The base of a triangle measures (3x − 6) units and the height measures (2x + 4) units. Part A: What is the expression that represents the area of the triangle? Show your work to receive full credit. (4 points) Hint: Area = 0.5bh Part B: What are the degree and classification of the expression obtained in Part A? (3 points) Part C: How does Part A demonstrate the closure property for polynomials? (3 points)

The expression x³ + 27 can be rewritten as (x + 3) (x² - 3x + 9).

The expression x³ + 27 can be rewritten using the formula for the sum of cubes, which is

a³ + b³ = (a + b) (a² - ab + b²)

In this case, we can recognize that 27 is equivalent to 3³

Therefore, we can set 'a' to be 'x' and 'b' to be '3' and apply the formula.

So, the expression x³ + 27 rewrites to:

x³ + 3³ = (x + 3)(x² - 3x + 9)

The expression x³+27 is rewritten as the sum of cubes by identifying a as x and b as 3, then applying the formula a³ + b³ = (a + b)(a² - ab + b²) to get (x + 3)(x² - 3x + 9).

The expression x³+27 can be rewritten using the sum of cubes formula. The sum of cubes formula is a³ + b³ = (a + b)(a² - ab + b²). In this case, x³ can be viewed as a^3 and 27 (which is 3³) as b³. Thus, to express x³ + 27 as a sum of cubes, we identify a as x and b as 3, and then apply the formula.

Using the sum of cubes formula:

Identify a = x and b = 3.

Substitute a and b into the sum of cubes formula: (x + 3)(x² - 3x + 9).

Simplify if necessary.

Therefore, x³ + 27 is rewritten as (x + 3)(x² - 3x + 9).

Answer:

y-3=-4(x+2)

Step-by-step explanation:

The solutions of the equations [tex]x^3-6x^2+9x=0[/tex] is:

                         x=0 or x=3

The solutions of the graph of the equation are the possible values of x at which the graph meets the x-axis at a point.

The function f(x) is given by:

[tex]f(x)=x^3-6x^2+9x[/tex]

The function f(x) could also be written as:

[tex]f(x)=x^3-3x^2-3x^2+9x\\\\i.e.\\\\f(x)=x^2(x-3)-3x(x-3)\\\\i.e.\\\\f(x)=(x^2-3x)(x-3)\\\\i.e.\\\\f(x)=x(x-3)(x-3)[/tex]

i.e. If

[tex]f(x)=0[/tex]

then

[tex]x(x-3)(x-3)=0\\\\i.e.\\\\x=0\ or\ x=3[/tex]

By the help of the graph we may observe that the function meets the x-axis at x=0 or x=3.

Final answer:

To calculate the mailing cost for an 8-pound package sent first class, convert to ounces, then apply the charges for the first ounce and each subsequent ounce. The total cost is $25.84.

Explanation:

The question involves calculating the total cost of mailing a package that weighs 8 pounds, first class, with specific charges for the first ounce and each additional ounce. To calculate the cost, we first need to convert the package's weight from pounds to ounces. There are 16 ounces in 1 pound, so an 8-pound package is equivalent to 8 x 16 = 128 ounces.

The post office charges $0.44 for the first ounce. Thereafter, it charges $0.20 for each additional ounce. The cost for the additional ounces is $0.20 x (128 - 1) = $25.40. Now, add the cost for the first ounce, so the total cost is $0.44 + $25.40 = $25.84.

Sample Answer: No, Ingrid is incorrect. The initial value in the scenario is 170 feet, which represents the y − intercept. Receding 4 feet in the scenario represents the rate of change, which is the slope. In slope-intercept form, y = mx + b, where m represents slope and b represents the y−intercept, so the correct equation is y = −4x + 170.

Answer:

No, Ingrid is incorrect. The initial value in the scenario is 170 feet, which represents the y − intercept. Receding 4 feet in the scenario represents the rate of change, which is the slope. In slope-intercept form, y = mx + b, where m represents slope and b represents the y−intercept, so the correct equation is y = −4x + 170.

Step-by-step explanation:

The solution is 3x+2y is less than or equal to 3 and 3x+4y is greater than or equal to 2

How to solve the inequality

Given the inequalities:

1. [tex]\( y < -\frac{3}{2}x + \frac{3}{2} \)[/tex]

2. [tex]\( y > -\frac{3}{4}x + \frac{1}{2} \)[/tex]

The inequality [tex]\( y < -\frac{3}{2}x + \frac{3}{2} \)[/tex] represents a dashed line with a slope of [tex]\( -\frac{3}{2} \)[/tex] and a y-intercept of [tex]\( \frac{3}{2} \)[/tex]. It does not include the points on the line (hence the dashed line).

3x + 4y = 2

The inequality [tex]\( y > -\frac{3}{4}x + \frac{1}{2} \)[/tex]

slope of [tex]\( -\frac{3}{4} \)[/tex]

y-intercept of [tex]\( \frac{1}{2} \)[/tex].

The region where both inequalities are simultaneously true

[tex]\( y > -\frac{3}{4}x + \frac{1}{2} \)[/tex]

and below the line

[tex]\( y < -\frac{3}{2}x + \frac{3}{2} \)[/tex]

This confirms the solution as the point (-2, 3) and the region it falls into on the graph.

Answer:

[tex]f(x)=x^3-2 x^2+9 x-18[/tex]

Step-by-step explanation:

The roots of the polynomial are [tex]2,3i,-3i[/tex].

This implies that [tex]x-2,x-3i,x+3i[/tex] are factors of the given polynomial.

The polynomial will have equation;

[tex]f(x)=(x-2)(x-3i)(x+3i)[/tex]

We expand using difference of two squares on the complex conjugates to get;

[tex]f(x)=(x-2)(x^2-(3i)^2)[/tex]

[tex]\Rightarrow f(x)=(x-2)(x^2-(-3)^2(i)^2)[/tex].

[tex]\Rightarrow f(x)=(x-2)(x^2-9(i)^2)[/tex].

Recall that;

[tex]\boxed{i^2=-1}[/tex]

[tex]\Rightarrow f(x)=(x-2)(x^2+9)[/tex].

Expand using the distributive property to get;

[tex]\Rightarrow f(x)=x^3+9x-2x^2-18[/tex].

We rewrite in standard form to obtain;

[tex]f(x)=x^3-2x^2+9 x-18[/tex]

Answer:

3

Step-by-step explanation:

.

.

.

Answer:

d 4 x 5

Step-by-step explanation:

round

The number of vertices will be 24. The correct option is D.

A prism is a polyhedron in geometry that has n parallelogram faces that connect the n-sided polygon basis, the second base, which is a translated duplicate of the first base, and the n faces. The bases are translated into all cross-sections that are parallel to them.

It is given that the number of vertices in a prism is twice the number of vertices of one of the bases. The number of the vertices of a regular prism with 14 faces and 36 edges is calculated as below:-

Use Euler's formula:-

V + F = E + 2

V + 14 = 36 + 2

V + 14 = 38

V = 38 - 14

V = 24

Therefore, the number of vertices will be 24. The correct option is D.

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Using Euler's formula, V + F = E + 2, with the given values for the faces (F = 14) and edges (E = 36), we determine there are 24 vertices in the prism. Considering a prism has twice as many vertices as one of its bases, it follows that one base has 12 vertices.

The question asks about the number of vertices on one of the bases of a regular prism that has 14 faces and 36 edges.

First, we use Euler's formula, which is V + F = E + 2, where V, F, and E represent the number of vertices, faces, and edges, respectively.

In this case, we have F = 14 and E = 36, so plugging these into Euler's formula gives us V + 14 = 36 + 2, therefore V = 24 vertices in total for the prism.

Since the prism has two congruent bases and the remainder of the faces are parallelograms connecting the bases, we can say that the number of vertices of the prism is twice the number of vertices of one of its bases.

So if V is 24, the number of vertices for one base is V/2, which is 12. That means each base of the prism has 12 vertices.

 Part A you would use 200-L ( since you have 400 feet total 200 feet would equal length plus width)

Part B 200 - 80 = 120

Part C 200-90 = 110

area = 90 x 110 = 9900 square feet

The domain of the function is x ≤ 40, representing the maximum number of weeks Natalie can take to repay the loan.

The situation can be modeled by the function f(x) = 1800 - 45x, where x represents the number of weeks. The domain of this function represents the possible values for x. Since Natalie borrowed $1800 and agrees to repay $45 per week, the maximum number of weeks she can take to repay the loan is given by:

Therefore, the inequality that represents the domain of the function is x ≤ 40. This means that Natalie can take up to 40 weeks to repay the loan without exceeding the borrowed amount.

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"Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is expressed from zero to one".

For the given situation,

The sample space for rolling two cubes,

s = { (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)

(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)

(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) }

⇒[tex]n(s)=36[/tex]

The event is getting the sum 8,

[tex]e=[/tex] { [tex](2,6), (3,5), (4,4), (5,3), (6,2)[/tex] }

⇒[tex]n(e)=5[/tex]

The formula to find the probability of event, [tex]P(e)=\frac{n(e)}{n(s)}[/tex]

⇒[tex]P(e)=\frac{5}{6}[/tex]

⇒[tex]P(e)=13.8\%[/tex] ≈ [tex]14\%[/tex]

Hence we can conclude that the probability that the sum is 8 is          5/36 ≈ 14%.

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The total distance covered by her is 0.81 miles.

Alyssa is jogging near Central Park.

She runs along 65th Street for about 0.19 miles, turns right, and runs along Central Park West for about 0.28 miles.

She then turns right again and runs along Broadway until she reaches her starting point.

We have to determine

How long is her total run to the nearest hundredth of a mile?

According to the question

She runs along 65th Street for about 0.19 miles, turns right, and runs along Central Park West for about 0.28 miles.

Alyssa's route can be considered a right triangle with legs of length 0.19 and 0.28 (hundredths).

The total distance run is determined by using the Pythagoras theorem.

Pythagorean theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Therefore,

The distance run by her is,

[tex]\rm x^2 = (0.19)^2 +(0.28)^2\\\\ x^2=0.0361 + 0.784\\ \\ x^2 = .1145\\ \\ x= \sqrt{.1145} \\\\ x= 0.34[/tex]

Therefore,

The total distance covered by her is,

= .34 + .19 + .28

= 0.81 miles

Hence, the total distance covered by her is 0.81 miles.

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The solution to the equation is y = 2 + 2x.

We have,

To solve the equation 6x - 3y = -6 for y, we can isolate the term with y by performing the following steps:

Start with the equation: 6x - 3y = -6.

To isolate the term with y, subtract 6x from both sides of the equation:

6x - 3y - 6x = -6 - 6x.

This simplifies to: -3y = -6 - 6x.

Next, divide both sides of the equation by -3 to solve for y:

(-3y) / -3 = (-6 - 6x) / -3.

This simplifies to: y = 2 + 2x.

Therefore,

The solution to the equation is y = 2 + 2x.

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Final answer:

The set G is a subset of the universal set U and contains the first seven natural numbers within U: {1,2,3,4,5,6,7}.

Explanation:

In mathematics, particularly in set theory, when we talk about the set G, we are referring to a collection of elements that are all part of a larger universal set, denoted by U. In your case, the universal set U is {1,2,3,4,5,6,7,8,9,10} and the set G is given as {1,2,3,4,5,6,7}. The set G contains the first seven natural numbers within the universal set.

Understanding the concept of the universal set and subsets is fundamental in set theory. The universal set is the set that contains all possible objects (usually numbers in this context) under consideration, and any other set is a subset if all its elements are also elements of the universal set. The subset G in your example is defined by explicitly listing its members, which are all part of the universal set U.

Answer:

Saturn has about 2,000 times more mass.

Answer: Hello!

In the figure, Q and P are on opposite ends of the circle, and the same is for R and S, which means that the line that connects Q and P, or R and S, divides the circle in two equal halves. From this, we know that the angle between QoP is 180°, and the same for the angle RoS = 180°.

We also know that the angle RoP = 125°, then the angle between Q and R is the same as the angle between Q and P minus the angle between R and P.

this is : QoR = QoP - RoP = 180° - 125° = 55°

then QoR and RoP are supplementary angles, wich means that the addition adds up to 180°.

And is easy to see that the angles SoQ and PoS are reflexes of RoP and QoR respectively, then:

SoQ = 125° and PoS = 55°, where this angles also are supplementary.

Answer:

A

Step-by-step explanation: