7-2 Enrichment
Multiplying Polynomials
An architect is preparing to design an office building. She creates a sketch of the basic shape of the structure and
the relative dimensions of each piece shown. The angles in the sketch are all right angles. Find the volume of the structure using each approach. Draw lines on the sketches and record your expressions.

1. Divide the structure into two different rectangular
prisms. Add the volumes of the two prisms to create
an expression for the total
Volume

2. Divide the structure into three different rectangular
prisms. Add the volumes of the three prisms to create
an expression for the total volume.

3. View the structure as one prism with 2 equal sized prisms. Find the volume of the large prism and subtract the volumes of the smaller pieces to find the expression for the total volume!!

Thanks for helping!! (We are learning polynomials right now, like (2+x)^2 for clarification for the problems!!!) :)

Answer:

60%

Step-by-step explanation:

12/20 = 0.6  multiply this by 100% = 60%

It depends on the length of the festival, the length of each band's performance, and other factors. There could be any number of ways to schedule their appearances.

The number of ways to schedule the bands' appearances at the weekend festival depends on a variety of factors such as the length of the festival, the length of each individual band's performance, and the number and type of stages available.

For example, if the festival runs for two days and each band's performance is one hour, there could be a total of 22 hours of music, so eleven bands could each perform twice, once on each day.

Alternatively, if the festival has three stages and two of the stages are dedicated to different genres of music, then the bands could be scheduled to perform on those stages, with each stage hosting five bands each day.

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The relationship between the radius and volume of a cylinder with a height of 4 feet is nonlinear because the volume of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height. In this case, the height is fixed at 4 feet. As we change the radius, the volume changes in a nonlinear manner.

The volume increases much faster as the radius increases. This can be seen from the exponential nature of the formula, as the radius is squared. Therefore, even a small increase in the radius results in a substantial increase in the volume, making the relationship between radius and volume nonlinear.

In contrast, if the height were to be proportional to the radius, then the relationship between radius and volume would be linear. This is because the volume would be proportional to the product of the radius and height, which is a linear relationship.

Therefore, the nonlinear relationship between radius and volume, in this case, is due to the exponential nature of the formula for the volume of a cylinder and the fixed height of 4 feet.

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--The given question is incomplete; the complete question is

"This function represents the relationship between the radius of cylinders with a height of 4 feet and their volumes. Why is this relationship between radius and volume nonlinear?"--

The existence, uniqueness, and minimum radius of convergence of a Taylor series expansion about a point depend on the function being expanded. In general, there is no guarantee that a Taylor series will exist or be unique for every function.

A Taylor series expansion of a function about a point is a representation of the function as an infinite sum of terms involving powers of the difference between the independent variable and the point. If the function is well-behaved near the point, its Taylor series will converge to the function within some interval around the point. The smallest such interval is called the radius of convergence.

If the function is infinitely differentiable (i.e. has derivatives of all orders) near the point, its Taylor series will converge to the function within the entire interval of convergence. In this case, the solution is said to be unique. However, if the function is not infinitely differentiable, it may not be possible to find a Taylor series that converges to the function, or there may be multiple Taylor series that converge to the function within different intervals.

The minimum radius of convergence of a Taylor series is difficult to determine in general, as it depends on the behavior of the function at infinity. In some cases, the radius of convergence can be estimated using techniques such as Cauchy-Hadamard theorem. However, in general, determining the exact radius of convergence requires knowledge of the behavior of the function at all points, not just near the point of expansion.

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The divisors of 60 are the numbers that evenly divide 60 without leaving a remainder. In other words, if we divide 60 by one of its divisors, the result will be a whole number.

For example, if we divide 60 by 2, we get 30. If we divide 60 by 3, we get 20. And so on.

When the teacher wants to form equal groups of students, she needs to ensure that the number of students in each group is one of the divisors of 60. This way, she can divide the class into groups with an equal number of students in each group.

So, in this case, the different groups the teacher can make are:

1 group of 60 students

2 groups of 30 students each

3 groups of 20 students each

4 groups of 15 students each

5 groups of 12 students each

6 groups of 10 students each

10 groups of 6 students each

12 groups of 5 students each

15 groups of 4 students each

20 groups of 3 students each

30 groups of 2 students each

60 groups of 1 student each

These are the 12 different options for forming equal groups of students.

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Answer:

Step-by-step explanation:

Let [tex]f(x) = y[/tex]

∴ [tex]y = \sqrt[3]{x + 2} -5[/tex]

x has to be isolated and made the subject of the formula:

[tex]y + 5 = \sqrt[3]{x + 2}[/tex]

Cube is applied to both sides of the equation to get rid of the cube root:

[tex](y + 5)^{3} = x + 2[/tex]

x = [tex](y + 5)^{3} -2[/tex]

Substitute [tex]f^{-1} (y) = x[/tex]:

[tex]f^{-1} (y) = (y + 5)^{3} - 2[/tex]

Replace y  with x:

[tex]f^{-1} (x) = (x + 5)^{3} -2[/tex]

When computing the area of a circle, the mathematical constant pi () is required. Pi is the circumference-to-diameter ratio of a circle and is approximately equal to 3.14159.

The area of a circle is calculated by multiplying pi by the radius squared (r2). To estimate the area of a circle, utilize pi in the calculation since it is a vital aspect of the relationship between the circumference and diameter of a circle.

Pi is not required when determining the area of a square, rectangle, or cube since the formulas for these shapes are different.

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The question is -

The mathematical constant pi is needed when calculating the area of a _.

a. circle

b. square

c. rectangle

d. cube

The expression (2x + 7)(x - 3) is a factor of 2x^2 - 11x - 21.

Expression is a combination of variables, numbers, and operators (such as +, -, *, /) that represents a mathematical value. For example, 2x + 3, x^2 - 5x + 6, and (x + 2)(x - 3) are all expressions. Expressions can be simplified or evaluated to find their numerical value for a given value of the variable(s).

The expression (2x + 7)(x - 3) is a factor of 2x^2 - 11x - 21.

To see this, we can expand the right-hand side:

(2x + 7)(x - 3) = 2x(x) + 2x(3) + 7(x) - 7(3)

= 2x^2 + 6x + 7x - 21

= 2x^2 + 13x - 21

As we can see, (2x + 7)(x - 3) is indeed a factor of 2x^2 - 11x - 21.

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The value of b, given the lines that are parallel in the figure, would be D. 27. 67.

Given that the lines are parallel, then we can infer that angles 92 degrees and ( 3b + 5 ) degrees are supplementary angles which means that they add up to 180 degrees.

We can therefore solve for b by equating the sum of both angles to 180 degrees as shown in the formula below :

92 + ( 3b + 5 ) = 180

92 + 5 + 3 b = 180

3b = 180 - 92 - 5

b = 83 / 3

b = 27. 67

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The cube has a surface area of 1536 inches².

A three dimensional shape can be defined as a solid figure or an object or shape that has three dimensions—length, width, and height.

Given [tex]$ \mathrm{S = 6V^{\frac{2}{3} }}[/tex]

S is surface Area ‘A’, in square units, of a cube

V is volume of cube in cubic units.

Now in the given question,

V = 4096 cubic inches

The formula below can be used to get the cube's surface area:

Surface area = [tex]$ \mathrm{ 6V^{\frac{2}{3} }}[/tex]

Surface area = 6 × [tex]4096^{\frac{2}{3} }[/tex]

Surface area = 6 × 16²  = 6  × 256

Surface area = 1536 inches²

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Answer:

2/ c, e

Step-by-step explanation:

A graph can only be a function if one x only has one y.

If you use the line test, you see that only graphs C and E pass the test.

So, the answers are C and E.

Considering all the given information, the probability that the subcommittee consists of 3 republicans is 0.0345, or 3.45%.

To find the probability that the subcommittee consists of 3 Republican’s, we first calculate the number of ways to choose 3 Republicans out of 5:

C(5, 3) = 5! / (3! (5 - 3)!)

=5! / (3! 2!)

= (5 ×4 × 3!) / (3 × 2 × 3!)

= (5 × 4) / (2 × 2)

= 10.

The total number of ways to choose 3 members out of a committee of 5 Republicans, 6 Democrats, and 2 Independents is:

C(13, 3) = 13! / (3! (13 - 3)!)

=13! / (3! 10!)

= (13 × 12 × 11 × 10!) / (3 × 2 × 1 × 10!)

= (13 × 12 × 11) / (3 × 2 × 1)

= 286

So the probability of choosing 3 Republicans is:

P(3 Republicans) = 10 / 286 = 1/29

Therefore the probability of the subcommittee consisting of 3 Republicans is approximately 0.0345, or 3.45%.

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Q: A Senate committee consists of 5 Republicans, 6 Democrats, and 2 Independents. A subcommittee of 3 members is randomly chosen. What is the probability that the subcommittee consists of three Republicans?

The function constructed is y = 90x + 60.

"Information available from the question"

Jahlil is saving money at a constant rate to buy a new TV. After saving for 3 months, After saving for 3 months, Jahlil has $330. After saving for 5 months, Jahlil has $510.

Let x represents the months

and y represents the dollars

To find the dollars/month and by finding the slope.

(3, 330), (5, 510)

[tex]\frac{510-330}{5-3}[/tex] = 90

Since its a constant rate, the function will be in the form, y = mx+b, and since we found the slope we can plug m=90.

y=90x+b

To find b, we simply substitute one of the points.

330 = 90(3) + b

b = 60

Thus, plugging it in,

y = 90x + 60

Hence, The function constructed is y = 90x + 60.

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Assuming that the order of the cities matters, the number of possible routes is 11! (11 factorial), which is 39916800.

The number of possible routes for an overnight express company that includes eleven cities on its route is 39916800. This is because when the order of the cities matters, the number of possible routes can be calculated by finding 11 factorial, which is:

This calculation results in 39916800, which means that there are 39916800 potential routes for an overnight express company with eleven cities on its route.

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Note that the line of Best fit is attached accordingly. See the definition below.

A line of best fit is a straight line that represents the trend of a set of data.

It is used to summarize and analyze the relationship between two variables in a scatter plot. Note that the line of best fit can be used to approximate the trend or pattern of the data and make predictions for new observations that fall within the range of the data.

The line is drawn in a way that minimizes the distance between the observed data points and the line.

The line of best fit can be used to make predictions and identify outliers. It is important because it helps to reveal patterns and trends in data and allows us to make informed decisions based on the information.

Note that the line takes a position that gives an approximately equal amount of variables or data points on either side.

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Options A and C are correct for the condition 5 x 4 = 20.

A factor is a number that completely divides another number. To put it another way, if adding two whole numbers results in a product, then the numbers we are adding are factors of the product because the product is divisible by them.

Factors can be found using either division or multiplication. Rules of divisibility can also be applied.

Given 5 x 4 = 20

factors of 20 are 2*2*5

so 5 is the factor of  20,

A is correct.

now multiples of 20 are 20, 40, 60.......

since all values are greater than 5,

so 5 is not a multiple of 20.

B is not correct.

now multiples of 4 are 4, 8, 12, 16, 20, 24 .......

so 20 is a multiple of 4,

C is correct.

Hence A and C are correct statements.

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The equations are H = 4x + 60 and P = 8x + 20. We need to rent 10 tables to ensure that the cost of the rental will be the same.

A system of equations in algebra consists of two or more equations and looks for common answers to the equations. "A set of equations satisfied by the same set of variables is called a system of linear equations."

Let us suppose tables as x.

Given that,

The happy party rents tables for $4 per table plus a delivery fee of 60. This is expressed as:

H = 4x + 60

Party Rental rents tables for $8 per table plus a $20 delivery fee.

P = 8x + 20

To get the same cost of the rent:

H = P

4x + 60 = 8x + 20

60 - 20 = 8x - 4x

40 = 4x

x = 10

Hence, we need to rent 10 tables to ensure that the cost of the rental will be the same.

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Answer:

[tex]y = -2x + 10[/tex]

Step-by-step explanation:

The slope-intercept form of equation of line:

y = mx + c

where:

m = slope

c = y-intercept

Slope, m:   [tex]\frac{y_{2} - y_{1}}{x_{2} -x_{1}}[/tex]

               = [tex]\frac{4 - 8}{3 - 1}[/tex]

               = [tex]-\frac{4}{2}[/tex]

               = [tex]-2[/tex]

[tex]y = -2x +c[/tex]

Substitute the coordinates of either of the two points in this equation to calculate the value of c:

8 = [tex]-2[/tex](1) + c

8 = [tex]-2[/tex] + c

c = 8 + 2

c = 10

Therefore, the equation is:

[tex]y = -2x +10[/tex]

Answer:

To enlarge a triangle by a scale factor of 1/2 with the center (2, 0), each vertex of the triangle should be multiplied by 1/2 and then shifted so that the center of the new triangle is (2, 0).

Step-by-step explanation:

For example, if the original triangle has vertices at (a, b), (c, d), and (e, f), the vertices of the new triangle would be at ((a-2)/2 + 2, (b-0)/2 + 0), ((c-2)/2 + 2, (d-0)/2 + 0), and ((e-2)/2 + 2, (f-0)/2 + 0).

The total number of possible outcomes is the product of the number of outcomes for each event.

The spinner can land on any of its 5 sections on the first spin, and on any of the 5 sections again on the second spin. So there are 5 x 5 = 25 possible outcomes for the spinner.

The dice can land on any of its 6 sides.

The coin can land either heads or tails.

Therefore, the total number of possible outcomes is:

25 (spinner) x 6 (dice) x 2 (coin) = 300 possible outcomes.

Answer: An isosceles triangle is a triangle with two sides that are congruent (equal in length). If DEF is isosceles with base DF, then DE and EF are congruent and have the same length. This means that the two sides of the triangle are equal in length, and that the angle formed by DE and EF is equal to each other (since they are congruent sides, they must also be congruent angles). The two equal angles are called the base angles of the isosceles triangle.

Step-by-step explanation:

Answer:

(-3, 9)

2

Step-by-step explanation:

Answer:

p = 150 - 65

Step-by-step explanation:

We know

p is the amount he has left to raise

The target is $150

He has $65

So, the equation is

p = 150 - 65

Answer:

The value of x =10

Step-by-step explanation:

2x+ 5 = 25

Subtract 5 from each side.

2x+5-5 = 25-5

2x = 20

Divide each side by 2.

2x/2 = 20/2

x =10

The value of x =10.

Answer:

10

Step-by-step explanation:

Move the 5 to the right so the equation is 2x = 25 - 5
Then its 2x = 20 and 20/2 is 10.

Answer:

Step-by-step explanation:

Here are the steps to write the polynomial -15y + 12y² - 9y as a product of polynomials in standard form:

Factor out the coefficient of the linear term: -15y = -15 * y

Factor out the coefficient of the quadratic term: 12y² = 12 * y^2

Add the two factored polynomials: -15 * y + 12 * y^2 - 9 * y

Simplify the expression: (-15 * y + 9 * y) + 12 * y^2 = -6 * y + 12 * y^2

And there you have it, the polynomial -15y + 12y² - 9y can be written as a product of two polynomials in standard form, -6 * y + 12 * y^2.

Answer:

Correct, the inequality y ≥ 0.8x + 1.5 accurately describes the relation between the duration of the hike (x) and the amount of water needed (y). This inequality means that for any duration of the hike (x), the amount of water needed (y) must be greater than or equal to 0.8x + 1.5.

Proverbs 21:5 says, “The plans of the diligent lead to profit as surely as haste leads to poverty.” This verse encourages us to be diligent in our work as it is the path to success and profit.

Profit is the money left over when all expenses are paid. It is the reward for running a successful business. Profit is the additional income that results from a business venture beyond what is initially invested. It is a financial gain that is realized when the amount of revenue generated from a business activity exceeds the expenses, costs, and taxes needed to sustain the activity. Profit is an important measure of a business’s long-term viability and overall financial health.

Diligence, or hard work and perseverance, is the key to achieving anything. As we practice diligence, it will lead us to success and material gain. We will be rewarded in the end for our hard work.

Not only will diligence lead to financial gain, but it will also bring many other benefits. Diligence will create good habits, increase self-discipline, and help us to develop our skills. We will become more organized and efficient in our work, leading to better quality results. Hard work and diligence will also give us a sense of satisfaction and accomplishment which will bring more joy and fulfillment. In addition, it will also help us to build strong relationships with those around us, as our commitment to our work will be seen and appreciated.

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Answer:

Step-byoooooooooooo

-step explanation:

18,995 × 0.82^(6) = $5,774 so he should not sell

if the percentage decrease is 18% then do 100% - 18% which is 82. divide by 100 = 0.82. if it does this for 6 years then do 18,995 × 0.82^(6) because it's 6 years. the number is larger than Manuel's offer therefore he shouldnt sell

Answer:

  g(x) = -1/4x²

Step-by-step explanation:

Given f(x) = x² and point (2, -1) on the graph of g(x), which is a scaled version of f(x), you want the equation for g(x).

The scale factor k will satisfy ...

  g(x) = k·f(x)

  g(2) = k·f(2) . . . . . . true at the given point

  -1 = k·2² = 4k . . . . use the coordinates of the given point

  k = -1/4 . . . . . . . . . divide by 4

The equation of g(x) is ...

  g(x) = -1/4x²