A heavy rope, 60 ft long, weighs 0.8 lb/ft and hangs over the edge of a building 160 ft high. How much work W is done in pulling the rope to the top of the building?

Considering that the larger triangle is the image of the smaller triangle after a dilation whose center is at (-2,1), the scale factor that was used in the dilation is 2.

Dilation is a technique for creating comparable figures that have their size reduced or increased by certain scaling factors.

Hence, the dilation of a triangle involves decreasing or increasing their sizes by a scale factor.

Considering the larger triangle which is the image of the smaller triangle after a dilation with center at (-2,1);

Scale factor = 5 - 1 / 1 - (-1)

Scale factor = 4 / 2

Scale factor = 2

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Answer:

given points are(-5,-6)and(-3,0),

m=0-(-6)/-3-(-5)

=6/2

=3

Hence,For every +1 unit increase in X axis results in +3 units increase in y axis.

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Answer:

median = 4.5

Step-by-step explanation:

the median is the middle value of the data set arranged in ascending order. If there is no exact middle value then it is the average of the values either side of the middle

express the values in ascending order ( smallest to largest )

2 , 3 , 4 , 5 , 6 , 8

           ↑

the middle is between 4 and 5 , then

median = [tex]\frac{4+5}{2}[/tex] = [tex]\frac{9}{2}[/tex] = 4.5

Dividing the quantities of whole recipe by 4 it can be made for one person.

Multiplication is the opposite of division. If 3 groups of 4 add up to 12, then 12 divided into 3 groups of equal size results in 4 in each group. Creating equal groups or determining how many people are in each group after a fair distribution is the basic objective of division.

Any number must have a divisor that is less than the dividend when we divide it. Every time, deduct the product from the dividend number before moving on to the next number in the sequence. Always write the product below the dividend number and the quotient above the line.

Dividing the quantities of whole recipe by 4 -

9/16 cups of pastry dough

5/8 cups vegetables, chopped

1/16 of a cup of butter

salt and pepper to taste

3/8 cups of chicken

*serves one

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Answer:The slope-intercept form is y=mx+b y = m x + b , where m m is the slope and b b is the y-intercept. Add 10 10 to both sides of the equation. Using the slope-intercept form, the y-intercept is 10 .

Step-by-step explanation:

Answer:

China = 6018 / 1339 = 4.49

US = 5093 / 307 = 16.59

Russia = 1704 / 140 = 12.17

India = 1293 / 1157 = 1.12

Japan = 1246 / 127 = 9.81

Germany = 858 / 82 = 10.46

Canada = 614 / 33 = 18.60

Since 1214 is a four digit number and you have fewer than 10 blocks it is not possible to represent 1214 in any other way.

With fewer than 10 place value blocks, it would not be possible to show the number 1214 using standard place value notation, as you would need at least 4 blocks to represent the ones, tens, hundreds, and thousands place. Each block would represent a different place value, such as ones, tens, hundreds, and thousands. Each block would have a different numerical value, such as 1, 2, 1, and 4, respectively. Since 1214 is a four digit number and you have fewer than 10 blocks it is not possible to represent 1214 in any other way.

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The expression -3c+ c² - 2 + c² + 5c +3  is simplified to 2c² + 2c - 1

Algebraic expressions are described as mathematical expressions that are comprised of coefficients, terms, factors, constants and variables.

These expressions are known to be made up of mathematical operations, which incudes;

From the information given, we have that;

-3c+ c² - 2 + c² + 5c +3

Now, collect like terms, we get;

c² + c² - 3c + 5c - 2 + 3

Now, add or subtract the like terms, we get;

2c² + 2c - 1

Hence, the expression is 2c² + 2c - 1

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Each system of equations given below has a unique solution 3x−y=13y+3=9x, y+4x=7−2y−4=8x, 4y=3x+48y−6x=8.

A mathematical equation can be defined in a variety of ways. The definition of an equation in algebra is a mathematical statement that proves two mathematical expressions are equal. Consider the equation 3x + 5 = 14, where 3x + 5 and 14 are two expressions that are separated by the symbol "equal." Mathematical variables are used in even the most fundamental and straightforward algebraic equations.

Given that;

The system of equations is,

⇒ 3x - y = 13

⇒ y + 3 = 9x

Now,

Since, The system of equation is,

⇒ 3x - y = 13

⇒ y + 3 = 9x

It can be written as,

⇒ 3x - y = 13

⇒ 9x - y = 3

Clearly, We have;

⇒ 3/9 = - 1 / - 1

⇒ 1/3 ≠ 1

Thus, It has a Unique solution.

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According to number line, x > 5 is correct,

A horizontal line with evenly spaced numerical increments is referred to as a number line. How the number on the line can be answered depends on the numbers present. The use of the number is determined by the question that it corresponds to, such as when plotting a point.

Here,

Given :

∵ The inequality is -3(2x - 5) < 5(2 - x)

At first simplify each side

∵ -3(2x - 5) = -3(2x) + -3(-5)

Remember (-)(-) = (+)

∴ -3(2x - 5) = - 6x + 15

∵ 5(2 - x) = 5(2) + 5(-x)

Remember (+)(-) = (-)

∴ 5(2 - x) = 10 - 5x

∴ - 6x + 15 < 10 - 5x

Subtract 15 from both sides

∴ - 6x < -5 - 5x

Add 5x to both sides

∴ - x < - 5

∵ The coefficient of x is -1

∴ Divide both sides by -1

∴ x > 5

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As per the unitary method, the cost of bag is $93.3 and the cost of pen is $64.45

The term unitary method in math is known as the process of solving the  the value of a single unit, and based on this value we can find the value of the required number of units.

Here we have given that the cost of pen and bag is $153.75.

Here we also know that the bag cost $28.85 more than the pen.

Let us consider that x be the cost of pen and here we know that the cost of bag is $28.85 more than the pen then it can be written as,

=> x + 28.85

Therefore, the equation is written as,

=> x + (x + 28.85) = 153.75

Now, we simplify the equation, then we get,

=> 2x + 28.85 = 153.75

Here we need to find the value of x, so, we have to move the other terms on the other side, then we get

=> 2x = 153.75 - 28.85

=> 2x = 124.9

Therefore, the value of x is 64.45.

Therefore, the cost of bag is calculated as,

=> x + 28.85

=> 64.45 + 28.85

=> 93.3

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Based on the information in the bar graph, the number of students of the next 140 asked will select football rather than track and field as their favorite sport is, 35

The percentage is a mathematical term, which means a number or a ratio which is represented in fractions of 100. We use the '%' symbol to represent the percentage.

Formula for percentage;

Percentage = (present quantity/whole quantity)×100

Given that,

Mike asked 200 students to select their favorite sport and then recorded the results in the bar graph

the percentage of students who selects football is,

Percentage = (50/200)×100

Percentage =  25%

If we consider in same percentage students will select football in next 140 students,

then,   number of those students will be,

25 = (a/140)×100

a = 35

Therefore, the number of students is, 35

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2 with an exponent of 7 is 128. It means 2 multiplied by itself 7 times.

When a number is raised to a certain power, it means that the number is multiplied by itself that many times. For example, 2 with an exponent of 7 means 2 multiplied by itself 7 times.

So, 2 with an exponent of 7 is 128.

It's important to note that the exponent is also known as the power, so 2^7 is also read as 2 to the power of 7.

Exponents are a way of shorthand for repeated multiplication. Understanding this concept is important for solving mathematical problems, especially in algebra and calculus.

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The function whose vertex is on the y-axis is f(x) = (x – 2)(x + 2).

Option C is the correct answer.

A function is a relationship between inputs where each input is related to exactly one output.

Example:

f(x) = 2x + 1

f(1) = 2 + 1 = 3

f(2) = 2 x 2 + 1 = 4 + 1 = 5

The outputs of the functions are 3 and 5

The inputs of the function are 1 and 2.

We have,

Write each of the functions in the form of ax² + bx + c

Vertex of the function.

= ( -b/2a, f(-b/2a) )

Now,

f(x) = (x – 2)²

x² - 2x + 4

a = 1, b = -2, c = 4

-b/2a = 2/2 = 1

f(1) = -2

Vertex = (1, -2)

f(x) = x(x + 2)

x² + 2x

a = 1, b = 2, c = 0

-b/2a = -2/2 = -1

f(-1) = -1(-1 + 2) = -1

Vertext = (-1, -1)

f(x) = (x – 2)(x + 2)

x² + 2x - 2x + 4

x² + 4

a = 1, b = 0, c = 4

-b/2a = 0

f(0) = (0 -2)(0 + 2) = -4

Vertex = (0, -4)

f(x) = (x + 1)(x – 2)

x² - 2x + x - 2

x² -  x + 2

a = 1, b = -1, c = 2

-b/2a = 1/2

f(1/2) = (1/2 + 1)(1/2 - 2) = 3/2 x -3/2 = -9/4

Vertex = (1/2, -9/4)

Thus,

The vertex (0, -4) is on the y-axis.

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A triangle is said to be congruent with another triangle if its three sides are all congruent. If JK XY, KL YZ, and JL XZ are true, then JKL XYZ is true.

Two triangles are said to be congruent if their three sides are identical to those of another triangle. If JKXY, KLYZ, and JLXZ are true, then JKLXYZ.

Do this Move the orange pointer to M. The two angles JKM and LKM in the preceding diagram create a linear pair. The two non-common legs of the straight line segment JKL are supplementary because they are adjacent and because they always add up to 180 degrees.

It is impossible for the two triangles to be created here since the sum of the two angles is already 18 1 181 circ 181, hence neither J K L triangle (JKL) nor X Y Z triangle (XYZ) can be formed.

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Answer:

1

Step-by-step explanation:

Anything (besides 0) to the 0 power is 1.

To break even or make money, the company should charge at least $15 for it's product but no more than $50.

We are given the function that shows the company's profit or loss as;

f(x) = -0.5x² + 32.5x - 375

This function represents a company's profit (or loss) in thousands of dollars.

x represents what it charges for it's product, x in dollars.

Thus, for it to break even, we have to equate f(x) to zero and solve for x to know the maximum and minimum values of x.

Using quadratic formula, we have;

x = [-b ± √(b² - 4ac)]/2a

x = [-32.5 ± √(32.5² - 4(-0.5 * -375))]/2(-0.5)

x = 15 or 50

Thus, the break even charge has a minimum of $15 and a maximum of $50

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Answer: yes

Step-by-step explanation:

The ratio A:D = 1:64

A ratio is the comparison of two amounts of the same unit which shows the amount of one present in the other.

Given that A:B = B:C = C:D if C:D is 1:4, we need to find A:D.

So, we proceed as follws

A:B = B:C = C:D

⇒ A/B = B/C = C/D = 1/4

So, we have that the equation

A/B = 1/4

This implies that

A = B/4

Also, we have that the equation
B/C = 1/4

This implies that

B = C/4

Finally, we have that

C/D = 1/4

This implies that

C = D/4

Since A = B/4, B = C/4 and C = D/4,

Substituting C into B, we have that

B = C/4

= (D/4)/4

B = D/16

Since A = B/4, substituting this B into A, we have that

A = B/4

= (D/16)/4

A = D/64

So, A/D = 1/64

⇒ A:D = 1:64

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Answer:

Step-by-step explanation:

Answer:

y=7

Step-by-step explanation:

Answer: y=7

Step-by-step explanation:

15-8=7

y=7

Problem 4

I recommend to draw a number line. Place the baseball park at 0 on the number line.

Then place Luke's house at -3. The distance from -3 to 0 is 3 units, to represent the 3 miles.

Place Logan's house at 2 on the number line.

To go from -3 to 2, move 5 units to the right. This is the longest distance possible because each person's house is on opposite sides of the baseball park.

To get the shortest distance possible, we'll place each person's house to the right of 0. This time each person's house is on the same side.

We'll move Luke from -3 to +3. Keep Logan at 2. The distance from 2 to 3 is 1 unit. This is the shortest distance possible.

Answers:

Shortest distance = 1 mile

Longest distance = 5 miles

===============================================

Problem 5

d = distance between Luke's house and Logan's house

Refer to the result of problem 4, where we found d = 1 the smallest possible value of d. And also d = 5 the largest possible value.

In other words: d is between 1 and 5, including each endpoint.

The compound inequality to set up is [tex]1 \le d \le 5[/tex]

The graph involves closed filled in circles at 1 and 5 on the number line. Shade the region between these endpoints. Closed endpoints visually tell the reader "include this endpoint".

Answers:

Inequality:  [tex]\boldsymbol{1 \le d \le 5}[/tex]

Graph: Closed filled in circles at 1 and 5; shading in between

===============================================

Problem 6

The largest distance we previously found was 5 miles.

Luke traveled 0.5 miles so far, meaning he has at most 5-0.5 = 4.5 miles left to go.

Answer:  

4.5 miles

Answer:

Q4  
Shortest distance = 1 mile
Longest distance = 5 miles

Q5
1 ≤ d ≤ 5

Q6
4.5 miles

Step-by-step explanation:

A diagram does help for these questions

Q4

If both Luke and Logan's houses are on the same straight line in the same direction then the difference between their individual distances from the field will be the shortest distance.

See the first diagram

Shortest distance = 3 - 2 = 1 mile

However, if they lived in exactly opposite directions on a straight line then the distance would be the longest and would be 2 + 3 = 5 miles

--------------------------------------------------------------------------------

Q5

The compound inequality would be 1 ≤ d ≤ 5 where d represents all possible distances from the two homes.

(We can also write this as [1, 5] which is called interval notation)

See second figure for how this would be represented on a number line

------------------------------------------------------------------------------

Q6

Since we have determined the greatest distance between the two houses as 5 miles, and Luke has already biked 0.5 mile he has 5 - 0.5 = 4.5 miles left to bike

We will find the costs for each possible buy, and see if it is smaller or larger than $600.

A) They can.

B) They can't.

C) They can.

First, we know that:

And we know that the math club must spend less than $600.

The cost of 250 rolls of wrapping paper will be:

cost = 250*$2 = $500

This is smaller than $600, so they can buy the 250 rolls of wrapping paper.

B) 250 rolls of ribbon will cost:

cost = 250*$3 = $750

This is more than the amount that they have to spend, then they can't buy 250 rolls of ribbon.

C) Here we have the case of 100 of each, the total cost will be:

cost = 100*$2 + 100*$3

cost = $200 + $300 = $500

This is less than their budget, so they can buy this.

More generally, if x = number of rolls of wrapping paper and y = number of rolls of ribbon, we can write the inequality:

x*$2 + y*$3 ≤ $600

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A closed quadrilateral has four sides, four vertices, and four angles. It is a form of polygon. In order to create it, four non-collinear points are joined. Quadrilaterals always have a total internal angle of 360 degrees.

The area of the quadrilateral is computed as follows if the diagonal and the length of the perpendiculars from the vertices are known: Area of quadrilateral = (12) diagonal length + sum of the length of the perpendiculars drawn from the remaining two vertices.

A quadrilateral is a two-dimensional shape with four sides. Quadrilaterals include the following 2D shapes: square, rectangle, rhombus, trapezium, parallelogram, and kite.

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The gradient or slope of the line is given by the equation m = -3

The equation of a line is expressed as y = mx + b where m is the slope and b is the y-intercept

And y - y₁ = m ( x - x₁ )

y = y-coordinate of second point

y₁ = y-coordinate of point one

m = slope

x = x-coordinate of second point

x₁ = x-coordinate of point one

The slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )

Given data ,

Let the equation of line be represented as A

Now , the value of A is

Let the first point be represented as P = P ( -1 , 0 )

Let the second point be represented as Q = Q ( 0 , -3 )

Now , the slope of the line passing through the points is given by

Slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )

Substituting the values in the equation , we get

Slope m = ( -3 - 0 ) / ( 0 - ( -1 ) )

On simplifying the equation , we get

Slope m = -3/1

Slope m = -3

Hence , the slope of the line is m = -3

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3*(28) + 7 can be rewritten as 316 + 7, which can be further simplified by using the associative property of multiplication and the commutative property of addition as (3*16) + 7 = 48 + 7 = 55

The commutative property of addition states that the order of the numbers being added does not matter, for example, a+b = b+a.

The associative property of multiplication states that the grouping of the numbers being multiplied does not matter, for example, (ab)c = a(bc).

Therefore, the expression that uses the commutative property of addition and the associative property of multiplication to rewrite the expression 3*(2*8)+7 is (3*16) + 7 = 48 + 7 = 55.

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Answer:

18.0864 cm^2

Step-by-step explanation:

The formula for the area of the circle is

A = 3.14 · r^2

Radius = D / 2

Radius = 4.8 / 2 = 2.4 cm

Now let's solve for the area of this circle

A = 3.14 · 2.4^2

A = 3.14 · 5.76

A = 18.0864 cm^2

So, the area of this circle is 18.0864 cm^2