PLEASE HELPPP 50 POINTS !!! 4b) Use the Fact that exponents are repeated multiplication to expand the expression 2x^3 • 5x^2 and prove why the Product Rule works. Does 4a match 4b?

Answer:

Step-by-step explanation:

(D)

The expression that represents the amount of canned food collected so far by the three friends is [tex]$4x^2 + 2xy - 3$[/tex] and the expression that represents the number of cans the friends still need to collect to meet their goal is [tex]7x^2 + 9$.[/tex]

What is an expression?

In mathematics, an expression is a combination of numbers, variables, operators, and/or functions, which when arranged in a meaningful way, represents a mathematical relationship or a quantity.

Part A:

To find the amount of canned food collected so far by the three friends, we need to add up the number of cans collected by each friend. So, we can write:

Amount collected [tex]$= \text{Jessa's collection} + \text{Tyree's collection} + \text{Ben's collection}$[/tex]

Substituting the given expressions, we get:

Amount collected [tex]$= (2xy + 4) + x^2 + (3x^2 - 7)$[/tex]

Simplifying the expression, we get:

Amount collected [tex]$= 4x^2 + 2xy - 3$[/tex]

Therefore, the expression that represents the amount of canned food collected so far by the three friends is [tex]$4x^2 + 2xy - 3$[/tex].

Part B:

To find the number of cans the friends still need to collect to meet their goal, we need to subtract the amount collected by the three friends from the goal expression. So, we can write:

Number of cans needed [tex]$= \text{Goal} - \text{Amount collected}$[/tex]

Substituting the given expressions, we get:

Number of cans needed [tex]$= (11x^2 - 2xy + 6) - (4x^2 + 2xy - 3)$[/tex]

Simplifying the expression, we get:

Number of cans needed [tex]$= 7x^2 + 9$[/tex]

Therefore, the expression that represents the number of cans the friends still need to collect to meet their goal is [tex]7x^2 + 9$.[/tex]

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Answer:

yes

d = -30.

the difference between each successive term will give you -30, that's the common difference.

E.g -62 - ( -32) = -62 + 32 = -30

Answer:

Yes  (choice 1)
d = - 30 (choice 4)

Step-by-step explanation:

It is an arithmetic sequence because the difference between successive terms is -30

-32 - (-2) = -32 + 2 = -30

-62 - (-32) = -62 + 32 = -30 etc.

Therefore the common difference is -30

The expression d^5/d^(-3) simplified as a single power with only positive exponents is d^8.

The expression we're given is d^5 / d^(-3), which can also be written as d^5 × d^3 (using the rule that a/b = a × ( 1/b )).

Recall that when we multiply two terms with the same base, we add their exponents. In this case, the base is d, so we add the exponents of 5 and -3 (or, equivalently, add 3 to 5 and subtract 3 from 5). This gives us:

d^5 × d^3 = d^ (5+3) = d^8

So the simplified expression with only positive exponents is d^8.

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We estimate the fraction that  1,080 / 1,500 equals to 0.72 or 72% of all students take more than 10 minutes to get to school.

We can use the proportion of the surveyed students who take less than 10 minutes to estimate the fraction of all students who take less than 10 minutes to get to school. Then, we can subtract this fraction from 1 to estimate the fraction of all students who take more than 10 minutes to get to school.

The fraction of surveyed students who take less than 10 minutes to get to school is:

21/75 = 0.28

Assuming that the surveyed students are representative of all students in the school, we can use this proportion to estimate the fraction of all students who take less than 10 minutes to get to school. If there are 1,500 students in the school, then we can estimate the number of students who take less than 10 minutes to get to school as:

0.28 x 1,500 = 420

Therefore, we can estimate the fraction of all students who take more than 10 minutes to get to school as:

(1,500 - 420) / 1,500 = 1,080 / 1,500 = 0.72

So, we estimate that 72% of all students take more than 10 minutes to get to school.

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____The given question is incomplete, the complete question is given below:

In a school of 1,500 students, 75 students were surveyed about their time to go to school. 21 of those students say they spend less than 10 minutes on to go to school. Estimate the fraction of students who take more than 10 minutes to get to school.

Answer:

To solve this problem, we can use the formula for exponential growth:

N = N0 * (1 + r)^t

Where:

N0 = the initial population size

N = the population size after t time periods

r = the growth rate per time period

t = the number of time periods

In this case, we know that the initial population (N0) is 6000, the growth rate (r) is 15% per day, and we want to find the population size (N) after 10 days (t = 10).

First, we need to convert the growth rate from a percentage to a decimal:

r = 15% / 100% = 0.15

Now we can plug in the values and solve for N:

N = 6000 * (1 + 0.15)^10

N = 6000 * 3.439

N ≈ 20,634

Therefore, the population of bacteria after 10 days would be approximately 20,634.

Answer:

4/7

Step-by-step explanation:

16/28= 8/14= 4/7

The equation [tex]x^2 + 4x + 4 = 0[/tex] can be solved by completing the square.

To solve an equation by completing the square, first move the constant term to the other side of the equation. So for the equation [tex]x^2 + 4x + 4 = 0[/tex], we can move the 4 to the right side to get [tex]x^2 + 4x = -4[/tex]. Then divide the coefficient of the x-term (4 in this case) by 2, and square the result. 4 divided by 2 is 2, and 2 squared is 4, so we add 4 to both sides: [tex]x^2 + 4x + 4 = -4 + 4[/tex], which simplifies to [tex]x^2 + 4x = 0[/tex]. Then divide the coefficient of the x-term (4 in this case) by 2, and square the result. 4 divided by 2 is 2, and 2 squared is 4, so we add 4 to both sides.

[tex]x^2 + 4x + 4 = 0[/tex]

[tex]x^2 + 4x = -4[/tex]

[tex]x^2 + 4x + 4 = -4 + 4[/tex]

[tex]x^2 + 4x = 0[/tex]

[tex]x^2 + 4x + 4 = 0[/tex]

[tex](x + 2)^2 = 0[/tex]

This gives us [tex]x^2 + 4x + 4 = 0[/tex]. Next, we can factor the left side of the equation: [tex](x + 2)^2 = 0[/tex]. Then we can take the square root of both sides and solve for x: x + 2 = 0 or x = -2.

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We may type the following expression into a calculator to calculate 13822 x 623/14: 13822 * 623/14 = 615079. 6 x 105 is equal to 1.4 x 104 x 4.4 x 101. Our response to part a) makes sense because 615079 is not far from 6 x 105.

We may type the following expression into a calculator to calculate 13822 x 623/14: 13822 * 623/14 = 615079

As a result, 615079 is the precise value of 13822 x 623/14.

Using approximations to 1 significant figure, we may determine whether the response makes logical.

The difference between 13822 and 623/14 is around 1.4 x 104 and 4.4 x 101, respectively.

Hence, 6 x 105 is equal to 1.4 x 104 x 4.4 x 101.

Our response to part a) makes sense because 615079 is not far from 6 x 105.

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Answer:The required length of FH is obtained as 64.68.

What is the criteria for similarity of two triangles?

Two triangles are said to be similar when either the ratio of their corresponding sides are equal or all the corresponding angles are equal.

Two similar triangles are the same in shape but not in size.

The given diagram shows two similar triangles as ΔFGH and ΔKJH.

Since the ratios of the corresponding sides of the two triangles are equal, the following equation can be written for the given case as,

(x + 8)/32 = (4x - 25)/52

=> 52(x + 8) = 32(4x - 25)

=> 52x - 128x = -32 × 25 - 52 × 8

=> -76x = -1716

=> x = 22.42

Thus FH = 4x - 25 = 4 × 22.42 - 25 = 64.68

Hence, the length of FH for the given case is 64.68.

Step-by-step explanation:

Answer:

Let x be the number of shakes you can buy.

The cost of 5 meals is:

5 * $4 = $20

The total cost of the meals and shakes is:

$20 + $3x = $30

Subtracting $20 from both sides of the equation:

$3x = $10

Dividing both sides of the equation by $3:

x = 3.33

Rounding down to the nearest whole number, you can buy 3 shakes.

I hope this helps!

Answer:

Let's use variables to represent the number of kittens each person has fostered.

Let x be the number of kittens Kim has fostered.

Then, according to the problem, Preston has fostered one less than one third as many kittens as Kim, so Preston has fostered:

(1/3)x - 1

We know that the total number of kittens fostered is 47. Therefore, we can write an equation:

x + (1/3)x - 1 = 47

Simplifying and solving for x:

(4/3)x = 48

x = 36

So Kim has fostered 36 kittens.

To find out how many kittens Preston has fostered, we can substitute x = 36 into the expression we found earlier:

(1/3)x - 1 = (1/3)(36) - 1 = 11

Therefore, Preston has fostered 11 kittens.

In summary, Kim has fostered 36 kittens and Preston has fostered 11 kittens.

Answer:

36, 11

Step-by-step explanation:

Preston and Kim 47 kittens

Kim has fostered X number of kittens

Preston has fostered X/3 - 1 ( 1/3 of Kim - 1)

Total number of kitten will be Preston +Kim

X +1/3X - 1 = 47

Add 1 to both sides

X + 1/3X = 48

If 4/3X is 48, X is 36

If Kim has 36kittens, Preston fostered 47- 36 =11

To check

1/3 of 36 = 12

Subract 1 = 11

Answer:Park rangers released 4 fish into a pond in year 0. Each year, there were three times as many fish as the year before. How many fish were there after x years? Write a function to represent this scenario.

Step-by-step explanation:

An function which best represent the given scenario is,  4 x 3ˣ. So option B is correct.

What is geometric progression?

In algebra, in sequence we study various progressions, one of the progression is geometric, in this progression for every two consecutive terms, the common ratio is the same.

Formula for nth term of G.P.,

Tₙ = a×rⁿ

where a is first term and r is common ratio

Given that,

Park rangers released 4 fish into a pond in year 0.

Each year, there were three times as many fish as the year before.

The number of fish after x years = ?

After 1 year,

the number of fish  =  4 x 3

After 2 year,

the number of fish  = 4 x 3 x 3

                               = 4 x 3²

After 3 year,

the number of fish = 4 x 3² x3

                              = 4 x 3³

Similarly, after x years

the number of fish =  4 x 3ˣ

Hence, the expression is  4 x 3ˣ

Answer:

To find the extreme value of the function f(x) = 6x^2 - 180x + 1000, we can first take the derivative of the function and set it equal to zero:

f'(x) = 12x - 180 = 0

Solving for x, we get x = 15. Substituting this value back into the original function, we get:

f(15) = 6(15)^2 - 180(15) + 1000 = 1250

So the extreme value of the function is 1250, which represents the maximum profit in thousands of dollars that can be earned by selling a certain number of items.

More specifically, the value x = 15 is the value of x that maximizes the profit, and the maximum profit is $1,250,000. This means that if the company sells 15 items, they will make the most profit possible.

Step-by-step explanation:

Answer:

8 burgers and 8 french fries

Step-by-step explanation:

8x+2y=80

x=8

y=13

8x8=64

80-64=16

16/2=8

An expression that represent the average rate of decrease in the value of the car over the first 5 years of ownership is: 3. 11,855.50 per year.

In Mathematics and Statistics, a population or physical asset that decreases at a specific period of time represent an exponential decay. This ultimately implies that, a mathematical model for any population or physical asset that decreases by r percent per unit of time is an exponential equation of this form:

P(t) = I(1 - r)^t

Where:

By substituting given parameters, we have the following:

P(t) = I(1 - r)^t

P(5) = 22,465(0.88)^5

P(5) = 22,465(0.5277319168)

P(5) = 11,855.50 per year.

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If the total amount given the original price and tax or tip are 123.28,7. 5%, the total amount including tax is $132.52.

To find the total amount given the original price and tax or tip, we need to add the amount of tax or tip to the original price.

In this case, the original price is $123.28 and the tax is 7.5%. To find the amount of tax, we can multiply the original price by the tax rate expressed as a decimal.

Tax = 0.075 x $123.28 = $9.24

Therefore, the amount of tax is $9.24. To find the total amount, we simply add the original price and the amount of tax:

Total amount = $123.28 + $9.24 = $132.52

Note that if the 7.5% was a tip instead of a tax, we would calculate it in the same way by multiplying the original price by the tip rate expressed as a decimal.

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Answer:

-116u

Step-by-step explanation:

-69-47 is 116. The varblie stays making it -116u

1. For your 24 cans of juice, the possibilities of organizing them in arrays can be:

2. For Teyo's 25 cans of juice, the possible arrays are:

You must first consider the factors of the total number of juice cans you own. For activity 1, you must consider the factors of the 24 cans of juice, that is, the factors of 24, which are 1,2,3,4,6,8,12 and 24, and then find the combinations of these factors to sort them into arrays.

Using the information above, you can effectively draw the arrays on a separate sheet of paper.

Therefore, matrices are meant to represent a set of objects or values in a rectangular shape, arranged in rows and columns and are intended to represent a variety of numerical data, text and images.

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Nοra has $8 saved, Helga has $28 saved, and Gertrude has $12 saved.

A ratiο is a way οf cοmparing twο quantities οr numbers by divisiοn. It is a relatiοnship between twο numbers that tells us hοw many times οne value is cοntained in the οther. Ratiοs can be written in different fοrms such as a:b, a tο b, οr a/b. In a ratiο, the first number is usually referred tο as the "antecedent" and the secοnd number is referred tο as the "cοnsequent".

Fοr example, if we have a ratiο οf 2:3, it means that the first number is twο-thirds οf the secοnd number. Ratiοs can alsο be simplified οr expressed in different units, but the relatiοnship between the twο quantities remains the same. Ratiοs are cοmmοnly used in many different areas οf mathematics, such as in geοmetry, prοpοrtiοn, and statistics.

Let x be the amοunt οf savings that Nοra has

Then, Helga has 7x/2 (since the ratiο οf Helga's savings tο Nοra's savings is 7:2)

And, Gertrude has 3/2x (since the ratiο οf Nοra's savings tο Gertrude's savings is 2:3)

We knοw that the sum οf their savings is $48, sο we can write an equatiοn:

x + 7x/2 + 3/2x = 48

Tο sοlve fοr x, we can first simplify the left side οf the equatiοn:

(2x + 7x + 3x)/2 = 48

12x/2 = 48

6x = 48

x = 8

Nοw that we knοw Nοra has $8 saved, we can find hοw much Helga and Gertrude have:

Helga has 7x/2 = 7(8)/2 = $28 saved

Gertrude has 3/2x = 3/2(8) = $12 saved

Therefοre, Nοra has $8 saved, Helga has $28 saved, and Gertrude has $12 saved.

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Answer:

(8x + 1) (8x - 1)

Step-by-step explanation:

64x² - 1

Rewrite 64x² as (8x)²

(8x)² - 1

Rewrite 1 as 1²

(8x)² - 1²

Since both terms are perfect squares, factor using the difference of squares formula, a² - b² = (a + b) (a - b) where a = 8x and b = 1

So, our answer is (8x + 1) (8x - 1)

Answer: The function y = 7(2/3)^x is an example of exponential decay.

This is because the base of the exponential function, 2/3, is between 0 and 1, which means that the function is decreasing as x increases. As x increases, the value of (2/3)^x becomes smaller and smaller, which causes the overall value of the function y to decrease over time.

Exponential decay is a type of exponential function where the value of the function decreases over time. In contrast, exponential growth is a type of exponential function where the value of the function increases over time, and the base of the function is greater than 1.

Step-by-step explanation:

The common attributes of a right pyramid and a sphere are that they both have an apex and a curved surface.

What is a shape?

Shape can be defined as the external physical outline, structure or form of an object or entity. It refers to the appearance or visual characteristic of an object or entity, and is often determined by its edges, contours, curves, angles, or other physical attributes.

A right pyramid and a sphere have some similarities and differences in their attributes.

Similarities:

Both shapes have a single point at the top which is called the apex.

Both shapes have a curved surface. In the case of the right pyramid, the curved surface is a set of triangles that converge to the apex. In the case of the sphere, the curved surface is a continuous, smooth surface.

Both shapes are three-dimensional objects that occupy space.

Differences:

The base of a right pyramid is a polygon, whereas a sphere has no edges or vertices.

The curved surface of a right pyramid is made up of flat triangles, whereas a sphere's curved surface is continuous and has no flat faces.

A right pyramid has edges where the flat faces meet, whereas a sphere has no edges or corners.

A right pyramid has a finite number of vertices and edges, whereas a sphere has an infinite number of points on its curved surface.

A right pyramid can have different base shapes and sizes, whereas a sphere has only one size and shape.

Therefore, the common attributes of a right pyramid and a sphere are that they both have an apex and a curved surface.

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The potential rational root of a polynomial f(x) at point p is the value of x for which the result of evaluation of f(x) at that point is zero.

A potential rational root of a polynomial f(x) at point p is a value of x where the result of evaluation of f(x) at that point is zero. By Rational Roots Theorem To find a potential rational root of a polynomial f(x), we first factorize f(x) into its factors and then identify the factors which are of the form (x-p). The value of x for which the result of evaluation of f(x) at that point is zero, is the potential rational root of the polynomial f(x).

For example, consider a polynomial f(x) = x^3 + 4x^2 + 7x +2.

To find a potential rational root of this polynomial at point p, we factorize f(x) as follows:

f(x) = (x - 1)(x^2 + 5x + 2)

The factor (x - 1) is of the form (x-p) and hence, 1 is a potential rational root of the polynomial f(x). To verify that 1 is indeed a potential rational root, we evaluate f(1):

f(1) = 1^3 + 4(1^2) + 7(1) + 2

= 1 + 4 + 7 + 2

= 14

Since f(1) ≠ 0, 1 is not a rational root of the polynomial f(x).

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Complete question:

What is a potential rational root of the polynomial f(x) at the point p?

In reality, [tex]96[/tex] miles distance separate the two cities.

We kept an eye on them from afar. Compared to earlier, she perceives a gap between her and her brother. They had been close friends earlier, but now there was a great distance between them. He desires to disassociate himself from his old employer.

Mileage is a vector variable that quantifies "how much space an object has travelled through" while in motion. An object's dependent and independent variables in position is referred to as displaced, a vector variable that measures "the extent to which place an item is."

Using the scale provided, we can establish the proportion:

[tex]2 cm / 40 miles = 4.8 cm / x[/tex]

where [tex]x[/tex] is the actual distance between the two cities in miles.

To solve for [tex]x[/tex], we can cross-multiply:

[tex]2 cm * x = 40 miles * 4.8 cm[/tex]

Simplifying:

[tex]2x = 192[/tex]

[tex]x = 96[/tex]

Therefore, the actual distance between the two cities is [tex]96[/tex] miles.

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To solve this equation, we need to consider two cases, one where 2x + 1 is positive and another where it is negative. We will then solve for x in each case and check our solutions to make sure they satisfy the original equation.

Case 1: 2x + 1 ≥ 0

In this case, we have:

|2x + 1| = 2x + 1

and

|3x - 11| = 3x - 11

Substituting these expressions into the original equation, we get:

2x + 1 = 3x - 11

Solving for x, we get:

x = 12

Checking our solution, we have:

|2x + 1| = |2(12) + 1| = 25

and

|3x - 11| = |3(12) - 11| = 25

Therefore, x = 12 is a valid solution to the equation.

Case 2: 2x + 1 < 0

In this case, we have:

|2x + 1| = -(2x + 1) = -2x - 1

and

|3x - 11| = -(3x - 11) = -3x + 11

Substituting these expressions into the original equation, we get:

-2x - 1 = -3x + 11

Solving for x, we get:

x = -10

Checking our solution, we have:

|2x + 1| = |2(-10) + 1| = 21

and

|3x - 11| = |3(-10) - 11| = 41

Therefore, x = -10 is not a valid solution to the equation.

Therefore, the only solution to the equation |2x + 1| = |3x - 11| is x = 12.

To convert the total time from seconds to minutes, we need to divide by 60. So, the total time in minutes would be: (720 + 420) / 60 = 1140 / 60 = 19 minutes.

To find the total time it takes Andrew to take a shower and eat breakfast, we need to add the time it takes for each activity. Andrew takes 720 seconds to take a shower and 420 seconds to eat breakfast. To convert seconds to minutes, we divide the total time in seconds by 60. So the total time it takes Andrew to take a shower and eat breakfast is (720 + 420) / 60 = 19 minutes. Therefore, it takes Andrew 19 minutes to take a shower and eat breakfast.

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Complete question:

It takes Andrew 720 seconds to take a shower. He spends an additional 420 seconds eating breakfast. How many minutes does it take Andrew to take a shower and eat breakfast?

The equation represents the amount of money, in dollars, Kenny will save in x weeks is y = $10*x

We know that Kenny saved 10 dollars every week.

Then, if we define the variable y as the savings, after one week we know that:

y = 10

After another week:

y = 20

And so on.

We can define x as the number of weeks, and now we know that after x weeks, the amount of money that Kenny has saved is x times $10, this gives the linear equation.

y = $10*x

That is the equation we wanted.

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Therefore , the solution of the given problem of function comes out to be

a.(5, 7) ,b.(4, 5). and c. (1, 2).

The midterm exam covers a wide range of topics, including building, geometry, integers, ones personal divisions, and both actual and hypothetical geographic locations. A work describes the relationships between different elements that all variable cooperate to create the same outcome. A utility is composed of numerous unique elements that work together to produce unique outcomes for each input.

Here,

(a) On the intervals (2, 4) and, the function is growing (5, 7).

(b) On the intervals (0, 2) and (c), the algorithm decrements (4, 5).

(c) On intervals (0, 1) and, the function is constant (1, 2).

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