Three students need to produce a prime factorization of
48. Donna states that the first factors in the tree should
be 6 and 8. Larry states that the first factors in the tree
should be 4 and 12. Trish states that the initial factors of
48 do not affect the prime factorization. Explain why Trish
is correct.

In conclusion the length of the rectangle is 319 feet.

How to find?

Let's denote the width of the rectangle as w.

According to the problem, the length of the rectangle is 154 feet more than the width, so we can express the length as w + 154.

The perimeter of a rectangle is given by the formula P = 2l + 2w, where P is the perimeter, l is the length, and w is the width.

Using the given perimeter of 968 feet, we can write:

968 = 2(w + w + 154)

Simplifying and solving for w, we get:

968 = 4w + 308

660 = 4w

w = 165

Therefore, the width of the rectangle is 165 feet.

We can find the length by using the expression we found earlier:

l = w + 154 = 165 + 154 = 319

Therefore, the length of the rectangle is 319 feet.

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Answer:

s=x-18

Step-by-step explanation:

The total of number of books they need to sell is

the total minus 18. And that will equal the amount of books they still need to sell.

Answer: x = -1

Step-by-step explanation:

Step 1: Distribute. 6 + 3x - 2x = 11 + 4 + 10x

Step 2: Subtract like-terms. x + 6 = 10x + 15

Step 3: Leave x by itself. x + (6-6) = 10x + (15-6)

Step 4: Subtract 10x to x, and simplify. (x - 10x) = (10x - 10x) + 9

Step 5: Divide. -9x/9 = 9/9

The answer is x = -1. Hope this helps

Answer:

Since the vertex is (0,-3), we know that the equation of the parabola will have the form:

y = a(x - 0)^2 - 3

where "a" is the coefficient that determines whether the parabola opens up or down.

To find the value of "a", we can use the fact that the parabola passes through the point (4, -7):

-7 = a(4 - 0)^2 - 3

-7 + 3 = 16a

-4 = 16a

a = -1/4

Substituting this value of "a" into the equation for the vertex form, we get:

y = -(1/4)x^2 - 3

So the equation of the parabola in vertex form is y = -(1/4)x^2 - 3, with vertex (0, -3) and passing through (4, -7).

Step-by-step explanation:

Using the formula for x which is x = - b/2a, we know that the vertex of the given parabola is 5 respectively.

A parabola is an approximately U-shaped, mirror-symmetrical plane curve in mathematics.

It corresponds to a number of seemingly unrelated mathematical descriptions, all of which can be shown to define the same curves.

A parabola can be described using a point and a line.

Three different parabolas exist.

Vertex form, standard form, and intercept form are the three forms.

You can access a unique key feature for the graph on each form.

So, we have the equation:

y = -2x² - 8x - 3

Now, we know that x = - b/2a

Solve for x:

x = - b/2a

x = - (-8)/2(-2)

x = 8/-4

x = -2

Now, insert x = -2 in the given equation:

y = -2x² - 8x - 3

y = -2(-2)² - 8(-2) - 3
y = -2(4) + 16 - 3

y = -8 + 16 - 3

y = 8 - 3

y = 5

Therefore, using the formula for x which is x = - b/2a, we know that the vertex of the given parabola is 5 respectively.

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According to given equations, any value of x less than 15.6 will result in the shopkeeper spending less than $100 on electricity.

What is an equation?

An equation is a mathematical statement that shows that two expressions are equal. It consists of two sides, a left-hand side and a right-hand side, separated by an equals sign (=). The left-hand side and the right-hand side of an equation may contain one or more variables, and the equation is true only for certain values of these variables.

Let E be the amount the shopkeeper spends on electricity. Then, we have:

Total monthly expenses = Rent + Electricity = 8x + 15

Rent = 3x - 7

Substituting the value of Rent in the equation of total monthly expenses, we get:

8x + 15 = (3x - 7) + E

Simplifying this equation, we get:

5x + 22 = E

Therefore, the shopkeeper spends 5x + 22 dollars on electricity.

To find the value(s) of x for which the shopkeeper spends less than $100 on electricity, we can set up an inequality:

5x + 22 < 100

Solving for x, we get:

5x < 78

x < 15.6

Therefore, for any value of x less than 15.6, the shopkeeper will spend less than $100 on electricity. To verify this, we can plug in a value of x less than 15.6 and see if the resulting amount spent on electricity is less than $100. For example, if we let x = 10, then:

5x + 22 = (5 * 10) + 22 = 72

Since 72 is less than $100, we have verified that any value of x less than 15.6 will result in the shopkeeper spending less than $100 on electricity.

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hope you understood the answer

The inverse of the graph of f is added as an attachment

To graph an inverse function from the a graph, follow these steps:

Using the above steps, we have the inverse graph added as an attachment

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The coordinates of the image if triangle ABC is translated 4 units down would be: A′(1, -6), B′(9, -6), C′(5, -2)

To translate the triangle 4 units down, we need to subtract 4 from the y-coordinate of each vertex.

Vertex A has coordinates (1, -2), so its image A' after the translation is (1, -2 - 4) = (1, -6).

Vertex B has coordinates (9, -2), so its image B' after the translation is (9, -2 - 4) = (9, -6).

Vertex C has coordinates (5, 2), so its image C' after the translation is (5, 2 - 4) = (5, -2).

Therefore, the coordinates of the vertices of triangle ABC after it is translated 4 units down are A′(1, −6), B′(9, −6), and C′(5, −2).

So, the answer is option A: A′(1, −6), B′(9, −6), C′(5, −2)..

How can you use the coordinates of the vertices of a figure to determine its image after a translation?

To determine the image of a figure after a translation, you can add or subtract the same amount from the x- and/or y-coordinates of each vertex, depending on the direction and distance of the translation.

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the concept of a sequence of transformations is a fundamental one in many different fields, as it allows us to understand how complex systems or objects can be broken down into smaller, more manageable parts that can be manipulated or analyzed in a systematic way.

What is a sequence?

The sequence is an ordered list of elements or events that follow a particular pattern or rule. In mathematics, a sequence typically refers to a set of numbers that are arranged in a specific order, with each element in the sequence being defined by a mathematical formula or recurrence relation.

A sequence of transformations refers to a series of steps or operations that are performed in a specific order to change or manipulate a given object or system. These transformations can take many different forms, depending on the context in which they are used.

For example, in mathematics, a sequence of transformations might refer to a series of algebraic manipulations that are used to simplify or solve an equation. In computer science, a sequence of transformations might refer to a series of functions or algorithms that are applied to data in order to transform it into a different format or structure.

In other fields, such as physics or chemistry, a sequence of transformations might refer to a series of physical or chemical changes that occur in a system over time. These transformations could include changes in temperature, pressure, or chemical composition, among other factors.

Therefore , the concept of a sequence of transformations is a fundamental one in many different fields, as it allows us to understand how complex systems or objects can be broken down into smaller, more manageable parts that can be manipulated or analyzed in a systematic way.

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[tex]\textit{volume of a sphere}\\\\ V=\cfrac{4\pi r^3}{3}~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ V=167\pi \end{cases}\implies 167\pi =\cfrac{4\pi r^3}{3}\implies \cfrac{3}{4\pi }\cdot 167\pi =r^3 \\\\\\ \cfrac{501}{4}=r^3\implies \sqrt[3]{\cfrac{501}{4}}=r\implies 5.0\approx r \\\\[-0.35em] ~\dotfill[/tex]

[tex]\textit{volume of a sphere}\\\\ V=\cfrac{4\pi r^3}{3}~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ V=683\pi \end{cases}\implies 683\pi =\cfrac{4\pi r^3}{3}\implies \cfrac{3}{4\pi }\cdot 683\pi =r^3 \\\\\\ \cfrac{2049}{4}=r^3\implies \sqrt[3]{\cfrac{2049}{4}}=r\implies 8.0\approx r[/tex]

An inequality which has solutions represented by the graph include the following: C. x > 4

In Mathematics and Geometry, an inequality simply refers to a mathematical relation that is typically used for comparing two (2) or more numerical data and variables in an algebraic equation based on any of the inequality symbols;

Next, we would determine the solution as follows;

5 - x > 4

-x > 4 - 5

x > 1 (False).

2x + 1 ≥ 9

2x ≥ 9 - 1

2x ≥ 8

x ≥ 4 (False).

2x - 5 > 3

2x > 3 + 5

2x > 8

x > 4 (True).

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(a) The local extrema of the polynomial are: Local maximum at x = -0.68 and Local minimum at x = -0.32

(b) There is no global minimum or maximum.

To find the local extrema and absolute extrema of the given polynomial 10x⁵ + 22x⁴ + 20x³ - 0x² + 6, we need to take the first and second derivatives of the polynomial and find the critical points.

The first derivative is:

50x⁴ + 88x³ + 60x²

The second derivative is:

200x³ + 264x² + 120x

Setting the first derivative to zero, we get:

50x⁴ + 88x³ + 60x² = 0

Factoring out x², we get:

x²(50x² + 88x + 60) = 0

Using the quadratic formula to solve for the roots of the quadratic equation 50x² + 88x + 60 = 0, we get:

x = -0.68 or x = -0.32

These are the critical points of the polynomial. To determine whether they correspond to local extrema or inflection points, we need to look at the sign of the second derivative at each point.

Substituting x = -0.68 into the second derivative, we get:

200(-0.68)³ + 264(-0.68)² + 120(-0.68) = -148.68

Since the second derivative is negative at x = -0.68, this critical point corresponds to a local maximum.

Substituting x = -0.32 into the second derivative, we get:

200(-0.32)³ + 264(-0.32)² + 120(-0.32) = 9.48

Since the second derivative is positive at x = -0.32, this critical point corresponds to a local minimum.

To find the absolute extrema, we also need to evaluate the polynomial at the endpoints of the interval we are interested in. Since the polynomial does not have any real roots, we can assume that the interval of interest is the entire real line.

Taking the limit of the polynomial as x approaches positive or negative infinity, we see that the leading term dominates and the polynomial goes to positive infinity as x approaches infinity and negative infinity.

Therefore, there is no global minimum or maximum.

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Hence, there are 43.5 square inches accessible on the award for text engraving.

A square is a geometric figure with four equal sides and four right angles. It is a particular kind of rectangle with equal-length sides. It is also possible to describe it as a regular quadrilateral with equal-length sides and right-angled corners. By multiplying one side of a square by itself, the area of the square can be calculated. A = s2 is the formula for calculating the area of a square, where A stands for the area and s for the length of one side. The perimeter of a square is calculated similarly by adding the lengths of its four sides. P = 4s is the formula for calculating a square's perimeter, where P stands for the perimeter.

given

The plaque's available engraving space is equal to the plaque's surface area less the rhombus's area in the centre.

The plaque's location is:

Area plaque is equal to 7.5 x 10 inches, or 75 square inches.

The rhombus's surface area is:

(diagonal1 x diagonal2) / 2 = (7 x 9) / 2 = 31.5 square inches is the area of a rhombus.

Thus, the following is the engraving area:

Area available equals Area plaque minus Area rhombus, or 75 minus 31.5, or 43.5 square inches.

Hence, there are 43.5 square inches accessible on the award for text engraving.

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Answer:

Step-by-step explanation:

If we input 4 into g(x) = 3(x/2), we get:

g(4) = 3(4/2) = 3(2) = 6

Therefore, the output of g(4) is 6.

Answer:

D

Step-by-step explanation:

hopes this helps .........

By answering the presented question, we may conclude that As a result, inequality the cost difference between one pencil and one rubber is £0.08.

In mathematics, an inequality is a non-equal connection between two expressions or values. As a result, imbalance leads to inequity. In mathematics, an inequality connects two values that are not equal. Inequality is not the same as equality. When two values are not equal, the not equal symbol is typically used (). Various disparities, no matter how little or huge, are utilised to contrast values. Many basic inequalities may be solved by altering the two sides until just the variables remain. Yet, a lot of factors contribute to inequality: Negative values are split or added on both sides. Exchange left and right.

To determine the cost difference between one pencil and one rubber, we must first determine the cost per item for each product.

The price of a pencil is 1.56 / 6 = £0.26.

The price of a rubber is 2.72 / 8 = £0.34.

Therefore the cost difference between one pencil and one rubber is:

£0.34 - £0.26 = £0.08

As a result, the cost difference between one pencil and one rubber is £0.08.

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Answer:

11

Step-by-step explanation:

y = -1/2x + 7

x = -8

A double negative equals a positive therefore it would be the same as 8 x 1/2 which is 4.

y = 4 + 7

y = 11

Step-by-step explanation:

y = -1/2 X -8 + 7

y = -0,5 X -8 + 7

y = 4 + 7

y = 11

The dimensions of each half rectangular brick are, length 15 cm, and height 10 cm.

Let the original dimensions of the rectangular brick be length L, breadth B, and height H.

According to the problem statement, each dimension of the original brick is 5 cm larger than the desired dimensions of each half brick.

Therefore, the dimensions of each half brick are;

(L-5)/2, (B-5)/2, and (H-5)/2.

Since the two half bricks have identical dimensions, we have:

(L-5)/2 = (B-5)/2 = (H-5)/2

Simplifying this equation, we get:

L-5 = B-5 = H-5

Therefore, L = B = H + 5.

Now, the volume of the original brick is L x B x H, and the volume of each half brick is [(L-5)/2] x [(B-5)/2] x [(H-5)/2].

Since the two half bricks have the same dimensions, their volumes are equal. Thus, we have:

(L-5)/2 x (B-5)/2 x (H-5)/2 = (L x B x H)/2

Substituting L = B = H + 5, we get:

(H+5-5)/2 x (H+5-5)/2 x (H-5)/2 = (H x H x (H+5))/2

Simplifying this equation, we get:

(H+5)/2 x (H-5)/2 x H/2 = (H x H x (H+5))/2

Cancelling out the factor of 1/2, we get:

(H+5)(H-5)H = H x H x (H+5)

Simplifying this equation, we get:

H³ - 125H = H³ + 5H²

Subtracting H³ from both sides, we get:

-125H = 5H²

Dividing both sides by 5H, we get:

H = -25 or H = 0

Since H represents the height of the brick, it must be positive. Therefore, we have H = 0, which implies that the original brick has zero height. This is not a valid solution, so we reject it.

Therefore, the only valid solution is:

H = 25 cm

Substituting this value of H in the equation L = B = H + 5, we get:

L = B = 30 cm

Therefore, the dimensions of the original brick are:

Length = L + 5 = 35 cm

Breadth = B + 5 = 35 cm

Height = H + 5 = 30 cm

And the dimensions of each half brick are:

Length = Breadth = 15 cm

Height = 10 cm

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The graphs of the exponential functions are introduced in the image attached below.

The solution to the system of two exponential functions is 35 representatives and 7.735 minutes of average wait time.

In this problem we must represent two exponential functions describing average wait time, in minutes, as a function of number of customer service representatives. The functions are described by:

A(x) = 15.7 · 0.98ˣ

B(x) = 11 · 0.99ˣ

Now we draw each of the two functions by the help of a graphing tool. The solution to the system of exponential functions is:

Number of Customer Service Representatives: 35

Average Wait Time: 7.735 minutes

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Answer:

Mean = 880.8

Median = 936

Mode = None

Step-by-step explanation:

1.

To find the mean of a list of numbers, simply add them all up and divide them by the number of numbers listed.

(655+891+949+954+923+958+622+877+986+993)/10 = 880.8

Mean = 880.8

2.

To find median, put the numbers in order and choose the one in the middle

622 665 877 891 (923 949) 954 958 986 993

Because we were given two numbers in the middle, find the mean of those two numbers

(923+949)/2 = 936

Median = 936

3.

To find mode, you find the number that is listed the most.

Since all the numbers listed are different, there is no mode.

Answer: The surface area of the rectangular prism is 140 and 3/16 square centimeters.

The volume of both book end is  1458 [tex]cm^3[/tex].

The space taken up by any three-dimensiοnal sοlid cοnstitutes a vοlume, tο put it simply. A cube, cubοid, cοne, cylinder, οr sphere can be οne οf these sοlids. Cubic units are used tο measure the vοlume οf sοlids. The vοlume will be given in cubic metres, fοr instance, if the dimensiοns are given in metres.

Here the side length of cube = 9cm

Now using volume of cube formula then,

=> Volume of cube V = [tex]v^3[/tex] cubic unit.

=> V = [tex]9^3[/tex]

=> V = 9*9*9 = 729 [tex]cm^3[/tex].

Here we had pair of book end then volume = 2V = 2*729 = 1458 [tex]cm^3[/tex].

Hence the volume of both book end is  1458 [tex]cm^3[/tex].

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Answer: 188 messages

Step-by-step explanation:

We will set up and solve a proportion.

      [tex]\displaystyle \frac{9.75}{65} =\frac{28.2}{x}[/tex]

Now, we will cross-multiply.

      9.75 * x = 65 * 28.2

      9.75x = 1,833

Lastly, we will divide both sides of the equation by 9.75.

      x = 188 messages

The tension in cable B is 400 N.

What is tension in cable ?

Tension is the force transmitted through a string, rope, cable, or wire when it is pulled taut by forces acting from opposite ends. It is the pulling force that exists between two objects that are connected by a rope or cable and is usually measured in newtons (N) or pounds (lbs). The tension in a cable or rope can be calculated using the principles of equilibrium, where the sum of the forces acting on the object is equal to zero. In other words, the tension in a cable is the force that must be applied to the cable in order to keep it taut and maintain its position or prevent it from breaking.

According to the question:
To solve this problem, we need to use the fact that the sum of the forces acting on the sign must be zero in order for the sign to be in equilibrium. We can break down the weight of the sign into its components parallel and perpendicular to the cables:

F_parallel = 800 N * sin(30°) = 400 N

F_perpendicular = 800 N * cos(30°) = 693.2 N

The tension in cable A must balance the perpendicular component of the weight of the sign, so we have:

Tension_A = 693.2 N

The tension in cable B must balance the horizontal component of the weight of the sign, so we have:

Tension_B = 400 N

Therefore, the tension in cable B is 400 N.

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You do not use lines to connect the dots.

There are 90 distinct arrangements of the letters in SPINNING.

In mathematics, permutation refers to the arrangement of objects in a specific order. A permutation of a set of objects is an ordered arrangement of those objects. For example, if we have a set of three objects {A, B, C}, then the permutations of this set would be {ABC, ACB, BAC, BCA, CAB, CBA}. The number of permutations of a set of n objects is denoted by n!. Permutations are commonly used in combinatorics, probability theory, and other areas of mathematics and science.

In the given question,

The word SPINNING has 8 letters, out of which two letters appear twice.

Let's first consider the distinct arrangements of the 6 unique letters (ignoring the repeated letters).

There are 6! ways to arrange these 6 letters in a row.

However, we need to adjust for the repeated letters. The letter "N" appears twice, so we divide by 2! to account for the fact that the two N's can be arranged in 2! = 2 ways without changing the arrangement of the other letters. Similarly, the letter "I" also appears twice, so we divide by another 2!.

Therefore, the total number of distinct arrangements of the letters in SPINNING is:

6! / (2! x 2!) = 360 / 4 = 90.

So there are 90 distinct arrangements of the letters in SPINNING.

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The measure of angle KLN in the given set of intersecting lines is 100°.

In geometry, intersecting lines are lines that cross or meet each other at a point. The point at which the lines intersect is called the point of intersection. Intersecting lines are a fundamental concept in geometry and are used to define many other geometric shapes and concepts. For example, the intersection of two lines can form an angle, and the measurement of this angle can provide information about the relative positions of the lines.

Here,

Since the sum of angles in a quadrilateral is 360 degrees, we can set up an equation:

M + 100 + (x-y) + m<KLN = 360

We also know that m<MOL = m<KLN, since they are opposite angles of a line. Therefore:

m<KLN = m<MOL

We can use the fact that the sum of angles in a triangle is 180 degrees to set up another equation:

m<MOL + m<MON + m<NOL = 180

Since m<MOL = m<KLN, we can substitute m<KLN for m<MOL:

m<KLN + m<MON + m<NOL = 180

Now we have two equations:

M + 100 + (x-y) + m<KLN = 360

m<KLN + m<MON + m<NOL = 180

We need to solve for m<KLN. Let's isolate m<KLN in the second equation:

m<KLN = 180 - m<MON - m<NOL

Now we can substitute this expression for m<KLN in the first equation:

M + 100 + (x-y) + (180 - m<MON - m<NOL) = 360

Simplifying, we get:

280 - m<MON - m<NOL + x - y = M

Rearranging, we get:

m<MON + m<NOL = 280 + x - y - M

100=x-y

y=2/7x

100=x-2x/7

x=700/5

x=140

y=2*140/7

y=40

x-y=100°

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Answer: the radius of the circle is 14 cm. :)

Step-by-step explanation:

To find,

The radius of the circle.

Solution,

The radius of the circle will be 14 cm.

We can easily solve this problem by following the given steps.

According to the question,

Length of a rectangular wire = 32 cm

The breadth of a rectangular wire = 12 cm

This wire is bent into the shape of a circle.

So,

The perimeter of the rectangle = Circumference of the circle

We know the formula for the perimeter of a rectangle is as follows:

P = 2(Length+Breadth)

P = 2(32+12) cm

P = 2(44) cm

P = 88 cm

We know that the formula to find the circumference of the circle is given as follows:

C =

2πr = 88 cm

2×22r/7 = 88

44r/7 = 88

Using the cross multiplication method,

44r = (88×7)

r = (88×7)/44

r = (2×7) cm

r = 14 cm

Hence, the radius of the circle is 14 cm.

Answer:

14.01 cm

Step-by-step explanation:

The length of the wire is equal to the perimeter of the rectangle. So, the length of the wire is:

When this wire is bent into a circle, its length becomes equal to the circumference of the circle. The formula for the circumference of a circle is:

(let r = radius)

2 × π × r

So, we can write:

Solving for r, we get:

So, the exact value of the radius of the circle formed by bending this rectangular wire is 14.01 cm.

Answer: 24 pieces of pizza

Step-by-step explanation:

If you have one pizza and cut that into 6 pieces (which is what it means by sixths) that means you have 6 pieces per pizza. Since you have 4 pizzas and 6 slices each that means you multiply 4 by 6 to get the answer. Which ends up being 24 pieces.

Mr. Davis has 24 pieces of pizza for the family dinner.

In mathematics, an expression is a combination of symbols and numbers that represent a mathematical quantity or relationship.

It can consist of variables, constants, operators, and functions, and may include arithmetic operations, exponents, and other mathematical operations.

We always combine the like terms in an expression when we simplify.

We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.

Example:

1 + 3x + 4y = 7 is an expression.com

3 + 4 is an expression.

2 x 4 + 6 x 7 – 9 is an expression.

33 + 77 – 88 is an expression.

We have,

To find the total number of pizza slices, we need to first determine the total number of slices in one pizza, and then multiply that by the number of pizzas.

Since each pizza is cut into sixths, there are 6 slices in one pizza.

So,

For 4 pizzas, the total number of slices is:

Total number of slices

= 4 x 6 = 24

Thus,

Mr. Davis has 24 pieces of pizza for the family dinner.

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