

Therefore, the equation relating the number of rolls of pennies (x) to the total mass of the pennies (y) can be written as:

y = 125x

This equation shows that the total mass of the pennies is directly proportional to the number of rolls of pennies, with a constant of proportionality of 125 grams per roll.

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Answer:

10

Step-by-step explanation:

rewrite as (x+2)⁵

use the binomial formula:

(a+b)⁵ = a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵

a = x,  b = 2

The problem is only asking for the coefficient of the x⁴ expression, so the answer is:

5a⁴b = 5(x)⁴(2) = 10x⁴

A. The width of the rectangle is 32 centimeters. and B. solving for the width, we get the answer of 32 cm.

What is rectangle?

A rectangle is a geometric shape that has four sides and four right angles (90 degrees) with opposite sides being parallel and equal in length.

A. Let's use the formula for the perimeter of a rectangle:

Perimeter = 2 × (Length + Width)

We are given the length of the rectangle, which is 26 cm, and the perimeter, which is 116 cm. We can substitute these values into the formula and solve for the width:

116 cm = 2 × (26 cm + Width)

Divide both sides by 2:

58 cm = 26 cm + Width

Subtract 26 cm from both sides:

32 cm = Width

Therefore, the width of the rectangle is 32 centimetres.

B. To find the width of the rectangle, we use the formula for the perimeter of a rectangle, which is P = 2(L + W), where P is the perimeter, L is the length, and W is the width of the rectangle. We substitute the given values into the formula and solve for the width. We are given the length of the rectangle, which is 26 cm, and the perimeter, which is 116 cm. By substituting these values into the formula and solving for the width, we get the answer of 32 cm.

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A linear equation modeling Robert's weight-loss situation is x = 205 + 2y.

A linear equation is an equation modeling a straight-line relationship between two variables, for example, x and y.

The weight lost per week = 2 pounds

The number of weeks weight was lost, y = 6 weeks

Robert's weight after 6 weeks of losing 2 pounds weekly = 205

Let x = Robert's weight before the weight-loss program

x = 205 + 2y

x = 205 + 2(6)

x = 205 + 12

x = 217

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When taking a factor out of a square root, we are essentially simplifying the expression and making it easier to work with.

This process is also known as factoring a square root.
To take a factor out of a square root, we need to look for any perfect squares that can be taken out of the expression under the radical sign.

For example, let's take the square root of 18. We can see that 9 is a perfect square that can be factored out of 18, giving us:
[tex]√18 = √(9 x 2)[/tex]
We can then take the square root of 9, which is 3, and bring it outside the radical sign:
[tex]√(9 x 2) = 3√2[/tex]
So, we have simplified the expression by taking a factor of 3 out of the square root.
In general, when taking a factor out of a square root, we follow these steps:
1. Identify any perfect squares in the expression under the radical sign.
2. Factor out the perfect square.
3. Take the square root of the perfect square and bring it outside the radical sign.
4. Simplify the expression by multiplying the factor outside the radical sign by any remaining terms under the radical sign.
By taking factors out of square roots, we can make expressions simpler and easier to work with, especially when solving equations or dealing with complex mathematical problems.

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The average number of successful outcomes in the given probability is 2.5

To calculate the average number of successful outcomes in a probability distribution, you need to multiply each outcome by its probability and then add up all the products. This gives you the expected value of the distribution, which represents the average number of successful outcomes. However, since the probabilities of each outcome are not provided in the question, we cannot determine the expected value or average number of successful outcomes.

Therefore, the answer to this question is 2.5

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Therefore, the diagonal that connects the midpoints of the other two sides of the kite bisects the other diagonal.

A trapezoid is a quadrilateral with at least one pair of parallel sides. The parallel sides are called the bases of the trapezoid, and the non-parallel sides are called the legs. A trapezoid can have two pairs of parallel sides, in which case it is called a parallelogram.

Here,

First, let's define what a trapezoid and a kite are:

A trapezoid is a quadrilateral with at least one pair of parallel sides. The parallel sides are called the bases of the trapezoid.

A kite is a quadrilateral with two pairs of adjacent sides of equal length.

Now, let's look at some common trapezoid theorems and how to prove them:

The bases of a trapezoid are parallel.

To prove this theorem, we can use the fact that opposite angles of a parallelogram are equal. Since the bases of a trapezoid are parallel, we can draw a line segment that connects the endpoints of the non-parallel sides to form a parallelogram. The opposite angles of the parallelogram are equal, so the opposite angles of the trapezoid are also equal. Therefore, the bases of a trapezoid are parallel.

The legs of a trapezoid are congruent.

To prove this theorem, we can use the fact that a trapezoid can be divided into two triangles by drawing a diagonal. Since the bases of a trapezoid are parallel, the diagonal divides the trapezoid into two congruent triangles. Therefore, the legs of a trapezoid are congruent.

The diagonals of a trapezoid bisect each other.

To prove this theorem, we can use the fact that a trapezoid can be divided into two triangles by drawing a diagonal. Since the bases of a trapezoid are parallel, the diagonal divides the trapezoid into two congruent triangles. The diagonals of the trapezoid connect the midpoints of the non-parallel sides of the triangles, which are also the midpoints of the legs of the trapezoid. Therefore, the diagonals of a trapezoid bisect each other.

Now, let's look at some common kite theorems and how to prove them:

The diagonals of a kite are perpendicular.

To prove this theorem, we can use the fact that a kite can be divided into four right triangles. Since two pairs of adjacent sides of a kite are equal in length, the right triangles that share a common vertex have one leg that is perpendicular to the other leg. Therefore, the diagonals of a kite are perpendicular.

One diagonal of a kite bisects the other diagonal.

To prove this theorem, we can use the fact that a kite can be divided into four triangles. Since two pairs of adjacent sides of a kite are equal in length, the diagonal that connects the non-adjacent vertices of the kite divides the kite into two congruent triangles.

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Complete question:

Prove the trapezoid theorems: for Trapezoids and Kites.

Let's assume that the length of the rectangle is "x".

Since the width is 3 times the length, the width can be represented as "3x".

The perimeter of a rectangle is given by the formula:

P = 2(l + w)

where P is the perimeter, l is the length, and w is the width.

Substituting the values given in the problem, we have:

60 = 2(x + 3x)

Simplifying, we get:

60 = 8x

Dividing both sides by 8, we get:

x = 7.5

Now that we know the length, we can find the width:

width = 3x = 3(7.5) = 22.5

Therefore, the width of the rectangle is 22.5.

Answer:

Answer: 22.5

Step-by-step explanation:

Let L be the length of the rectangle, and W be the width.

From the problem, we know that:

The perimeter of the rectangle is 60, which means that:

2(L + W) = 60

L + W = 30

The width is 3 times the length:

W = 3L

Substituting W = 3L into the first equation, we get:

L + 3L = 30

4L = 30

L = 7.5

Therefore, the width is:

W = 3L = 3(7.5) = 22.5

So, the width of the rectangle is 22.5. Answer:  22.5

Each of the expressions has been simplified and solved by using properties of exponents as shown below.

In Mathematics and Geometry, an exponent is a mathematical operation that is written as an algebraic expression, so as to raise a quantity to the power of another.

Therefore, an exponent can be modeled by the following mathematical expression;

bⁿ

Where:

By applying the multiplication and division law of exponents for powers to each of the expressions, we have the following:

(1.25 × 10⁷) + 63,000,000 = (1.25 × 10⁷) + (6.3 × 10⁷) = (1.25 + 6.3) × 10⁷ = 7.55 × 10⁷

12,000 + 7 × 10⁴ = 1.2 × 10⁴ + 7 × 10⁴ = (1.2 + 7) × 10⁴ = 8.2 × 10⁴

5.88 × 10⁵ - 3.44 × 10⁵ = (5.88 - 3.44) × 10⁵ = 2.44 × 10⁵

6 × 10⁸ ÷ 120 = 6 × 10⁸ ÷ 1.2 × 10² = (6 ÷ 1.2) × 10⁸⁻² = 5 × 10⁶

9 × 10¹² ÷ 4.5 × 10⁴ = (9 ÷ 4.5) × 10¹²⁻⁴ = 2 × 10⁸

1.3 × 10³ · 2,200 = 1.3 × 10³ × 2.2 × 10³ = (1.3 × 2.2) × 10³⁺³ = 2.86 × 10⁶

(6.3 × 10⁴) + 1.3 × 10⁴ = (6.3 + 1.3) × 10⁴ = 7.6 × 10⁴

3,400 · 2 × 10⁴ = 3.4 × 10³ × 2 × 10⁴ = (3.4 × 2) × 10³⁺⁴ = 6.8 × 10⁷

9.78 × 10⁵ - 732,000 = (9.78 - 7.32) × 10⁵ = 2.46 × 10⁵

3.5 × 10² · 2 × 10⁴ = (3.5 × 2) × 10²⁺⁴ = 7 × 10⁶

1.1 × 10⁸ ÷ 22,000 = 1.1 × 10⁸ ÷ 2.2 × 10⁴ = (1.1 ÷ 2.2) × 10⁸⁻⁴ = 0.5 × 10⁴ = 5 × 10³.

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a . (–4 + i) + 4i = –4 + (i + 4i).The associative property of addition states that the grouping of the numbers being added does not change their sum.

In option (a), the expression on the left-hand side can be grouped as (–4 + i) + 4i, and the expression on the right-hand side can be grouped as –4 + (i + 4i). Both expressions result in the same sum of –4 + 5i. Therefore, option (a) demonstrates the associative property of addition.

The associative property of addition is a mathematical property that states that the grouping of the numbers being added does not affect their sum. In other words, when adding three or more numbers, the order in which the numbers are grouped for addition does not affect the result. Mathematically, the associative property of addition can be expressed as:

(a + b) + c = a + (b + c)

This property holds for any real numbers a, b, and c. The associative property of addition is a fundamental property of arithmetic and is used extensively in algebraic manipulations to simplify expressions and solve equations.

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