4.7.4 Modeling Pumpkin launch, we were covering graphs of quadratic functions please help! Thank you

To find the volume of a rectangular prism, multiply its 3 dimensions: length x width x height

V1=LWH= 32

Ex

2*4*4=32

Let's multiply all dimension by 2

4 * 8 * 8 =256

Answer:B. The volume is multiplied by 8.

In general, the linear regression formula is shown below

When the pile is 22 feet high, the height of the pile changes to 20/1089π.

Given (dV)/(dt) = 20, h = 22 feet , dh /dt =?

The volume of cone is V = 1/3 * pi * r ^ 2 * h

d=3h

2r = 3h

r = (3h)/2

V = 1/3 * pi * r ^ 2 * h

= 1/3 * pi * ((3h)/2) ^ 2 * h

= 1/3 * pi((9 * H ^ 2 )/4) * h

= (9pi)/12 * h ^ 3

Differentiate w.r.to t

(dV)/(dt) = (3pi)/4 * (3h * h^ 2) * (dh)/(dt)

(dV)/(dt) = (9pi)/4 * (h ^ 2) * (dh)/(dt)

(dV)/(dt)=20, h=22 feet

20 = (9pi)/4 * (22 ^ 2) * (dh)/(dt)

(dh)/(dt) = 80/ (9pi * (22^ 2))

h^ t = 20/ 1089 *pi   ft / min

Therefore, The height of the pile changing when the pile is 22 feet high is 20/1089π

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When b²−4ac=0 there is one real root.

When b²−4ac>0 there are two real roots.

When b²−4ac<0 no real roots or two complex roots

First equation

-x²-4x+7

b²−4ac>0, then equation -x²-4x+7 has two real roots.

Second equation

-2x²+9x-11

b²−4ac<0, then equation -2x²+9x-11 has two complex roots.

Third equation

3x²-6x+3

b²−4ac=0, then equation 3x²-6x+3 has one root.

The equation of each line in slope-intercept form are:

1. y = -2x + 3

2. y = -3/5x - 6

3. y = 1/4x + 8

If we are given a point (a, b) and the slope of a line (m), we can write the linear equation in slope-intercept form by first plugging in the given values into y - b = m(x - a), then rewrite it in the slope-intercept form as y = mx + b.

1. Slope (m) = -2

Point (a, b) = (0, 3)

Plug in the values into y - b = m(x - a):

y - 3 = -2(x - 0)

y - 3 = -2x + 0

y = -2x + 0 + 3

y = -2x + 3 [slope-intercept form]

2. Slope (m) = -3/5

Point (a, b) = (-5, -3)

Plug in the values into y - b = m(x - a):

y - (-3) = -3/5(x - (-5))

y + 3 = -3/5x - 3

y + 3 - 3 = -3/5x - 3 - 3

y = -3/5x - 6 [slope-intercept form]

3. Slope (m) = 1/4

Point (a, b) = (-8, 6)

Plug in the values into y - b = m(x - a):

y - 6 = 1/4(x - (-8))

y - 6 = 1/4(x + 8)

y - 6 = 1/4x + 2

y - 6 + 6 = 1/4x + 2 + 6

y = 1/4x + 8 [slope-intercept form]

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The laws of exponents are useful to solve algebraic expression as shown below.

A branch of mathematics known as algebra deals with symbols and the mathematical operations performed on them. Variables are the name given to these symbols because they lack set values. We frequently observe constant change in specific values in our day-to-day situations. But the need to depict these shifting values is ongoing. These values are frequently represented in algebra by symbols such as x, y, z, p, or q, and these symbols are known as variables. In order to determine the values, these symbols are also subjected to various addition, subtraction, multiplication, and division arithmetic operations.

The laws of exponents (sometimes referred to as the rules of exponents) allow us to simplify expressions by using our understanding of what an exponent implies.

The goal is to write algebraic expressions as simply as you can in both situations.

Recall that an exponent indicates the number of times to multiply a number, variable, or expression by itself.

[tex]x^4=x \cdot x\cdot x\cdot x[/tex]

and, [tex](x+5)^3=(x+5)(x+5)(x+5)[/tex]

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Answer:

180 degrees converted is 3.14159  i think

Step-by-step explanation:

Answer:

< y = 17 degrees.

Step-by-step explanation:

x + y = 180 - 112 = 68 (as there are 180 degrees in a triangle)

Also, x = 3y, so substituting for x in the first equation:

3y + y = 68

4y = 68

y = 17.

So,

x = 3*17 = 51,

The positive exponent for the given expression is [tex]\frac{1}{4^6}[/tex].

The given expression is [tex](4^{-2})^3[/tex].

A positive exponent tells us how many times to multiply a base number, and a negative exponent tells us how many times to divide a base number. We can rewrite negative exponents like x⁻ⁿ as 1 / xⁿ.

The given expression can be simplified as follows

[tex](4^{-2})^3[/tex] = [tex]4^{-2\times3}=4^{-6}[/tex]

So, the positive exponent = [tex]\frac{1}{4^6}[/tex]

Therefore, the positive exponent for the given expression is [tex]\frac{1}{4^6}[/tex].

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The equation for the cost when the bowling alley charges $14 for the first game $10 for the second game is 24 + 5x.

An equation is the statement that illustrates that the variables given. In this case, two or more components are taken into consideration to describe the scenario

In this case, the bowling alley charges $14 for the first game $10 for the second game and $5 per game for every game after.

Let the number of games be illustrated as x.

Therefore, the equation will be:

14 + 10 + (5 × x)

= 24 + 5x

This illustrates the cost.

An overview was given as the information given is incomplete.

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The number is 234.

Step-by-step explanation:

Let the 3 digits be H , T and U so the number is HTU.

H + T + U = 9

T = 0.5H + 0.5U

U = 2H

So  from the last 2 equations

T = 0.5H + 0.5(2H)

T = 0.5H + H

T = 1.5H

So substituting for T and H is the first equation:

H + 1.5H + 2H = 9

4.5H = 9

H = 2.

So T = 1.5*H = 1.5 *2 = 3.

Answer:

0.321    repeating 1

Step-by-step explanation:

Volume of the cone = 50.29 mm³

radius = r = ?

height = 3mm

slant height = 5mm

To get the radius, we would apply pythagoras' theorem:

Hypotenuse² = opposite² + adjacent²

hypotenuse = 5mm

opposite = 3mm

adjacent = radius = ?

5² = 3² + radius²

25 = 9 + radius²

25 - 9 = radius²

16 = radius²

radius = √16

radius = 4

The structure must be designed so that there are no risks to the health and safety of the patients, staff, and general public. The main goal of layout exists to make clear that work, materials, and information flow through a system smoothly.

Layout design generally seeks to arrange organizational units inside a building so that the space is used to its fullest potential and total distances are kept to a minimum.

The structure must be designed so that there are no risks to the health and safety of the patients, staff, and general public. It must be able to endure the pressure and environmental conditions that they might encounter.

A facility's layout and design are crucial to a company's overall operations because they may both suit the needs of employees and maximize the efficiency of the production process. The main goal of layout exists to make clear that work, materials, and information flow through a system smoothly.

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Answer: C = 35 + 15 x 3 = 80

Step-by-step explanation:

15h = 15 x (h) = 15 x 3 = 45 + 35 = $80

C = (15 x 3) + 35 = 80

Answer:

[tex]4^{6}[/tex] or 4096

Step-by-step explanation:

When the bases are the same, you add the exponents.

Answer:

Step-by-step explanation:

[tex]\boxed{\bf\\a^n*a^m=a^{n+m}}[/tex]

[tex]\displaystyle(4)^2\cdot(4)^4\cdot(4)^0=(4)^{2+4+0}=4^6=4096[/tex]

Answer:

12.5 cups

Step-by-step explanation:

10/4 = 2.5

2.5 × 5 = 12.5

12.5 cups

The transformation from the graph of f(x) to g(x) is horizontal shift right 0.333 units and vertical shift is up 5 units.

The given functions are f(x) = x² and g(x)= 4(3x-1)²+5.

Functions are the fundamental part of the calculus in mathematics. The functions are the special types of relations. A function in math is visualized as a rule, which gives a unique output for every input x.

The transformation is

To find the transformation, compare the function to the parent function and check to see if there is a horizontal or vertical shift, reflection about the x-axis or y-axis, and if there is a vertical stretch.

Horizontal Shift: Right 0.333 Units

Vertical Shift: Up 5 Units

Reflection about the x-axis: None

Reflection about the y-axis: None

Vertical Compression or Stretch: Stretched

Therefore, the transformation from the graph of f(x) to g(x) is horizontal shift right 0.333 units and vertical shift is up 5 units.

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Part A

[tex]D(1,0) \longrightarrow D'(1-8, 0-3)=D'(-7, -3)\\\\E(7,2) \longrightarrow E'(7-8, 2-3)=E'(-1, -1)\\\\F(8,-1) \longrightarrow F'(8-8, -1-3)=F'(0, -4)\\\\G(2,-3) \longrightarrow G'(2-8, -3-3)=G'(-6, -6)[/tex]

Part B

Reflecting over the x axis means [tex](x, y) \longrightarrow (x, -y)[/tex].

[tex]D'(-7, -3) \longrightarrow D''(-7, 3)\\\\E'(-1, -1) \longrightarrow E''(-1, 1)\\\\F'(0, -4) \longrightarrow F''(0, 4)\\\\G'(-6, -6) \longrightarrow G''(-6, 6)[/tex]

The required solution of the expressions is (a) 7x⁵ + 13³ - 11x⁴ - x, (b) 5p² - 7p + 1.

Given that,
To arrange the like terms in columns and add them. - a) 2x³ - 7x⁴ + 7x⁵ and 11x³ - 4x⁴ – x b) -2p - 3p² and 1 - 5p + 8p²​.

The algebraic expression consists of constant and variable. eg x, y, z, etc.

Here,
(a)
= 2x³ - 7x⁴ + 7x⁵ + 11x³ - 4x⁴ – x
= 7x⁵ - 7x⁴  - 4x⁴+ 2x³ + 11x³– x
=  7x⁵ + 13³ - 11x⁴ - x,

(b)
=  -2p - 3p² +  1 - 5p + 8p²
= 1  -2p - 5p -3p² + 8p²
=  1 - 5p + 8p²​

Thus, the required solution of the expressions is (a) 7x⁵ + 13³ - 11x⁴ - x, (b) 5p² - 7p + 1.

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The required solutions to the given polynomials are 13x³- 11x⁴+ 7x⁵  – x and - 7p + 5p²​+ 1.

A polynomial is defined as a mathematical expression that has a minimum of two terms containing variables or numbers.

The polynomials are given in the question :

2x³ - 7x⁴ + 7x⁵, and 11x³ - 4x⁴ – x

-2p - 3p² and,1 - 5p + 8p²​

According to the question,

⇒ 2x³ - 7x⁴ + 7x⁵ + 11x³ - 4x⁴ – x

Rearrange the terms and apply the arithmetic operation,

⇒ 2x³ + 11x³- 7x⁴- 4x⁴ + 7x⁵  – x

⇒ 13x³- 11x⁴+ 7x⁵  – x

⇒ -2p - 3p² + 1 - 5p + 8p²​

Rearrange the terms and apply the arithmetic operation,

⇒ -2p - 5p - 3p²  + 8p²​+ 1

⇒ - 7p + 5p²​+ 1

Thus, the required solutions to the given polynomials are 13x³- 11x⁴+ 7x⁵  – x and - 7p + 5p²​+ 1.

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The number of feet from the tent pole to the rope along the ground will be 12 Feet.

Let:

        RQ = x

According to Pythagoras Theorem:

[tex]Hypotenuse^{2} = Perpendicular^{2} + Base^{2} \\PQ^{2} = PR^{2} + RQ^{2} \\15^{2} = 9^{2} + x^{2} \\225 = 81 + x^{2} \\x^{2} = 225 - 81\\x^{2} = 144\\x = 12[/tex]

Therefore, the number of feet from the tent pole to the rope along the ground will be 12 Feet.

Pythagoras Theorem is widely used to establish the relation between the three sides of a right-angled triangle. The applications of Phythagoras Theorem are the angle of depression, angle of elevation, etc. Even the Eiffel tower is one of its applications.

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The trend line is linear and the trend line has an equation of y = 500/9x  + 100

From the graph, we can see that:

The points appear to be on a straight line and it increases along the x-axis

This means that the points can appear on an approximate straight line

So, we can conclude that the trend is a linear trend

On the line, we have the following points

(x, y) = (0, 100) and (9, 600)

We start by calculating the slope of the points using the following slope equation

Slope = (y₂ - y₁)/(x₂ - x₁)

Where x and y are defined above

So, we have

Slope = (600 - 100)/(9 - 0)

Evaluate

Slope = 500/9

The equation is then calculated as

y = Slope(x - x₁) + y₁

Where

(x₁, y₁) = (0, 100) and Slope = 500/9

So, we have

y = 500/9 * (x - 0) + 100

Evaluate

y = 500/9x  + 100

Hence, the equation is y = 500/9x  + 100

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The largest number when the sum of 4 consecutive integers is 170 will be 44.

According to the question,

We have the following information:

Sum of four consecutive integers = 170

Now, let's take the first integer to be x.

So, the next 3 integers will be (x+1), (x+2) and (x+3) respectively.

Now, the largest number will be (x+3).

Adding 4 consecutive integers:

x+ (x+1) + (x+2) + (x+3) = 170

4x+6 = 170

4x = 170-6 (Moving 6 from the left hand side to the right hand will result in  the change of the sign from plus to minus.)

4x = 164

x = 164/4

x = 41

So, the next three consecutive integers will be 42, 43 and 44.

Hence, the largest of the four consecutive integers is 44.

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Answer:

Explanation:

Here, we want to determine the type of correlation relationship

From science, we understand that engaging in exercise helps internal organs to function better. These are the sources of endurance

Exercising for a couple of more times, all things being equal means there will be more endurance. We can largely say that, for most people, it is expected that the higher the number of time spent exercising, the higher the endurance being built

Thus, we can conclude there is a positive correlation between the two

The probability that the coin lands heads up six times in a row is 0.016

Probability is the term used in mathematics to understand how likely an event is to happen or can also be described as how likely it is that a proposition is true.

It can be determined by dividing the favorable outcomes or events by the total number of outcomes.

As the coin has two sides, the probability that when the coin lands, it lands heads up is determined as follows;

Probability = 1 / 2

Now the probability that the coin lands heads up six times in a row can be calculated as follows;

Probability = (1^6) / (2^6)

Probability = 1 / 64

Probability = 0.015625

Rounding it to three decimal places as follows;

Probability = 0.015625 = 0.016

Therefore, the probability that the coin lands heads up six times in a row is calculated to be 0.016

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Answer: [tex]6\sqrt{6}+4 \times 6\sqrt{6}-4[/tex] yards

Step-by-step explanation:

Let the length of the walkway be x yards. Then,

[tex](18-2x)(26-2x)=200\\\\4x^2 -88x+468=200\\\\4x^2-88x+268=0\\\\x^2 -22x+77=0\\\\x=\frac{22 \pm \sqrt{22^2 -4(77)}}{2}\\\\=11 \pm 3\sqrt{6}[/tex]

However, [tex]x=11+3\sqrt{6} > 18[/tex], which is illogical, so we take [tex]x=11-3\sqrt{6}[/tex].

So,

[tex]18-2x=18-2(11-3\sqrt{6})=6\sqrt{6}-4\\\\26-2x=26-2(11-3\sqrt{6})=4+6\sqrt{6}[/tex]

Distances Sandra recorded in order from least to greatest is 1/9, 5/18, 11/36, 7/18, 17/36, 2/3.

Given that, the fractions represent the distances that Sandra recorded is 7/18, 2/3, 1/9, 11/36, 5/18, 17/36.

In Mathematics, fractions are represented as a numerical value, which defines a part of a whole. A fraction can be a portion or section of any quantity out of a whole, where the whole can be any number, a specific value, or a thing.

Here, distance is 10/36

So, the equivalent fractions for the given fractions are

7/18 = 14/36

2/3 = 24/36

1/9 = 4/36

11/36

5/18 = 10/36

17/36

So, least to greatest is

4/36 < 10/36 < 11/36 < 14/36 < 17/36 < 24/36

1/9 < 5/18 <11/36 <7/18 <17/36 <2/3

Therefore, distances Sandra recorded in order from least to greatest is 1/9, 5/18, 11/36, 7/18, 17/36, 2/3.

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Answer:

30y^5 x^8

Step-by-step explanation:

3x^8 * 5y^4 * 2y

Add 4 and 1 and you'll get 5 so it'll be:

3x^8 * 5y^5 * 2

Which it'll look like

15x^8 y^5 *2=

30x^8 y^5