ben and david plot the locations of their houses and the park on the grid show below. they r planning to each start at home and travel in a straight line to the park. how many times greater is the distance ben travels to get to the park than the distance david travels?

An expression which is the simplest form of 4(3x + y) + (2x - 5y) + x² include the following: x² + 14x - 6y.

In Mathematics and Geometry, an expression simply refers to a type of mathematical equation which is typically used for illustrating the relationship that exist between two (2) or more variables and numerical quantities (number), without an equal to sign (symbol).

In this scenario and exercise, the simplest form of the given expression 4(3x + y) + (2x - 5y) + x² can be determined or calculated by simplifying it as follows;

4(3x + y) + (2x - 5y) + x²

12x + 4y + 2x - 5y + x²

14x + - y + x²

By rearranging, we have;

x² + 14x - 6y.

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Complete Question:

Which expression is the simplest form of 4(3x+y)+2x-5y)+x²

The two numbers given by the algebraic expressions are x = 5 and y = 2.

A system of equations is a set of two or more equations that are to be solved simultaneously. In other words, a system of equations represents multiple relationships between variables that need to be solved for at the same time. A system of equations can be solved using various methods, such as substitution or elimination. The goal of solving a system of equations is to find the values of the variables that satisfy all of the equations in the system. One common method for solving systems of linear equations is called Gaussian elimination, which involves manipulating the equations in the system to eliminate one variable at a time until a solution is found.

Let us suppose the numbers as x and y.

From the given statement the mathematical expression can be set as:

y = 2x - 8

x + y = 7

Now. substituting the value of y in equation we have:

x + (2x - 8) = 7

3x - 8 = 7

3x = 15

x = 5

Now, for x = 5 the value of y is:

y = 2(5) - 8

y = 2

Hence, the two numbers are x = 5 and y = 2.

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The complete question is:

The number y is 8 less than 2 times another number x, their sum is 7. Find the number.

A histogram would be the most effective graphic depiction to compare the distribution of the hours spent online among teenagers, young adults, and the elderly.

What is a histogram ?

A histogram is a graph that displays the distribution of a continuous variable. It divides the data into a set of intervals (called bins) and counts the number of data points that fall into each bin.

This creates a bar-like representation of the data, with the height of each bar indicating the frequency of data points in that bin. By using a histogram, we can see the shape of the distribution, as well as any patterns or trends that may exist. We can compare the distribution of the hours spent online by teenagers, young adults, and the elderly by placing the histograms of their data side by side. This will allow us to visually compare the shape, spread, and central tendency of the distributions for each group. Additionally, we can use different colors for each group to make the comparisons even clearer. This will enable us to quickly see if there are any significant differences in the hours spent online among the three groups.

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The integer number that represents an 800 ft gain in elevation is given as follows:

+800.

Integer number are numbers that can have either positive or negative signal, but are whole numbers, meaning that they have no decimal part.

For altitudes, the signs are given as follows:

Hence the integer number that represents an 800 ft gain in elevation is given as follows:

+800.

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Answer:

x<1

Step-by-step explanation:

looking at the graph, the x values are 0,-1,-2,-3,-4,-5.

therefore, all the values of x are below 1.

x<1 is the suitable inequality that matches the graph.

The function y = √x+2 is undefined for any value of x that makes the expression inside the square root negative, which in this case is x ≤ -2.

What is an inequality equation?

An inequality equation is a mathematical statement that compares two expressions using an inequality symbol such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).

The function that is undefined for x = 0 is y = √x+2.

When x = 0, we have:

y = √0+2

y = √2

Since there is no real number whose square is negative, the square root of a negative number is undefined in real numbers.

Therefore, the function y = √x+2 is undefined for any value of x that makes the expression inside the square root negative, which in this case is x ≤ -2.

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Answer:

Step-by-step explanation:

Area= pi times radius²

The radius in the last one is 7, so 7² is 49.

Multiply 3.14 by 49 and you get 153.86. Round it to the nearest tenth and you get 153.9 as the area

Hope this helps!

The ratio of whites to blacks is 20 to 3 ( or 20/3) in United States in 1990.

From a survey of the population of the United States, a circle graph (or, pie chart) was created on the information,

Population of the United States in 1990 :

White consists of 80%

American Indian, Eskimo, and Aleut consists of 1%

Black consists of 12%

Asian and Pacific Islander consists of 3%

Other Races consists of 4%

Let say the total population of the United States in 1990 was x.

By applying the formula of percentage to numbers we get,

The number of people who were white in 1990 in United States = 80% of x

= [tex][\frac{80}{100} ]x[/tex] = [tex]\frac{4}{5} x[/tex]

The number of people who were black in 1990 in United States = 12% of x

= [tex][\frac{12}{100} ]x[/tex] = [tex]\frac{3}{25} x[/tex]

Thus, we can calculate ratio as,

Ratio of whites to blacks = [tex]\frac{\frac{4}{5} x}{\frac{3}{25} x}[/tex] = 20/3

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The situation is modeled by the linear equation:

t = $5*x

We know that Brenda earns $5 each week for chores. The variable x represents the number of weeks brenda did her chores, and t represents the total dollars she earned.

Then if she works for x weeks, the amount that she earns is x times $5, then the equation will be a linear one:

t = $5*x

That is the equation.

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The true expression about the account is D.) It is the sum of the initial amount and the additional amount after d days.

The answer is D because it represents the sum of the initial amount and the additional amount after d days. The initial amount is $10 and Owen adds $0.50 to the account each day for a number of days, represented by d.

By subtracting 1 from d, the expression ensures that on the first day (when d=1), only the initial $10 is counted. On each subsequent day, the additional amount of $0.50 is added to the total. Therefore, the expression 10+0.5(d−1) gives the amount of money in Owen's account after d days.

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Answer:  The solution to the inequality is x is less than or equal to 4. We can write it in interval notation as (-∞, 4].

Step-by-step explanation: To solve the inequality 1/2x + 8 ≤ 10, we need to isolate the variable x on one side of the inequality sign.

Subtracting 8 from both sides, we get:

1/2x ≤ 2

To isolate x, we need to multiply both sides of the inequality by 2:

x ≤ 4

The circumference of the larger circle is approximately 9.4 units.

The circumference is the distance around the edge or boundary of a closed two-dimensional shape, such as a circle or an ellipse.

Let C be the circumference of the larger circle. Then the circumference of the smaller circle is 0.25C, according to the given information.

We know that the circumference of a circle is given by the formula C = 2πr, where r is the radius of the circle. Therefore, we can write:

C = 2πr for the larger circle

0.25C = 2πr for the smaller circle

Dividing the second equation by 0.25, we get:

C = 8πr for the smaller circle

Now we can equate the two expressions for C and solve for r:

2πr = 8πr

6πr = C

r = C/(6π)

Finally, we can substitute this value of r into the expression for C in terms of r for the larger circle:

C = 2πr = 2π(C/(6π)) = (1/3)Cπ

Solving for C, we get:

C = (3/π)C

C = 3π ≈ 9.4 (rounded to the nearest tenth)

Therefore, the circumference of the larger circle is approximately 9.4 units.

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The rate of the plane in still air is 810 km/h and the rate of the wind is 90 km/h.

Explain the term rate

Rate is a measure of the speed or frequency of change of a quantity over time or space. It is expressed as a ratio between two quantities, such as distance travelled per unit of time or number of occurrences per unit of time. Rates are commonly used in mathematics, science, and economics to analyze and compare different phenomena.

According to the given information

Let's let x be the rate of the plane in still air and y be the rate of the wind. When flying against the wind, the plane's effective speed is (x - y) km/h, and when flying with the wind, its effective speed is (x + y) km/h.

From the information given in the problem, we can set up two equations to represent the plane's distance travelled when flying against and with the wind:

6390 = 9(x - y)

2970 = 3(x + y)

Solving for x and y in these equations, we find that x = 810 and y = 90. So the rate of the plane in still air is 810 km/h and the rate of the wind is 90 km/h.

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Answer:

Part A:

To find the total length of the two sides of the triangle, we need to add Side 1 and Side 2:

Side 1: 3x² - 4x - 1

Side 2: 4x - x² + 5

Total length: (3x² - 4x - 1) + (4x - x² + 5)

Simplifying and combining like terms, we get:

Total length: -x² + 7x + 4

Therefore, the total length of the two sides of the triangle is -x² + 7x + 4.

Part B:

To find the length of the third side of the triangle, we need to subtract the sum of Side 1 and Side 2 from the perimeter of the triangle:

Length of Side 3 = Perimeter - (Side 1 + Side 2)

Length of Side 3 = (5x³ - 2x² + 3x - 8) - (3x² - 4x - 1 + 4x - x² + 5)

Simplifying and combining like terms, we get:

Length of Side 3 = 2x³ - 2x² + 7x - 12

Therefore, the length of the third side of the triangle is 2x³ - 2x² + 7x - 12.

Part C:

The answers for Part A and Part B show that the polynomials are closed under addition and subtraction. The sum of Side 1 and Side 2 is a polynomial (-x² + 7x + 4), and the difference between the perimeter of the triangle and the sum of Side 1 and Side 2 is also a polynomial (2x³ - 2x² + 7x - 12). In both cases, the result is a polynomial, which means that the set of polynomials is closed under addition and subtraction.

Answer:0.125

Step-by-step explanation:

The amount (coupon interest) that is earned off the $1,000 bond with 4.3% coupon every 6 months is $21.50.

The coupon interest represents the periodic payment received by a bondholder from the date of issuance until the date of maturity of a bond.

The coupon interest is computed by applying the coupon rate on the face value of the bond.

Face value of the bond = $1,000

Coupon interest rate = 4.3%

Maturity period = 30 years

Coupon (bond) interest for 6 months = $21.50 ($1,000 x 4.3% x 6/12)

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The C-intercept of the equation means that she will have to pay $15 per day for renting the car.

From the question, we have the following parameters that can be used in our computation:

C=0.32m+15

In the equation C=0.32m+15, the C-intercept represents the fixed cost of renting a car per day, which is $15 in this case.

This means that regardless of how many miles Janelle drives, she will have to pay $15 per day for renting the car.

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The x-coordinates of the points on the curve where the tangent line is horizontal will be option A.

(x² + y²)² = 4(x² – y²)

x⁴ + 2x²y² + y⁴ = 4x² - 4y²

y⁴ + 2x^2y² + 4y² - 4x² = 0

Now we can use implicit differentiation to find the derivative of y with respect to x:

4y³(dy/dx) + 4xy² + 4y(dy/dx)x² + 8y(dy/dx) - 8x = 0

Simplifying this expression, we get:

4y³(dy/dx) + 4xy² + 4xy²(dy/dx) + 8y(dy/dx) - 8x = 0

(dy/dx)(4y³ + 4x²y²) + 8y(dy/dx) = 8x - 4xy²

(dy/dx)(4y³ + 4x^2y² + 8y) = 4x(2-y²)

(dy/dx) = 4x(2-y²)/(4y³ + 4x²y² + 8y)

Based on the information, x-coordinates of the points on the curve where the tangent line is horizontal will be option A.

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Answer:

13

Step-by-step explanation:

First of all riley saved 200.73$ she needs 1,290.78$ so,

1290.78$-200.73$=1090.95$

she earns 83.85$ a week

then, 1090.95÷83.85= 13

13 weeks

$[tex]240[/tex] is saved by Isaiah from his allowance within the space of [tex]20[/tex] weeks.

The link between two or more variables is depicted using an equation. Mathematical operations may be used in equations.

Isaiah is saving  $[tex]12[/tex] per week from his pay.

Amount of money saved by Isaiah  in [tex]20[/tex] weeks

Money saved [tex]=[/tex] $[tex]12[/tex] ×[tex]20[/tex]

Money saved = $[tex]240[/tex]

Therefore, $[tex]240[/tex] is saved by Isaiah from his allowance within the space of [tex]20[/tex] weeks.

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the calculated total savings matches the given total savings of $240, we can say that the statement is true

What is the arithmetic operation?

The four fundamental operations of arithmetic are addition, subtraction, multiplication, and division of two or more quantities. Included in them is the study of numbers, especially the order of operations, which is important for all other areas of mathematics, including algebra, data management, and geometry. The rules of arithmetic operations are required in order to answer the problem.

To check whether this statement is true or false, we can use the following formula:

Total Savings = Savings per Week × Number of Weeks

Substituting the given values, we get:

Total Savings = $12/week × 20 weeks = $240

Since the calculated total savings matches the given total savings of $240, we can say that the statement is true.

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There are a total of 2000 workers in​ all in the company

From the question, we have the following parameters that can be used in our computation:

Women percentage = 80%

People not women = 400

Using the above as a guide, we have the following:

People not women percentage = 1 - 80%

Evaluate

People not women percentage = 20%

So, we have

People not women = People not women percentage * Total

substitute the known values in the above equation, so, we have the following representation

20% * Total = 400

Divide

Total = 2000

Hence. there are 2000 people in the company

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Answer:

C or D it’s hard to see if the bottom of the shape is a square or rectangle.

Step-by-step explanation:

If rectangle C if squared D

I would go with C

The surface area is 653.1 square centimeters.

What is surface area?

A three-dimensional object's surface area is the space it takes up when viewed from the outside.

Here the given inner cylinder diameter d = 7m then

Radius r= d/2 = 7/2 = 3.5m

Now outer cylinder diameter = 9 cm then

Radius R= d/2 = 9/2 = 4.5 cm.

Height h = 26 cm.

Now using surface area of cylinder formula then,

=> Surface area A = [tex]\pi (R^2-r^2)h[/tex] square unit.

=> A = [tex]3.14(4.5^2-1.5^2)26[/tex]

=> A = [tex]3.14\times(20.25-12.25)\times26[/tex]

=> A = 3.14×8×26

=> A = 653.1 square centimeters.

Hence the surface area is 653.1 square centimeters.

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The requried  [tex]\int_a^{oo} g(x) dx[/tex] must be convergent. Option C is correct.

Since f(x) is greater than or equal to g(x) for all x, we know that:

[tex]\int_a^{oo} f(x) dx\geq\int_a^{oo} g(x) dx[/tex]

If [tex]\int_a^{oo} f(x)[/tex]dx is convergent, then by comparison test, [tex]\int_a^{oo} g(x) dx[/tex] must also be convergent. Therefore, the correct answer is (C) g(x)dx is convergent.

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The original numbers multiplied by 10² are:

968

0.002

3.33

When a number is multiplied by 10², the position of the decimal point will move to the right by 2 steps. If the number does not have decimal point, two zeros is added to the back of the number .

6 * 10³  = 6,000

968 * 10² = 96,800

0.002 * 10² = 0.2

70,000 ÷ 10² = 700

3.33 * 10² = 333

Thus, the original numbers multiplied by 10² are 968, 0.002 and 3.33

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We can see here that from the available information it is clear that imports of Benin were:

While the exports are:

A locally produced commodity or service that is sold to another nation is referred to as an export. To put it another way, it's a good or service that is sold or sent to another nation to be utilized or consumed there.

Exports may help a nation's economy expand by bringing in money and creating jobs.

Imports are actually known to be goods and services that are brought into a country/nation.

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The two solutions to the simultaneous equations are; y1 = -0.87  and y2 = 2.87. x1 = 2.87, x2 =  -0.87

To solve these simultaneous equations, we can use the method of substitution. We can rearrange the second equation to isolate one of the variables, for example, y:

y = 2 - x

Now, substitute this expression for y into the first equation:

x^2 + (2 - x)^2 = 9

Expand the equation:

x^2 + (4 - 4x + x^2) = 9

Combine like terms:

2x^2 - 4x + 4 = 9

Move all terms to one side to set the equation to equal zero:

2x^2 - 4x - 5 = 0

Now, we can solve this quadratic equation using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

In our equation, a = 2, b = -4, and c = -5:

x = (4 ± √((-4)^2 - 4(2)(-5))) / (2 x 2)

x = (4 ± √(16 + 40)) / 4

x = (4 ± √56) / 4

The two possible values for x are:

x1 ≈ (4 + 7.48) / 4 = 2.87

x2 ≈ (4 - 7.48) / 4 = -0.87

Now, we'll find the corresponding y values using the equation y = 2 - x:

For x1 ≈ 2.87:

y1 ≈ 2 - 2.87 ≈ -0.87

For x2 ≈ -0.87:

y2 ≈ 2 - (-0.87) ≈ 2.87

So, the two solutions to the simultaneous equations are approximately (2.87, -0.87) and (-0.87, 2.87), both correct to 2 decimal places.

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The possible values of the random variable P are 0, 1, or 2.

A triangle is a polygon with three straight sides and three angles. It is one of the most fundamental shapes in geometry and is used extensively in mathematics, science, engineering, and art. The sum of the interior angles of a triangle is always 180 degrees, and the length of the sides and the size of the angles determine the shape of the triangle.

Triangles can be classified according to the length of their sides and the size of their angles. A scalene triangle has three unequal sides and three unequal angles. An isosceles triangle has two equal sides and two equal angles. A equilateral triangle has three equal sides and three equal angles, each measuring 60 degrees. Triangles can also be classified based on the size of their angles: an acute triangle has three acute angles (less than 90 degrees), a right triangle has one right angle (90 degrees), and an obtuse triangle has one obtuse angle (greater than 90 degrees).

Triangles have many important properties and applications in geometry. For example, the Pythagorean theorem is a fundamental relationship between the sides of a right triangle, which states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The area of a triangle can be calculated using the formula 1/2 x base x height, and this formula can be used to solve many practical problems involving triangles.

The random variable P can take on four possible values, depending on how many heads are obtained:

P = 0 if both coins show tails.

P = 1 if exactly one of the coins shows heads.

P = 2 if both coins show heads.

Therefore, the possible values of the random variable P are 0, 1, or 2.

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F(x) is an antiderivative of f(x) because F'(x) = f(x)

An anti-derivative is the integral of a function

To verify that F is an antiderivative of f where

F(x) = √(3x² - 4) and f(x) = 3x/√(3x² - 4)

This means that F(x) = ∫f(x)

This implies that F'(x) = f(x)

So, we verify as follows

F(x) = √(3x² - 4)

Taking the derivative with respect to x, we have that

dF(x)/dx = d√(3x² - 4)/dx

Let u = 3x² - 4

So, dF(x)/dx = d√u/dx

= d√u/du × du/dx

= 1/(2√u) × du/dx

Now,du/dx = d(3x² - 4)/dx

= 6x

So,

dF(x)/dx  = 1/(2√u) × du/dx

= 1/(2√(3x² - 4)) × 6x

= 1/(√(3x² - 4)) × 3x

= 3x/(√(3x² - 4))

= f(x)

So, F(x) is an antiderivative of f(x) because F'(x) = f(x)

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In linear equation, 170 adult and 380 children's tickets were sold.

What is a linear equation in mathematics?

A linear equation in algebra is one that only contains a constant and a first-order (direct) element, such as y = mx b, where m is the pitch and b is the y-intercept.

                         Sometimes the following is referred to as a "direct equation of two variables," where y and x are the variables. Direct equations are those in which all of the variables are powers of one. In one example with just one variable, layoff b = 0, where a and b are real numbers and x is the variable, is used.

It usually works well to let a variable represent the higher-value item in the mix. Here, we can let 'a' represent the number of adult tickets sold. Then the total revenue is ...

 7a + 2(550 -a) = $1950

  7a + 1100 - 2a = 1950

     5a =  1950 -  1100

      5a = 850

        a = 170

 c = 550 - 170  = 380

170 adult and 380 children's tickets were sold.

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