Which equation shows how to rewrite the sum of 42 and 54 using their greatest common factor?

Answer:

$3.25 per hotdog

Step-by-step explanation:

This question can be written as a system of equations:

x = price of hotdogs

y = price of drinks

First family:

4x + 4y = 18

Second family

5x + 3y = 20

Multiply each equation so that one of the variables can cancel out:

-3(4x + 4y = 18)

4(5x + 3y = 20)

First equation:

-12x - 12y = -54

Second equation

20x + 12y = 80

Add the equations together

-12x - 12y = -54

20x + 12y = 80

8x = 26

Divide

x = $3.25

Insert the x into one of the original equations to find y:

4(3.25) + 4y = 18

13 + 4y = 18

4y = 5

y = $1.25

The height of the tree to the nearest hundredth of a meter is 10.77m.

Detailed explanation:

An illustration of identical triangles is shown in the diagram. View the illustration I included with my response to see how triangles can be used to solve this issue.

Because they both share an internal angle, the triangles made by Alyssa and her shadow and the triangle created by the tree and its shadow are examples of comparable triangles.

The ratios of the comparable sides on both sides are proportionate according to the similar triangles theorem.

Therefore, in order to determine the height of the tree, we must make a percentage of the corresponding sides of both triangles. Let x equal the tree's height.

[tex]\frac{BE}{BC} =\frac{DE}{AC}[/tex]

BE = 4.25 meters

DE = 1.65 meters

BC = 27.75 meters

AC = x meters

[tex]\frac{4.25}{27.75} =\frac{1.65}{x}[/tex]

Cross multiply is now used to find x.

4.25x= 1.65(27.75)

x = 10.77

Therefore, the height of the tree is 10.77 meters.

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Complete question:

Alyssa is 1. 65 meters tall. At 11 a. M. , she measures the length of a tree's shadow to be 27. 75 meters. She stands 23. 5 meters away from the tree, so that the tip of her shadow meets the tip of the tree's shadow. Find the height of the tree to the nearest hundredth of a meter.

Answer:

Step-by-step explanation:

The formula for the circumference of a circle is C = 2 * π * r, where r is the radius of the circle. We can find the radius of the circle by rearranging the formula to r = C / (2 * π).

Substituting the given values into the formula, we get:

r = 21.98 inches / (2 * 3.14) = 21.98 inches / 6.28 = 3.50 inches

The diameter of the circle is equal to twice the radius, so we can find the diameter by multiplying the radius by 2:

d = 2 * r = 2 * 3.50 inches = 7.00 inches.

To find the diameter of a circle with a given circumference, you can use the following steps:

Define the formula for the circumference of a circle: The circumference of a circle is given by the formula C = 2πr, where π (pi) is a mathematical constant approximately equal to 3.14, and r is the radius of the circle.

Substitute the given circumference into the formula: If the circumference of the circle is 21.98 inches, we can substitute that into the formula to get 21.98 = 2πr.

Solve for the radius: To solve for the radius, we can divide both sides of the equation by 2π: 21.98 / 2π = r. Using 3.14 for π, this gives us 21.98 / (2 * 3.14) = r, which can be simplified to r = 2.01 inches.

Calculate the diameter: The diameter of a circle is twice the radius, so the diameter can be calculated as 2 * r = 2 * 2.01 = 4.02 inches.

So, the diameter of the circle with a circumference of 21.98 inches is 4.02 inches.

The intervals of each behavior of the function are given as follows:

a) Increasing: (-∞, -1).

b) Decreasing: (-1, 1).

c) Constant: (1, 3).

Function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable .

here, we have,

to classify the function as increasing, decreasing or constant:

The function is increasing when the graph moves right and up.

The function is decreasing when the graph moves right and down.

The function is constant when the graph of the function is an horizontal line.

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A. The indifference curve of a consumer is negatively sloped to the right because it represents the consumer's preference for one good over another. As the consumer's income increases, their willingness to substitute one good for the other increases.

B. The indifference curve is convex to the origin because it shows the diminishing marginal rate of substitution of the two goods. As the consumer reaches the optimal combination of goods, the marginal rate of substitution between the two goods decreases.

C. Using the Lagrange method, the optimum combination of goods can be determined as follows:

Let L(X,Y) be the Lagrangian of this problem:

L(X,Y) = √X² + Y² - λ(200 - 3X - 4Y)

Where λ is the Lagrange multiplier.

The optimal solution is obtained by setting the derivatives of L with respect to X and Y to 0 and solving the equations simultaneously.

∂L/∂X = 0 -> X = 11.85

∂L/∂Y = 0 -> Y = 8.57

Therefore, the optimum combination of goods is X = 11.85 and Y = 8.57.

D. The slope of the indifference curve is equal to the slope of the budget line. This can be shown by comparing their respective equations and noting that their slopes are the same. The equation for the indifference curve is Y/X = (-2/3) and the equation for the budget line is Y/X = (-3/4). Since these equations are the same, it follows that the slopes of the two lines are equal.

Answer: $272.60

Step-by-step explanation:

9.75x5.4= 52.65

52.65+15.50=68.15

68.15x4=272.6

Answer:

48ft²

Step-by-step explanation:

Visualize the trapezoid as consisting of a rectangle with two triangles on either side. To find its total area, find the area of each and add them up.

Note that the formula for the area of a square is (base*height) and the formula for the area of a triangle is (0.5*base*height). The bottom of the trapezoid is 15ft and the top is 9ft, which makes the combined bases of the two triangles 15-9=6ft, meaning each triangle's base is 3ft.

Area of left triangle = 0.5*3*4 = 6ft²

Area of square = 4*9 = 36ft²

Area of right triangle = 0.5*3*4 = 6ft²

Therefore the total area is: 6ft² + 36ft² + 6ft² = 48ft²

Answer:

48ft?

Visualize the trapezoid as consisting of

a rectangle with two triangles on either

side. To find its total area, find the area of

each and add them up.

The perimeter and area of the original square are 4s and s², respectively, and the perimeter and area of the dilated square are 192 and 2304, respectively.

What is area of square ?

Area of square can be defined as the square of given side.

Given ,

Let's assume that the side length of the original square is 's'. The perimeter of the original square is then 4s and its area is s².

The perimeter of the dilated square will be 4 times longer than the original square, so it will be 4 * 4s = 16s.

The area of the dilated square will be 4 times greater than the original square, so it will be 4^2 * s² = 16s².

Here s is 12

so ,

16s = 16 * 12 = 192

16s*s = 16 * 12 * 12 = 2304

Therefore, The perimeter and area of the original square are 4s and s², respectively, and the perimeter and area of the dilated square are 192 and 2304, respectively.

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Answer:

a= 12, b=4, Choices B and F

Step-by-step explanation:

2/2=1

8/2=4

12/2=6

Step-by-step explanation:

Since every single mile it descends the same amount (which is 500ft), that means it's linear, so we'll use that they gave us, the h(x) = mx+b

To find m, we use the following equation

[tex] \frac{y2 - y1}{x2 - x1} [/tex]

to find b, we have to find out the starting point. luckily they gave it to us, which is 4,000ft. B = 4000

Now every 500 feet, we go down 1 mile

Which means at 1000 feet, we go down 2 miles

500 and 1000 can be our y1 and y2

1 and 2 can be our x1 and x2

[tex] \frac{1000 - 500}{2 - 1} = \frac{500}{1} = 500 = m[/tex]

Now that we have our m and b, we just put it together

[tex]h(x) = - 500x + 4000[/tex]

A negative was placed before the 500 because they are descending, not ascending.

The same shape and similar are relationship between ΔXYZ and ΔUVW options (A) and (D) are correct.

What is the similarity law for triangles?

It is defined as the law to prove that two triangles have the same shape, but it is not compulsory to have the same size. The ratio of the corresponding sides is in the same proportions and the complementary angles are congruent.

We have given two triangles with dimensions.

Take the ratio of the corresponding sides:

12/24 = 5/10 = 13/26 = 1/2

The ratios of the corresponding sides are same which means the triangle are similar and having same shape.

Hence, the same shape and similar are relationship between ΔXYZ and ΔUVW option (A) and (D) are correct.

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The complete question is in the attached image.

The x-components  and y-components of the vector [tex]\mathbf{\hat v}[/tex] are -3.0 cm/s and 0 cm/s, respectively.

In mathematics, an axis is a reference line used to locate points in space. It is often used to define a coordinate system, which is a system for representing points and geometric shapes in space using numbers or coordinates.

To find the x- and y-components of the vector [tex]\mathbf{\hat v}[/tex] = (3.0 cm/s, −x-direction), we need to determine the direction of the vector. The notation "-x-direction" means that the vector points in the opposite direction of the positive x-axis. Therefore, the angle between the vector and the negative x-axis is 180 degrees.

The magnitude (or length) of the vector is given by the first component, which is 3.0 cm/s.

The x-component of the vector can be found by multiplying the magnitude by the cosine of the angle:

x-component = magnitude x cos(angle) = 3.0 cm/s x cos(180°) = -3.0 cm/s

The negative sign indicates that the x-component is in the opposite direction of the positive x-axis.

The y-component of the vector can be found by multiplying the magnitude by the sine of the angle:

y-component = magnitude x sin(angle) = 3.0 cm/s x sin(180°) = 0 cm/s

Therefore, the x-components  and y-components of the vector [tex]\mathbf{\hat v}[/tex] are -3.0 cm/s and 0 cm/s, respectively.

Expressed as an ordered pair, the components are (-3.0 cm/s, 0 cm/s).

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Answer:

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

Given inequality:

Consider x ≠ 0 and multiply all terms by x, 4 and 3:

The x-intercepts are:

This quadratic function has a positive leading coefficient and two zeros, and hence is positive when:

The matching choice is C.

The new triangle with vertices A'(0, 6), B'(12, -6), and C'(6, -6) would be larger than the original triangle by a factor of 120%.

A point is transformed when it is moved from where it was originally to a new location. Translation, rotation, reflection, and dilation are examples of different transformations.

To graph a polygon with vertices A(0,5), B(10,-5), and C(5,-5), we can plot these points on a coordinate plane and connect them to form a triangle.

Then, to find the image of this triangle after a dilation with scale factor k = 120%, we need to multiply each coordinate of the vertices by k = 1.2.

For vertex A, the new coordinates would be (0 × 1.2, 5 × 1.2) = (0, 6).

For vertex B, the new coordinates would be (10 × 1.2, -5 × 1.2) = (12, -6).

For vertex C, the new coordinates would be (5 × 1.2, -5 × 1.2) = (6, -6).

So, the new triangle with vertices A'(0, 6), B'(12, -6), and C'(6, -6) would be larger than the original triangle by a factor of 120%.

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A. The value of cos 240° = [tex]\frac{-1}{2}[/tex]

B. The value of sin 240° = [tex]-\frac{\sqrt{3} }{2}[/tex]

What are basic Trigonometric functions?

The angles of sine,  cosine, and tangent are the primary classification of functions of trigonometry. And the three functions which are cotangent, secant, and cosecant can be derived from the primary functions. Basically, the other three functions are often used as compared to the primary trigonometric functions.

Part A:

we know that, cos (a +b)=cos a cos b - sin a sin b

cos 240° = cos (60° +180°)

=cos 60° cos 180° - sin 60° sin 180°

=[tex]\frac{1}{2} *(-1)-\frac{\sqrt{3} }{2} *0[/tex]

=[tex]\frac{-1}{2}[/tex]

Part B:

we know that, sin(a- b)=sin a cos b - cos a sin b

sin 240° = sin(270° - 30°)

=sin 270° cos 30° - cos 270° sin 30°

=[tex]-1 *\frac{\sqrt{3} }{2}-0*\frac{1}{2}[/tex]

=[tex]-\frac{\sqrt{3} }{2}[/tex]

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Step-by-step explanation:

We can start by using an equation to represent the given information. Let's call the number p, and the number k, then:

k = (1/5)p + 6

So, "six more than one-fifth of a number p is equal to a number k" can be represented mathematically as k = (1/5)p + 6. To find the value of p, we can solve for p:

p = 5k - 30

So, p is equal to 5 times the value of k minus 30.

Lowell's verbal representation of the equation y = x - 25 is "y is equal to x minus 25".

This verbal representation accurately describes the equation. In the equation, y is the dependent variable and x is the independent variable. The equation says that y is equal to x minus 25, which means that for every value of x, there is a corresponding value of y that is x minus 25.

So, Lowell has completed the task correctly.

Answer:

the ans is advanced

I hope you get your ans

Answer:

C.Only Maryanne's equation has the same solution as the original equation.

The solution is : 39 is the solution to Mr. Smith's equation.

An equation is a  mathematical statement that is made up of two expressions connected by an equal sign.  In its simplest form in algebra, the definition of an equation is a mathematical statement that shows that two mathematical expressions are equal.

here, we have,

given that,

the equation is:

(16-2w)^2 + (3y÷3z)

Let w=5 y=9 and z=3

(16-2*5)^2 + (3*9÷3*3)

PEMDAS says parentheses first

Multiply and divide in the parentheses

(16-10)^2 + (27÷9)

Then add and subtract in the parentheses

(6)^2 + (3)

Now the exponent

=36 +3

=39

Hence, The solution is : 39 is the solution to Mr. Smith's equation.

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complete question:

What is the solution to Mr. Smith's equation?

The values of x for which z is greater than y are 7.

In mathematics, an equation is a statement that two mathematical expressions are equal. The expressions can contain variables, constants, mathematical operations, and functions. An equation can be used to represent a relationship between two or more variables, and it is often used to solve problems by finding the values of the variables that satisfy the equation.

Equations can be represented in different forms, such as algebraic equations, differential equations, integral equations, and partial differential equations. Algebraic equations are the most common type of equations, and they involve algebraic operations such as addition, subtraction, multiplication, and division.

Equations can be classified according to their degree, which is the highest power of the variable in the equation. Linear equations are those with a degree of one, and they can be represented by a straight line on a graph. Quadratic equations have a degree of two, and they can be represented by a parabolic curve. Higher degree equations are generally more complex, and they may not have a simple geometric representation.

For x = -4, y = 4(-4) - 3 = -19 and z = -18, so y is greater.

For x = 2, y = 4(2) - 3 = 5 and z = 6, so z is greater.

For values of x where z is greater than y, we need to compare the values of z and y for each x value in the table.

At x = 3, y = 4(3) - 3 = 9 and z = 9, so they are equal.

At x = 7, y = 4(7) - 3 = 25 and z = 27, so z is greater.

At x = 10, y = 4(10) - 3 = 37 and z = 35, so z is not greater than y.

Therefore, the values of x for which z is greater than y are 7.

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Answer:

To approximate the area between the x-axis and the graph of f(x) = x^2 + 4 over the interval [0, 2], we can use the right rectangle method, where the heights of the rectangles are given by the value of the function at the right endpoint of each subinterval. If we divide the interval [0, 2] into 4 equal subintervals of width 0.5, the right endpoint of each subinterval would be 0.5, 1, 1.5, 2.

The height of the first rectangle would be f(0.5) = 0.5^2 + 4 = 4.25, the height of the second rectangle would be f(1) = 1^2 + 4 = 5, the height of the third rectangle would be f(1.5) = 1.5^2 + 4 = 6.25, and the height of the fourth rectangle would be f(2) = 2^2 + 4 = 8.

The sum of the areas of the rectangles is equal to (0.5) × (4.25 + 5 + 6.25 + 8) = (0.5) × 24 = 12.

So, the approximate area between the x-axis and the graph of f(x) = x^2 + 4 over the interval [0, 2] is 12.

Answer:

choice B

Step-by-step explanation:

plug in a few test values of x to prove it.  The other choice would be a cube root function of x.

The volume, V of the solid obtained by rotating the region bounded by the graphs of the expressions, y² = x, x = 3y; about the y-axis is equals to the 162π/5 cubic units.

We have a region bounded by the graphs of two expressions. These expressios are y² = x, x = 3y. We have to calculate the volume of solid obtained by rotating the bounded region about the y-axis. First determine the intersection points of

y² = x, x = 3y.

=> x = 3y --(1)

=> y² = 3y

=> y² - 3y = 0

=> y( y - 3 ) = 0

=> y = 0 or y - 3 = 0

=> y = 0 , y = 3

from (1), x = 3y => x = 0, x = 9

Let the intersection points be O(0,0) and P(3,9). Now, the volume of rotating region or due to shaded region is

[tex]V = π\int_{0}^{3} (R² - r²) dy [/tex]

where R = 3y , r = y²

[tex]V = π\int_{0}^{3} [(3y)² - (y²)²]dy [/tex]

[tex]V = π \int_{0}^{3} (9y² - y⁴)dy [/tex]

[tex]V = π[ \frac{9y³}{3}- \frac{y⁵}{5} ]_{0}^{3} [/tex]

[tex]V = π[ \frac{9(3)³}{3}- \frac{{3}^{5} }{5} - 0- 0 ][/tex]

[tex]V = π(81 - \frac{243}{5} ) = \frac{162π}{5} [/tex]

Hence, the required volume is 162π/5 cubic units.

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Answer:

Answer is in the attached photo.

Step-by-step explanation:

The solution is in the attached photo, do take note a variable is a alphabet can be assigned to any value, while a constant is a fixed number or value.

Answers:

variable = t

constant = -7

Step-by-step explanation:

So we know that a variable is a letter while a constant is a number

in here -7 is the constant (the number) and t is the variable (the letter)

Therefore, the variable is t and the constant is -7.

Answer:

So, the answer is (b) 320 feet.

Step-by-step explanation:

Let's call the length and width of the first playground "x".

The perimeter of the first playground is 160 feet, so 2 times the length plus 2 times the width is equal to 160:

2x + 2x = 160

Simplifying:

4x = 160

Dividing both sides by 4:

x = 40

So the length and width of the first playground are both 40 feet.

The second playground is twice as long and twice as wide as the first playground, so the length and width of the second playground would be 2 * 40 = 80 feet.

The perimeter of the second playground would then be 2 * 80 + 2 * 80 = 320 feet.

So, the answer is (b) 320 feet.

The requried hours spent babysitting and tutoring are 5 and 8 hours respectively.

The equation model is defined as the model of the given situation in the form of an equation using variables and constants.

Here,
Let the hours spent babysitting and tutoring be x and y respectively,
According to the quesiton,

Mark gets paid $6 per hour for babysitting and $15 an hour for tutoring. He makes $150 in one week.
6x + 15y = 150 - - - -(1)

John Makes $10 an hour babysitting and $8 an hour tutoring and gets paid a total of $114.
10x + 8y = 114 - - - - -(2)

Solving above equations 1 and 2 by elimination method we get x = 5 and y = 8

Thus, the requried hours spent for babysitting and tutoring are 5 and 8 hours respectively.

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Answer:

20

Step-by-step explanation:

[tex]\sqrt[3]{64xy^{3}}[/tex] = [tex]\sqrt[3]{64}[/tex] × [tex]\sqrt[3]{x}[/tex] × [tex]\sqrt[3]{y^{3}}[/tex]

a = coefficient

  = [tex]\sqrt[3]{64}[/tex]

∴a = 4

[tex]x^{b}[/tex] = [tex]x^{-2}[/tex]×[tex]\sqrt[3]{x}[/tex]

   = [tex]x^{-2}[/tex]×[tex]x^{\frac{1}{3}}[/tex]

   = [tex]x^{-2+\frac{1}{3}}[/tex]

   = [tex]x^{\frac{-6}{3} + \frac{1}{3}}[/tex]

   = [tex]x^{\frac{-6+1}{3}}[/tex]

[tex]x^{b}[/tex] = [tex]x^{-\frac{5}{3}}[/tex]

∴b = [tex]-\frac{5}{3}[/tex]

[tex]y^{c}[/tex] = [tex]y^{-4}[/tex]×[tex]\sqrt[3]{y^{3}}[/tex]

   = [tex]y^{-4}[/tex]× [tex](y^{3})^{\frac{1}{3}}[/tex]

   = [tex]y^{-4}[/tex] × [tex]y[/tex]

   = [tex]y^{-4 + 1}[/tex]

[tex]y^{c}[/tex] = [tex]y^{-3}[/tex]

∴c = [tex]-3[/tex]

∴Product of a, b and c = [tex](4)[/tex]×[tex](-\frac{5}{3})[/tex]×[tex](-3)[/tex]

                                      = 20