4 12. DISCUSS MATHEMATICAL THINKING Recall that a perfect square is a number with integers as its
square roots. Is the product of two perfect squares always a perfect square? Is the quotient of two
perfect squares always a perfect square? Explain your reasoning.
Please help will give Brain liest

Answer:

[tex]\frac{1}{16}[/tex]

Step-by-step explanation:

We will start by calculating the probability of drawing a spade from one deck, and then square it since we need to get the chance of getting two spades. We know there are 13 spades in a deck, out of 52 cards, giving us:

[tex]\frac{13}{52}[/tex]

Which can be simplified to :

[tex]\frac{1}{4}[/tex]

Then, we will square this fraction since we need two spades in a row:

[tex]\frac{1}{4} * \frac{1}{4} = \frac{1}{16}[/tex]

So the probability is [tex]\frac{1}{16}[/tex].

Hope this helped!

To solve a function, you must determine the value of the function for a given input. This requires finding the equation for the function and then plugging in the given input to find the output. For example, if the function is f(x) = x2 + 2x - 5, and the given input is 3, then the output of the function is f(3) = 12.

1. Substitute the given input, x = 3, into the equation.

2. Simplify the equation:  f(3) = 32 + 2(3) - 5

3. Solve the equation:  f(3) = 9 + 6 - 5

4. Simplify the equation: f(3) = 12

5. Therefore, the output of the function is f(3) = 12.

To solve a given function, you must first determine the equation for the function. Then, you must substitute the given input into the equation and solve it. In this example, the given input was 3 and the equation for the function was f(x) = x2 + 2x - 5. After plugging in the input of 3, the equation was simplified to f(3) = 9 + 6 - 5, which was further simplified to f(3) = 12. Therefore, the output of the function is f(3) = 12.

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Answer:

do it

Step-by-step explanation:

The projected population of Haleyville in 2020 is 4139.

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,

Population in 2010 = 4403

The decline in population per year = 0.06%

Now,

The population in 2020.

There are 10 years between 2010 and 2020.

The function used to find the population after x years.

= 4403 x [tex]0.994^x[/tex]

Now,

x = 10

= 4403 x [tex]0.994^{10}[/tex]

= 4403 x 0.94

= 4139

Thus,

The population in 2020 is 4139.

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Answer: x=4

Step-by-step explanation: x+3=2x-1, then you do simple algebra

Answer:

59

Step-by-step explanation:

The area of circle A to the area of circle B ratio is 100: 64, and the ratio of e: f is 13: 10.

The area of the circle is equal to the multiplication of the square of the circle's radius and π.

A = πr²

where 'r' is the radius

Circumference of circle A × 0.8 = circumference of circle B

So, taking r as the radius of A,

2πr =  circumference of circle A

2πr × 0.8 = circumference of circle B

So the radius of B is r × 0.8.

Now we can substitute that in the area equations :

Area of A = π×r²

Area of B = π × (r × 0.8)² = π × r² × 0.64

The circumferences of the A to B ratio is

Circumference of A : Circumference of B = (2 × π × r)/(2 × π × r × 0.8)

= 1: 0.8 = 100: 80,

The areas of circle A to circle B ratio is

Area of A : Area of B = (π × r²)/(π × r² × 0.64) = 1: 0.64 = 100: 64

e² = f² × 1.69

69% larger shows that f² is the 100% to which we must add 69% to obtain e². 100% + 69% Equals 169%, obtaining a factor of 1.69.

We have this equation:

e² = f² × 1.69

Apply the square root on both sides.

e = f × √(1.69) = f × 1.3

e/f = 1.3 = 1.3/1 = 13/10

the ratio e:f = 13: 10

Thus, the ratio of circle A's area to circle B's area is 100: 64 and the ratio of e: f is 13: 10.

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On solving the provided question, we can say that - here in the given equation we got x- 1 = 0; x = 1, y = 2

An equation is a formula in mathematics that joins two statements with the equal symbol = to represent equality. The definition of an equation in algebra is a mathematical statement proving the equality of two mathematical expressions. In the equation 3x + 5 = 14, for instance, the terms 3x + 5 and 14 are separated by an equal sign. The link between two phrases on either side of a letter is expressed mathematically. There is often only one variable, which is also the symbol. instance: 2x - 4 Equals 2.

y = 2x +1

y = 3x - 1

x- 1 = 0

x = 1, y = 2

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Answer:

Step-by-step explanation:

7(9w)7(9w) = 11

(7 ×7)(9×9)(w×w) = 11

49 ×81 w² = 11

3969w² = 11

[tex]\frac{3969w^{2} }{3969}[/tex] =  [tex]\frac{11}{3969}[/tex]

w²   =  [tex]\frac{11}{3969}[/tex]

[tex]\sqrt{w^{2} }[/tex]   =   [tex]\sqrt{11/3969}[/tex]

w = ± [tex]\frac{\sqrt{11} }{63}[/tex]

Because the solution contains a square root the answer contains both a negative and a positive solution.  

The number of terms in the expansion of [tex](2x^3y^2)(2x^2y^3)[/tex] is 12.

When expanding the product of two polynomials, the number of terms in the expansion is equal to the product of the number of terms in each polynomial. In this case, the first polynomial has 1 term (2x^3y^2) and the second polynomial has 1 term (2x^2y^3), so the number of terms in the expansion will be 1*1=1.

A polynomial function is one in which the variable's non-negative integer powers or positive integer exponents are the only parts of the equation. Examples of polynomial functions include quadratic, cubic, and other equations. An example of a polynomial with an exponent of 1 is 2x+5.

However, the expression provided is not a polynomial, it is a number.

[tex]2x^3y^2 * 2x^2y^3 = 4x^5y^5[/tex]

It is just one term.

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The actual dimensions are 9.75 feet and 7.5 feet. The area will be 73.125 square feet.

The ratio of the scale of an original thing to a new object that is a representation of it but of a different size is known as a scale factor (bigger or smaller).  The scale factor is defined as the proportion of the new image's size to that of the previous image.

Here the scale factor for the given image is 2 inch = 3 ft.

The measure of the actual dimensions will be calculated as,

2 in = 3 ft

1 in = 3/2 ft

6.5 in = 6.5 x 3 / 2 = 9.75 ft

5 in = 5 x 3 / 2 = 7.5 ft

The actual area will be calculated as,

Area = 9.75 x 7.5

Area = 73.125 square feet

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The longer ladder can reach 3 8/7 feet higher than the shorter one.

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Answer:

Step-by-step explanation:

Divide: 18 ÷3=?

The answer is 6

There are several ways to make a graph to compare two sets of data, but one common method is to use a line graph.

The steps to create a line graph to compare two sets of data:

Organize the data: Arrange the data into two columns, one for each set of data. Each column should have a label at the top to indicate what the data represents.

Choose a scale: Decide on an appropriate scale for the x- and y-axes that will allow you to clearly display the data. The x-axis should represent the categories being compared, and the y-axis should represent the values of the data.

Plot the data: Using the chosen scale, plot the data points for each set of data on the graph. Use different colors or symbols for each set of data to make it easy to distinguish between the two.

Add a title: Give the graph a title that describes what the graph is showing.

Add a legend: Add a legend to the graph to indicate what each set of data represents.

Add labels: Label the x- and y-axes to indicate what they represent.

By following these steps, you can create a clear and informative line graph that allows you to easily compare two sets of data.

Other types of graph that can be used to compare two sets of data are bar charts, scatter plots, and stacked bar charts. The choice of graph depends on the type of data you have, and what you want to compare or highlight.

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Using the Pythagoras Theorem the value of c=11.40 and the number that goes beneath the radical is 130.

What is Pythagoras Theorem?

In a right-angled triangle, the square of the hypotenuse side is equal to the sum of the squares of the other two sides, according to Pythagoras's Theorem. These triangle's three sides are known as the Perpendicular, Base, and Hypotenuse. Due to its position opposite the 90° angle, the hypotenuse in this case is the longest side.

The rectangle is divided into two triangles which have exactly same sides.

So take any one triangle.

Apply the Pythagoras theorem since it is a right-angled triangle with base = 9 and perpendicular = 7.

So, according to Pythagoras theorem -

(Hypotenuse)^2=(Base)^2+(Perpendicular)^2

Plugging in the values in the equation -

c^2=(9)^2+(7)^2

c^2=81+49

c^2=130

c=[tex]\sqrt{130}[/tex]

c=11.40

Therefore, the value of c is obtained as 11.40.

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Answer:

Step-by-step explanation:

the answer is 152

Answer:  Part A:

To represent the cost per night x and cost per meal y in a system of equations, we can use the information provided in the question:

First Special:

3x + 4y = 233

Second Special:

3x + 3y = 226.50

Where x is the cost per night and y is the cost per meal.

Part B:

To find the cost per night and the cost per meal, we can use the information provided in the question and system of equations.

We can solve this system of equations using several methods: substitution or elimination method.

One common method to find the solution is by substitution method, where we find the value of x in terms of y from one of the equations and then substitute it into the other equation to solve for y.

x = (233 - 4y)/3

then substitute it into the other equation:

3(233 - 4y) + 3y = 226.50

now solve for y:

699 - 12y + 3y = 226.50

-9y = -472.50

y = 52.50

and we can substitute it back into the first equation:

3x + 4(52.50) = 233

3x = 80.50

x = 26.83

So the cost per night is $26.83 and the cost per meal is $52.50

Step-by-step explanation:

Distance and time here is directly proportional to each other and the term '3' shows that the distance is increasing, is 3 times of the time, increased.

Given,

A snail is moving away from a rock.

The equation d=3t represents the relationship between the distance, d , in inches that the snail is from the rock and time, t, in minutes.

The number '3' is constant term, which basically represents the the distance covered by the snail, in terms of time or, the rate of increment of distance is terms of time.

Distance and time here is directly proportional to each other and the term '3' shows that the distance is increasing, is 1/3 of the time, increased.

For example if the snail is moving for 1 unit time, the t will be considered as 1. so, the d will be 3*1 = 3, i.e. 3 unit distance will be covered by the snail in 1 unit time.

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Answer: x=-1/4 and x=-1

Step-by-step explanation:

The answer choices you give are extremely hard to figure out, so I hope my explanations and answers help.

The quadratic formula is [tex]x=\frac{-b\pm\sqrt{b^2-4ac} }{2a}[/tex]. Once we figure out a, b, c, we can plug them into the formula and solve for x.

The standard form equation is ax²+bx+c=0. The equation given matches this so we know that:

a=4

b=5

c=1

Plug them into the formula. There is a [tex]\pm[/tex] symbol, which stands for "plus or minus". This means we solve the formula with addition and subtractions separately. We should have 2 solutions at the end. Let's do them separately to show work.

Solution 1: addition

[tex]x_1=\frac{-5+\sqrt{5^2-4(4)(1)} }{2(4)}[/tex]             [exponents]

[tex]x_1=\frac{-5+\sqrt{25-4(4)(1)} }{2(4)}[/tex]             [multiply]

[tex]x_1=\frac{-5+\sqrt{25-16} }{8}[/tex]                    [inside square root]

[tex]x_1=\frac{-5+\sqrt{9} }{8}[/tex]                          [square root]

[tex]x_1=\frac{-5+3}{8}[/tex]                            [add]

[tex]x_1=\frac{-2}{8}[/tex]                                [divide top and bottom by 2]

[tex]x_1=-\frac{1}{4}[/tex]

Solution 2: subtraction

[tex]x_2=\frac{-5-\sqrt{5^2-4(4)(1)} }{2(4)}[/tex]             [exponents]

[tex]x_2=\frac{-5-\sqrt{25-4(4)(1)} }{2(4)}[/tex]             [multiply]

[tex]x_2=\frac{-5-\sqrt{25-16} }{8}[/tex]                    [inside square root]

[tex]x_2=\frac{-5-\sqrt{9} }{8}[/tex]                          [square root]

[tex]x_2=\frac{-5-3}{8}[/tex]                            [subtract]

[tex]x_2=\frac{-8}{8}[/tex]                                [divide]

[tex]x_2=-1[/tex]

I am not able to find the answer choices, but we know that x=-1/4 and x=-1.

Yes, because as x increases, y increases and because all of the pairs represent the same ratio.

When two sets of provided numbers change in proportion, the ratios are said to be directly proportional if they do so in the same direction.

If two ratios or fractions are equivalent, it may be shown using the proportion formula. By dividing the supplied numbers, we may determine the value that is missing. You may write the percentage formula as a: A and d are the extreme terms, and b and c are the middle terms, according to the formula b::c: d = a/b = c/d.

Yes, because as x increases, y increases.

Yes, because all of the pairs represent the same ratio.

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In function notation, the function is represented as f(x) = (3/4)x + 7.

In function notation, the function is represented by the letter "f" followed by the input value inside parentheses. The input value is typically represented by the letter "x". In this case, the function takes "x" as its input and performs the operation of multiplying it by 3/4 and then adding 7 to get the output. We can represent this function using the equation f(x) = (3/4)x + 7.

It's worth noting that the function notation is a way to express a mathematical relationship between two variables, the input variable (x) and the output variable (f(x)), in which every input value of x corresponds to exactly one output value. This equation represents the same function regardless of the input value, and it can be used to find the output for any input value by simply substituting it in the equation.

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The value of DX for solving simultaneous equations 3x 2y =- 11 and 7x 4y 9 by Cramer's rule is -13/5.

The general formula for Cramer's rule is

DX = detX/detA

Where detX is the determinant of the matrix formed by replacing the x-column of the original matrix with the constant column vector, and detA is the determinant of the original matrix.

For the given equations,

A = [3, 2; 7, 4]

X = [-11; -9]

Therefore,

detA = 3*4 - 2*7 = -10

detX = 3*(-9) - 2*(-11) = 13

Hence, DX = detX/detA = 13/-10 = -13/5

The value of DX for solving simultaneous  

3x 2y =- 11 and 7x 4y 9 by Cramer's rule equation is -13/5.

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The example of an associative property is a+(b+c) = (a+b) + c for additional operation and a × (b × c) = (a × b) × c for the multiplication operation. The associative property is used to solve an equation or expression of real numbers.

What are the properties of real numbers?

In the number system, real numbers are only the fusion of rational and irrational numbers. These numbers can generally be used for all arithmetic operations and can also be expressed on a number line.

The four primary characteristics of real numbers are as follows:

Commutative property

Associative property

Distributive property

Identity property.

Associative property: The associative property is applicable on two arithmetic operations. They are multiplication and addition.

For multiplication operations: a × (b × c) = (a × b) × c

For additional operation: a+(b+c) = (a+b) +

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The value of x is 45 degree and value of y is 49 degree.We can find by using supplementary and corresponding angles.

Supplementary angles are those angles that sum up to 180 degrees. To find the angle which is supplementary to another angle subtract the given angle from 180 degrees. For example if one angle is 60 degrees then another angle is 180 – 60 = 120 degrees.

Corresponding angles are the angles that are formed when two parallel lines are intersected by the transversal. The opening and shutting of a lunchbox, solving a Rubik's cube, and never-ending parallel railway tracks are a few everyday examples of corresponding angles.

Now we find

according to supplementary angles

3x+45=180

3x=180-45

x=45

According to corresponding angles

y-4=45

y=45+4

y=49

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-20w-35

I used the distributive property to distribute terms, and then simplified

[tex](\stackrel{x_1}{3}~,~\stackrel{y_1}{-2})\qquad (\stackrel{x_2}{5}~,~\stackrel{y_2}{-2}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{-2}-\stackrel{y1}{(-2)}}}{\underset{\textit{\large run}} {\underset{x_2}{5}-\underset{x_1}{3}}} \implies \cfrac{-2 +2}{2} \implies \cfrac{ 0 }{ 2 } \implies \text{\LARGE 0}[/tex]

Answer:

slope = 0

Step-by-step explanation:

calculate the slope m using the slope formula

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

with (x₁, y₁ ) = (3, - 2 ) and (x₂, y₂ ) = (5, - 2 )

m = [tex]\frac{-2-(-2)}{5-3}[/tex] = [tex]\frac{-2+2}{2}[/tex] = [tex]\frac{0}{2}[/tex] = 0

Using simple unitary method and with the total and per unit value provided, The answer is 8 boxes maximum can be bought.

The unitary method is a method used in mathematics to solve problems involving units of measurement. It involves expressing the units of measurement as a fraction or multiple of a single unit and then using this fraction or multiple to solve the problem. The method is particularly useful in solving problems involving conversions between different units of measurement. The unitary method is particularly useful in solving word problems where the conversion of units is required. It involves expressing the given information in terms of a single unit and then using that information to find the unknown quantity.

The unitary method is a useful tool for solving problems involving units of measurement, it simplifies the problem and improves understanding of conversions. It also helps to develop critical thinking skills and is a useful tool for problem-solving in a variety of subjects.

each box costs  = 2.5

total money =20

maximum that can be bought = 20/2.5 =8

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Answer: 6

Step-by-step explanation:

y = -3x + 6 is the linear equation of the form y = mx + c

where m is the gradient of the graph and c is the y-intercept of the graph.

Hence,

m = -3

c = 6

y-intercept of the graph = c

                                        = 6

Answer:

B. -5/2

Step-by-step explanation:

(0, -7) and (-4, 3)

m = 3+7/-4-0

m = 10/-4

m = -10/4

m = -5/2

Answer:

b

Step-by-step explanation:

320/31 gallons of hot water and 300/31 gallons of cold water are needed.

Here we are required to fill a 20-gallon bathtub and the water should be precisely 100° F.

It is given that hot water has a temperature of 175° F and cold water has a temperature of 20° F.

Let the amount of hot water used be x gallons.

Let the amount of cold water used be y gallons.

Since the bathtub has a capacity of 20 gallons,

x + y = 20                                                                                  [1]

or, y = 20 - x                                                                             [2]

The temperature after the water is mixed will be a weighted average of the temperature and amt of water used.

Hence we get,

(175x + 20y) / (x + y) = 100

from equation [1] we get

175x + 20y = 2000                                                                   [3]

from equations [2] and [3] we get

155x - 400 = 2000

or, x = 1600/155 = 320/31                                                        [4]

Similarly,

y = 20 - 320/31

= 300/31

Hence 320/31 gallons of hot water and 300/31 gallons of cold water are needed.

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Answer:

6[tex]\frac{2}{3}[/tex]

Step-by-step explanation:

In this equation I will do the elimination method.

3x-y=6

3x+y=34

6x=40

Simplify

x=6[tex]\frac{2}{3}[/tex].

Check:

3(6.666...)-y=6

20-y=6

-20 from 6

-y=-14

y=14

20-14=6

6=6

3x(6.666...)+y=34

20+14=34

34=34