a square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. what is the smallest possible value of the length of the side of the original square?

All of the graphs and associated functions and equations serve as examples of quadratic functions.

The locations where the graph's line or curve crosses the x-axis are known as the zeros of a function. The line dividing a graph into equal halves is known as the axis of symmetry.

The minimum or greatest point on a graph is the vertex of a function. If a quadratic function expands upward, its vertex is lowest; if not, it is maximum.

The worksheet is given below:

Using the stated definitions, the worksheet's answers are:

Graph 1

Zeros: -4 and 0

Axis of symmetry: x = -2

Min or Max: Maximum

Vertex: (-2, 4)

Graph 2

Zeros: 2

Axis of symmetry: x = -2

Min or Max: Minimum

Vertex: (-2, 0)

Graph 3

Zeros: -3 and 3

Axis of symmetry: x = 0

Min or Max = Minimum

Vertex: (0, -9)

Graph 4

Zeros: -5 and -3

Axis of symmetry: x = -4

Min or Max = Maximum

Vertex: (-4, -1)

Graph 5

Zeros: 3 and 7

Axis of symmetry: x = 5

Min or Max = Minimum

Vertex: (5, -4)

Graph 6

Zeros: None

Axis of symmetry: x = 24

Min or Max = Minimum

Vertex: (2, 5)

Graph 7

Zeros: 1 and 5

Axis of symmetry: x = 3

Min or Max = Minimum

Vertex: (3, -4)

Graph 8

Zeros: -2 and 0

Axis of symmetry: x = -1

Min or Max = Maximum

Vertex: (-1, 1)

Graph 9

Zeros: 5

Axis of symmetry: x = 5

Min or Max = Minimum

Vertex: (5, 0)

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The equation relating the number of hours worked and the total amount earned will be y = 14x.

A connection between a number of variables results in a linear model when a graph is displayed. The variable will have a degree of one.

The linear equation is given as,

y = mx + c

Where m is the slope of the line and c is the y-intercept of the line.

From the table, the two points will be written as (1.25, 17.50) and (3.75, 52.50). Then the equation of the line is given as,

(y - 17.50) = [(52.50 - 17.50) / (3.75 - 1.25)](x - 1.25)

y - 17.50 = 14(x - 1.25)

y - 17.50 = 14x - 17.50

y = 14x

The equation relating the number of hours worked and the total amount earned will be y = 14x.

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The missing table is attached below.

After 30 cm of the belt sewn, Aspen will have equal numbers of green and silver beads left.

Aspen has 306 green beads and 210 silver beads.

For every 5 cm of a belt that Aspen sews, they use 40 green beads and 24 silver beads.

We can set up an equation to determine when Aspen will have equal numbers of green and silver beads left.

Let x be the number of centimeters of the belt Aspen will have sewn when they have equal numbers of green and silver beads left.

We can use the following equation:

306 - (40/5)x = 210 - (24/5)x

Solving for x, we get:

x = (306-210)/(40/5-24/5) = (96)/(16/5) = 96*5/16 = 30 cm

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The trinomial expression that represents C is

C = -3x² - 18x + 19

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

C = G - 3F ______(1)

F = 2x² + 6x - 5

G = 3x² + 4

Substituting F and G in (1)

C = 3x² + 4 - 3(2x² + 6x - 5)

C = 3x² + 4 - 6x² - 18x + 15

C = -3x² - 18x + 19

Thus,

The trinomial expression is

C = -3x² - 18x + 19

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The final step in solving the inequality is: C. x < 3.

To solve an inequality means to find the possible value of a variable in in inequality statement, which will also make it true.

Given the inequality, –2(5 – 4x) < 6x – 4, we can solve as follows:

Step 1: Apply the distributive property

–10 + 8x < 6x – 4

Step 2: Subtract 8x from both sides

–10 < –2x – 4

Step 3: Add 4 to both sides

–6 < –2x

Step 4: Divide both sides by -2

-6/-2 > -2x/-2

3 > x

x < 3

The final answer of the inequality is: C. x < 3.

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The value of the coefficient of the variable 'x' is 3. Then the correct option is C.

The allocation of weights to the important variables that produce the calculation's optimum is referred to as a direct consequence.

The equation is given below.

(x + 7)(x - 4) = x² + gx - 28

Simplify the equation, then we have

(x + 7)(x - 4) = x² + gx - 28

x² + 7x - 4x - 28 = x² + gx - 28

x² + (7 - 4)x - 28 = x² + gx - 28

3x = gx

g = 3

The value of the coefficient of the variable 'x' is 3. Then the correct option is C.

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The complete question is given below.

Answer:

To find the function f(x), we need to use the Remainder Theorem. The Remainder Theorem states that if we divide a polynomial by x - a, the remainder will be equal to the value of the polynomial at x = a.

We know that f(x) divided by 3x + 1 leaves a remainder of 7, so we can set up the equation:

f(x) = (3x + 1)(2x^2 + 2x - 9) + 7

Expanding the left side of the equation, we get:

f(x) = 6x^3 + 2x^2 - 27x - 9 + 7

Simplifying, we get:

f(x) = 6x^3 + 2x^2 - 27x + 2

The function f(x) is in standard form.

Answer:

When the function f (x) is divided by 3x – 1, the quotient is 2x² – 3x + 6 a

Step-by-step explanation:

Step 1 Devisor = 3x - 1Quotient = 2x2 - 3x + 6Remainder = 7Dividend = f(

The length of US in the rhombus is 16 units

The complete question is added as an attachment

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

RT = 6x + 4

SO = 7x - 6

By the properties of rhombus, we have

RT = 2 * SO

So, we have

6x + 4 = 14x - 12

Evaluate the like terms

8x = 16

So, we have

x = 2

This means that

US = 6 * 2 + 4

US = 16

Hence, the length is 16 units

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MNOP is a parallelogram and ∠MQN ≅ ∠ORP.  MN ≅ PO. MN ║ PO. Using these statements, we can prove that the ΔMQN ≅ ΔORP.

A quadrilateral with two sets of parallel sides is referred to as a parallelogram. In a parallelogram, the opposing sides are of equal length, and the opposing angles are of equal size. Additionally, the interior angles that are additional to the transversal on the same side. 360 degrees is the sum of all interior angles.

Given that the quadrilateral MNOP is a parallelogram and ∠MQN ≅ ∠ORP.

To prove that ΔMQN ≅ ΔORP, the correct statements are:

MNOP is a parallelogram and ∠MQN ≅ ∠ORP.

MN ≅ PO

MN ║ PO

Using these statements, we can prove that the ΔMQN ≅ ΔORP.

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The total clay required is 376.2 cm³and area to be painted is given by 300.7 cm²

The volume of a Right Circular Cylinder. In general, the volume of a right cylinder is the area of the base times the height of the cylinder. The area of the circular base is given by the formula A = πr2. Substitute to get V = πr2h.

Given here: The dimensions of the edge of the cube as 5cm and the cylinder diameter is 4cm with a height of 20cm

We know the volume of the cube is given by a³ where a is the measure of the length of the cube

Thus a³=5³

            =125 cm³

and surface area of the cube is given by 6a²=6×25

                                                                          =150 cm²

Again the volume of the cylinder and surface area of the given cylinder is as follows:

Volume= 22/7×2²×20

            =251.2 cm³

surface area = 2·π·2·10+2·π·22≈150.79645cm²

Thus the clay required to make the toy is=125+251.2

                                                                    =376.2 cm³

And the area to be painted is given by =150.7+150

                                                                 =300.7 cm²

Hence, The total clay required is 376.2 cm³and area to be painted is given by 300.7 cm²

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The value of the bulldozer at year 2.5 is 93593.67 and at year 5 is 66451.16.

A construction company bought a new bulldozer for $125,000. The bulldozer depreciates exponentially and after 2 years the value of the bulldozer is $80,000. What is the value at year 2. 5 and 5Let V0 be the initial value of the bulldozer, and k be the depreciation rate. We know that the value of the bulldozer after 2 years is V2 = V0 * e^(-k * 2) = 80000.

So, k = -ln(V2/V0) / 2.

The value of the bulldozer at year 2.5 can be calculated as:

V2.5 = V0 * e^(-k * 2.5)

The value of the bulldozer at year 5 can be calculated as:

V5 = V0 * e^(-k * 5)

Plugging in V0 = 125000 and k = -ln(V2/V0) / 2, we get:

V2.5 = 125000 * e^(-k * 2.5) = 93593.67

V5 = 125000 * e^(-k * 5) = 66451.16

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Samuel pay 5.81 for the 3 bottles of juice.

A number or ratio that can be expressed as a fraction of 100 or a relative value indicating hundredth part of any quantity is called percentage.

We have to given that;

Samuel buys 3 bottles of juice that each have an original price of 2.80.

And, He uses a coupon for 35% off. Then, he is changed sales tax at a rate of 6.5%.

Here, The original cost of 3 bottles of juice = 3 × 2.80

                                                                  = 8.40

Now, He uses a coupon for 35% off.

Hence, The off price = 35% of 8.40

                               = 35/100 × 8.40

                               = 2.94

So, The amount of 3 bottles of juice = 8.40 - 2.94

                                                       = 5.46

And, He is changed sales tax at a rate of 6.5%.

Thus, The sales tax on bottles = 6.5% of 5.46

                                              = 6.5/100 × 5.46

                                              = 0.35

Thus, The actual amount of 3 bottles of juice = 5.46 + 0.35

                                                                    = 5.81

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The rise and run may be used to determine the slope. Discover the y-intercept.

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For the given graph, the slope is 2/9, t the U.S. box office revenue in 2018 is 12  billion.

Given a graph depicting approximations of US box office earnings (in billions of dollars) from 2000 to 2012, what would you conclude?

We must calculate the graph's slope and y-intercept.

In order to overcome the issue, we are employing the point-slope form. Two points are obtained from the graph: (x1,y1) = (0,8) and (x2,y2) = (9, 10).

Consequently, slope is m = y2 - y1 / x2 - x1

⇒ 10 - 8/ 9 - 0 = 2/9

This is the graph's straight line's slope.

The point-slope form is given by y-y1 = m. (x-x1)

Meaning: y - 8 = m(x - 0)

The line's equation is then y = 2x/9 + 8, with 8 serving as the y-intercept.

• In this case, the y-intercept provides the beginning value, or 8 when x = 0. Here, the year 2000 is represented by the number 0, and the U.S. box office took in $8 billion in that year.

2 • The graph's straight line shape and slope (m) indicate that the U.S. box office's revenue is rising at a steady rate of $2 billion every two years. 2 year 9 is on the x-axis.

• Using the equation, y = 2, we may get the precise income in any year from 2000 through 2012.

The x axis in the graph corresponds to the year;

hence, the point at x = 0 denotes the year 2000 and the one at x = 12 denotes the year 2012.

In order to get the income for 2018, we must determine the value of y when x equals 18.

The formula is y = 2x/9 + 8

y = 2(18)/9 + 8

y = 12

That is, 12 billion dollars will be made at the U.S. box office in 2018.

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Complete Question -

The condensed form of 6 ln x + 4 ln y is ln ( x⁶ y ⁴ ).

The opposite of an exponential function is a logarithmic function. A log function and an exponential function both use the same base. An exponent is a logarithm. f(x) = bx is how the exponential function is expressed. The formula for the logarithmic function is f(x) = log base b of x.

Given a logarithmic function  6 ln x +4 ln y.

Using:

ᵃ log m = log mᵃ and

log a + log b − log c = log ( a b/ c )

In our case,

6 ln x + 4 ln y

=>  ln x⁶ + ln y⁴

=> ln ( x⁶ y⁴ )

Therefore,  ln ( x⁶ y ⁴ ) is The condensed form of 6 ln x + 4 ln y.

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The ratio of the lengths of the smaller quadrilateral to the larger quadrilateral will be 4:5.

The quadrilateral has four sides and four angles. The sum of internal angles is 360 degrees.

The ratio of the matching sides will remain constant if two triangles are comparable to one another.

The quadrilateral MLKJ is similar to the quadrilateral ABCD. Then the ratio of the lengths of the smaller quadrilateral to the larger quadrilateral will be given as,

Ratio = 12 / 15

Ratio = 4/5

The ratio of the lengths of the smaller quadrilateral to the larger quadrilateral will be 4:5.

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The value of x in the above triangle, according to the supplied assertion, is 21.

The Pythagorean Theorem states that the squared on the tangent line (the side across from the right angle) of a right triangle, or, in standard mathematical notation, a2 + b2, are equal to the numbers on the legs. A triangle is a two-half polygon in geometry that has 3 components and three vertices. The most important attribute of a triangle is that its inside angles sum to 180 degrees. The right triangle angle sum attribute is what is known as.

x+x+7/x+7 = 9+12/12

2x+7/x+7 = 27/12

8x+28 = 7x+49

8x-7x = 49-28

x = 21

Therefore, the of x is 21

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The solution to the equation 2-x³ = 4x + 3 is x=1.

What is an equation?

A mathematical definition of an equation is a claim that two expressions are equal when they are joined by the equals sign ("=").

To find the solution to the equation 2-x³=4x+3, we can set the two sides of the equation equal to each other and solve for x.

2-x³ = 4x + 3

x³ - 4x - 1 = 0

To solve this equation, factor it or use the cubic formula.

One possible solution is x = 1, which can be verified by substituting it back into the original equation.

Another way to find the solution graphically, we can plot the graphs of y = 2-x^3 and y = 4x+3 on the same coordinate plane. The points at which the two graphs intersect are the solutions to the equation.

The graph of y = 2-x^3 is a downward facing parabola and the graph of y = 4x+3 is a line with slope 4 and y-intercept 3.

The intersection point of the parabola and the line is the solution x=1

It is not possible to find the other solutions graphically because the parabola doesn't intersect with the line again.

Therefore, the solution is obtained as x = 1.

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We can start solving the problem using systems of equations. Let x be the cost of each balloon and y be the cost of each banner. We know from the problem that:

1x + 1y = 3 (the cost of 1 balloon and 1 banner is $3)

5x + 2y = 9 (the cost of 5 balloons and 2 banners is $9)

We can use the first equation to solve for one of the variables in terms of the other, and substitute that into the second equation. Then we can solve for the remaining variable.

From equation 1:

y = 3 - x

We can substitute this into equation 2:

5x + 2(3 - x) = 9

Which simplifies to:

5x + 6 - 2x = 9

3x = 3

x = 1

So each balloon costs $1

We can now substitute x=1 in the first equation:

1*1 + 1y = 3

y = 2

So each banner costs $2

Step-by-step explanation:

Let us assume the cost of

In first case , cost of 1 1 balloon and 1 banner is $3 . We can set up a equation as ,

1*x + 1*y = 3

x + y = 3 . . . . . (i)

In second case, cost of 5 balloons and 2 banners is $9 . So ,

5*x + 2*y = 9

5x + 2y = 9 . . . . . (ii)

From equation i and ii , we have ,

5(3-y) + 2y = 9

15 - 5y + 2y = 9

15 - 3y = 9

3y = 15 -9

3y = 6

y = 2

And we had ,

x + y = 3

x + 2 = 3

x = 3 - 2

x = 1

Hence,

and we are done!

Answer:

Step-by-step explanation:

bro ima bout to drop out of highschool because of math '_'

I dont care what happeneds with me anymore

here we are given that a 6ft high man is flying a kite with a string of length 100ft at an angle of elevation 52° . We would like to calculate the height of the kite above the ground. For figure refer to the attachment. We would have to make use of trigonometric ratios here as ,

In ∆ ABE ,

[tex]\longrightarrow \sin 52^o = \dfrac{AB}{AE} \\ [/tex]

[tex]\longrightarrow 0.8 =\dfrac{y}{100ft} \\ [/tex]

[tex]\longrightarrow y =0.8(100ft) \\ [/tex]

[tex]\longrightarrow y = 80ft \\ [/tex]

Now the total height above the ground will be ,

[tex]\longrightarrow h = y + h_{man} \\ [/tex]

[tex]\longrightarrow h = (80 + 6)ft\\ [/tex]

[tex]\longrightarrow\underline{\underline{ h = 86\ ft. }} \\ [/tex]

And we are done!

Substitute the  sign for the symbol to graph the provided inequality.

Replace the symbol with the = sign to graph the specified inequality.

Create a table now.

y= 8x - 6.

Assume a random integer for x to determine y.

x y= 8x - 6.

-2 -4

0 -4

-6 8

We are now graphing the points. Connect the spots with the dotted line.

Now, we'll do shading.

Use the test point (8,6)

Therefore, we darken the area that doesn't include (8,6)

We are now graphing the points. Connect the spots with the dotted line.

Substitute the = sign for the symbol to graph the provided inequality.

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a) The slope equation representing Maya's spending situation is x - 3n = 68, where x is the initial amount Maya has and n is the number of days she has spent her money.

b) Using the above equation, after 25 days, Maya will have $35 left.

c) Using the above equation, after 9 days, Maya will have $83 left.

d) The slope is the average rate Maya spends per day, which is $3 while the y-intercept represents the initial amount she has.

A slope equation is written in the slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.

Maya's average spending rate per day = $3

The number of days Maya has spent money = 14 days

The amount left after 14 days = $68

Let the initial amount Maya has = x and the number of days she has spent her money = n

a) x - 3n = 68

x = 68 + 3n

x = 68 + 3(14)

x = 68 + 42

x = 110

= $110

b) 110 - 3n = 35

-3n = -75

n = 25

c) 110 - 3(9)

= 110 - 27

= 83

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Ivanna earns $2,400 if she doesn't sell any copies of history is fun.

Ivanna makes an extra $120 ($120) for each additional copy she sells and a loss of $120 ($120) for each fewer copy.

If Ivanna doesn't sell any copies of History is Fun, her total compensation is $2,400.

Ivanna makes a total salary of y = 2,400 + 120x, where x is the quantity of copies of her book History is Fun she sells.

We compare the compensation for two successive values of y to determine the change in Ivanna's pay.

We must estimate y when x exists of consecutive values i.e. 1, 2, 3, 4

We compute the value of y when x = 1, 2 and 3.

When x = 1, y = 2,400 + 120 (1) = 2,400 + 120 = $2,520.

When x = 2, y = 2,400 + 120 (2) = 2,400 + 240 = $2,640.

When x = 3, y = 2,400 + 120 (3) = 2,400 + 360 = $2,760.

To estimate the increase in her pay for a copy, we subtract y when x = 2 from y when x = 3.

(y when x = 3) - (y when x = 2) = $2,760 - $2,640 = $120.

So for every extra copy, Ivanna gets an extra $120.

To estimate the decrease in her pay for a copy, we subtract y when x = 3 from y when x = 2.

(y when x = 2) - (y when x = 3) = $2,640 - $2,760 = -$120.

So for every less copy, Ivanna loses $120.

To estimate her pay when she doesn't sell any copies of history is fun, we substitute the value of x as 0 in y.

When x = 0, y = 2,400 + 120 (0) = $2,400.

So Ivanna earns $2,400 if she doesn't sell any copies of history is fun.

The complete question is:

Ivanna is a software saleswoman. Let y represent her total pay (in dollars). Let x represent the number of copies of History is Fun she sells. Suppose that x and y are related by the equation 2400+120x = y.

Answer the questions below.

Note that a change can be an increase or a decrease.

For an increase, use a positive number. For a decrease, use a negative number.

What is the change in Ivanna's total pay for each copy of History is Fun she sells?

What is Ivanna's total pay if she doesn't sell any copies of History is Fun?

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The the height of the rectangle 6.3cm

A rectangle is a polygon of four sides whose opposite sides are equall and parallel

The given parameters that will help us to solve for the height are

the area of a rectangle is the space the rectangle occupies

The area is denoted by :

Area= Lenght * Height

245 = 39 * h

h= 245/39

Height = 6.3cm

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The price of the stock models exponential is 500, The rate of exponential growth or exponential decay is 19%.

A relation is a function if it has only One y-value for each x-value.

The given function is f(x) = 500(1 − 0.19 )x² models the price of a share of stock over time.

The quantity decreases slowly at regular intervals by a fixed percent.

The price of the stock models exponential is 500.

The rate of exponential growth or exponential decay is 0.19×100=19%

f(x) = 500(1 − 0.19 )x²

f(x)=500(0.81)x²

f(x)=500(0.81)x² is the equivalent function of  f(x) = 500(1 − 0.19 )x²

Hence, The price of the stock models exponential is 500, The rate of exponential growth or exponential decay is 19%.

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a^6 + b^6= (a^2 +b^2)(a^4-a^2b^2+b^2) is an identity. The reasoning is done below.

The term "operation" in mathematics refers to the process of computing a value utilizing operands and a math operator. For the specified operands or integers, the math operator's symbol has predetermined rules that must be followed. In mathematics, there are five basic operations: addition, subtraction, multiplication, division, and modular forms.

Given an expression  a^6 + b^6= (a^2 +b^2)(a^4-a^2b^2+b^2) we need to assure if its right or not

Let "x" = a² and "y" = b² thus the equation

x³ + y ³ = ( x + y)(x² + y² - xy)

Take L.H.S.

=> x³ + y³ = (x + y)³ - 3xy (x + y)

» Take (x + y) common

=> (x + y) [(x + y)² - 3xy (1)]

=> (x + y) [(x² + 2xy + y²) - 3xy]

Because (a + b)² = a² + 2ab + b²

=> (x + y) (x² + y² + 2xy - 3xy)

=> (x + y) (x² + y² - xy)

L.H.S. = R.H.S.

Take R.H.S.

=> (x + y) (x² - xy + y²)

=> x(x² - xy + y²) + y(x² - xy + y²)

=> x³ - x²y +xy² + x²y - xy² + y³

=> x³ + y³

L.H.S. = R.H.S.

therefore, a^6 + b^6= (a^2 +b^2)(a^4-a^2b^2+b^2) is an identity .

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The probability that you win the game on your second draw is 1/1140.

We have an urn containing 17 balls marked LOSE and 3 balls marked WIN. You and the opponent take turns of a random selection of a single ball from the urn without replacement.

The person who picks the third WIN ball wins the game. It does not matter who picked the first two WIN balls.

If you first, we need to find the probability that you will win the game on your second draw.

Here we consider Y: you draw W first, O: opponent draws W first and

Y: you draw W second.

Then,

P(YOY) = P(W W W )

Where W is win.

P(YOY) = 3/20 × 2/19 × 1/18

P(YOY) = 1/1140

Therefore the probability that you win the game on your second draw is 1/1140.

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The measures are given as follows:

1. Length of 36 cm.

2. Height of 1.44 cm.

The measures are obtained applying the proportions in the context of this problem.

The ratio between the length and the width is given as follows:

Length/Width = 180/75

Length/Width = 2.4

Length = 2.4 Width.

For a width of 15 cm, the length is given as follows:

15 x 2.4 = 36 cm.

The ratio between the length and the height is given as follows:

Length/Height = 180/20

Length/Height= 9

Length = 9Height

Height = Length/9

Hence the height if the length is of 13 cm is given as follows:

13/9 = 1.44 cm.

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Answer:

x = 4 cm

Area= 36 cm2

Step-by-step explanation:

x is the measurement of one side of the triangular shape above (4cm)

Area = (4)(9) = 36 (shaded face)

Hope this helps

The Ruiz family's laptop has the greater initial resale value. During the first four years, the resale value of the Sawyer family's laptop decreases at an average rate faster than​ the resale value of the Ruiz family's laptop.

The initial resale value of the laptops for both families are the same, as the function f(x) models the resale value for both families and does not differentiate between them.

During the first four years, the average rate of change in the resale value of the Sawyer family's laptop can be found by finding the average rate of change of the function f(x) over the interval [0, 4]. The average rate of change during this interval can be approximated by finding the average rate of change over a smaller interval within [0, 4]. For example, finding the average rate of change over the interval [0, 1] and then multiplying by 4 to find the average rate of change over the entire interval [0, 4].

Correct Question :

The Sawyer and Ruiz families each bought a laptop computer at the same time resale values in dollars of the laptops are modeled by the given function where X is the number of years that each family has owned the laptop.

The ____ family's laptop has the greater initial resale value. During the first four years, the resale value of the Sawyer family's laptop decreases at an average rate ____ the resale value of the Ruiz family's laptop.

1st blank:

Ruiz

Sawyer

2nd blank:

slower than

equal to

faster than​

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