If 2y=x³ + 2x², find dy/dx when x=2.

[tex]\large\displaystyle\text{$\begin{gathered}\sf \left(\frac{4}{10}\times x\right)-(2 \times x)+\frac{8}{5}=\frac{4}{5} \end{gathered}$}[/tex]

Reduce the fraction 4/10, to its minimum expression, extracting and canceling 2.

Combine [tex]\bf{\frac{2}{5}x }[/tex] and -2x to get [tex]\bf{-\frac{8}{5}x}[/tex].

Subtract 8/5 from both sides.

Since 4/5 and 5/8 have the same denominator, join their numerators to subtract them.

Subtract 8 from 4 to get -4.

Multiply both sides by [tex]\bf{-\frac{5}{8}}[/tex], the reciprocal of [tex]\bf{-\frac{5}{8}}[/tex].

Multiply -4/5 by -5/8 (to do this, multiply the numerator by the numerator and the denominator by the denominator).

Reduce the fraction 20/40 to its lowest expression by extracting and canceling 20.

The first ten terms of the sequence are 1, 2, 8, 33, 148, 765, 4626, 32431, 259512, 2335689.

The n-th term of the sequence is aₙ ₊ ₁ = (aₙ + 1) · n.

A sequence is a set of elements generated by at least one condition, usually an equation. In this case, the sequence is generated by a recurrence formula:

a₁ = 1, aₙ ₊ ₁ = (aₙ + 1) · n        (1)

The first ten terms of the sequence are:

n = 1

a₂ = (a₁ + 1) · 1

a₂ = 2

n = 2

a₃ = (a₂ + 2) · 2

a₃ = 4 · 2

a₃ = 8

n = 3

a₄ = (a₃ + 3) · 3

a₄ = 11 · 3

a₄ = 33

n = 4

a₅ = (a₄ + 4) · 4

a₅ = 37 · 4

a₅ = 148

n = 5

a₆ = (a₅ + 5) · 5

a₆ = 153 · 5

a₆ = 765

n = 6

a₇ = (a₆ + 6) · 6

a₇ = (765 + 6) · 6

a₇ = 4626

n = 7

a₈ = (a₇ + 7) · 7

a₈ = 4633 · 7

a₈ = 32431

n = 8

a₉ = (a₈ + 8) · 8

a₉ = 32439 · 8

a₉ = 259512

n = 9

a₁₀ = (a₁₀ + 9) · 9

a₁₀ = 259521 · 9

a₁₀ = 2335689

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Answer:

5) y = 1x + 2

6) y = -0.5x + 6

Explanation:

5)

Given points are (-3, -1), (2, 4)

[tex]\sf slope \:formula: \dfrac{y_2 - y_1}{x_2- x_1} = \dfrac{\triangle y}{\triangle x} \ \ \ where \ (x_1 , \ y_1), ( x_2 , \ y_2) \ are \ points[/tex]

Here, find slope:

[tex]\rightarrow \sf slope \ (a) = \dfrac{4-(-1)}{2-(-3)} = \dfrac{5}{5} = 1[/tex]

Find Equation:

y = ax + q

Here found that a = 1, take (x, y) = (-3, -1)

[tex]\sf -1 = 1(-3) + q[/tex]

[tex]\sf q - 3 = -1[/tex]

[tex]\sf q = -1 + 3[/tex]

[tex]\sf q = 2[/tex]

So, in total equation:

y = 1x + 2

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

6)

Given points are (-2, 7), (2, 5)

[tex]\sf slope \:formula: \dfrac{y_2 - y_1}{x_2- x_1} = \dfrac{\triangle y}{\triangle x} \ \ \ where \ (x_1 , \ y_1), ( x_2 , \ y_2) \ are \ points[/tex]

Here, find slope:

[tex]\rightarrow \sf slope \ (a) = \dfrac{5-7}{2-(-2)} = -0.5[/tex]

Find Equation:

y = ax + q

Here found that a = -0.5, (x, y) = (-2, 7)

[tex]\sf 7 = -0.5(-2) + q[/tex]

[tex]\sf 7 = 1 + q[/tex]

[tex]\sf q = 7-1[/tex]

[tex]\sf q = 6[/tex]

So, in total equation:

y = -0.5x + 6

Answer:

Answer:

undefined

Step-by-step explanation:

because

the figure does not appear

Answer:

y-intercept: (0,3)

Step-by-step explanation:

To find the y-intercept, we set x to equal 0, giving the equation:

-3(0)+2y = 6

0+2y = 6

y = 3

The linear speed based on the information is 1326 km/h.

The radius will be:

= 6378 + 252

= 6630 km

The angular speed is 0.2. The linear speed will be:

= 6630 × 0.2

= 1326 km/h.

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answer:

x = - 9/2

y = -10

Step-by-step explanation:

make both equations look like y =

2x-3y=21 -> -3y= -2x+21 -> y = (-2x+21)/-3 -> y = 2/3x - 7

-6x + 2y = 7 -> 2y = 6x + 7 -> y = 3x + 7/2

now make the equations equal to each other (since they both equal y)

get x

2/3x - 7 = 3x + 7/2

-7 -7/2 = 3x - 2/3x

-14/2 -7/2 = 9/3x - 2/3x

-21/2 = 7/3x

(3/7) -21/2 = x

x = (3/7) -21/2

x = - 63/14

x = - 9/2

put it back in an equation to get y

y = 3x + 7/2

y = 3(-9/2) + 7/2

y = -27/2 + 7/2

y = -20/2

y = - 10

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Answer: B

Step-by-step explanation: Trust me bro

The equation of the parabola is [tex]y=\frac{1}{24} x^2+3[/tex]

None of the given options is correct

Given:

Focus: (0, 3)

Directrix: y = -3

Note that:

f - k = k - (-3)

f - 3 = 3 + 3

f = 6 + 3

f = 9

The equation of the parabola is of the form:

[tex]y=\frac{1}{4(f-k)} (x-h)^2+k[/tex]

Substitute f = 9, k = 3, h = 0 into the equation

[tex]y=\frac{1}{24} (x-0)^2+3\\\\y=\frac{1}{24}x^2+3[/tex]

The equation of the parabola is [tex]y=\frac{1}{24} x^2+3[/tex]

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The quantity of rainfall that fell in the second half of the month is; ¹/₆ of an inch.

We are given;

Rainfall during first half of the month = 1/2 inch

Total rainfall for the month = 2/3 inch

Now, to find the rainfall for the second half of the month, we just subtract the one for the first half from the one for the month to get;

Rainfall in second half of the month = ²/₃ - ¹/₂ = ¹/₆ inches

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The parabola vertex is (1,5), the focus of the parabola is (1,6), and the directrix y = 4.

The graph of a parabolic equation is a U-shape curve graph that is established from a quadratic equation.

From the given information:

[tex]\mathbf{y=\dfrac{1}{4}(x-1)^2+5}[/tex]

The vertex of an up-down facing parabolic equation takes the form:

y = ax² + bx + c  is [tex]\mathbf{x_v = -\dfrac{b}{2a}}[/tex]

Rewriting the given equation:

[tex]\mathbf{y = \dfrac{x^2}{4}-\dfrac{x}{2}+\dfrac{21}{4}}[/tex]

[tex]\mathbf{x_v = -\dfrac{b}{2a}}[/tex]

[tex]\mathbf{x_v = -\dfrac{(-\dfrac{1}{2})}{2(\dfrac{1}{4})}}[/tex]

[tex]\mathbf{x_v =1}[/tex]

Replacing the value of x into the equation, y becomes:

[tex]\mathbf{y_v = 5}[/tex]

Thus, the parabola vertex is (1,5)

From the vertex, the focus of the parabola is (1,6), and the directrix y = 4.

The graphical representation of the parabola is seen in the image attached below.

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The couple are owning income tax of $22,996.

What is tax?

Tax is the obligatory payment payable to the government on earnings.

In this case, the amount of tax to be paid is the sum of $9,235 plus 22% of the excess of the combined taxable income of the couple over $80,250.

Excess over $80,250=$142,800-$80,250

Excess over $80,250=$62,550

Initial tax amount=$9,235

extra tax amount=$62,550*22%

extra tax= 13,761

total tax amount=initial tax amount+ extra tax

total tax amount=$9,235+$13,761

total tax amount=$22,996

In essence, the couple would pay an initial tax of $9,235 plus an extra tax of $13,761 which gives a total tax liability of $22,996

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[tex] {\qquad\qquad\huge\underline{{\sf Answer}}} [/tex]

Let's get started ~

[tex]\qquad \sf  \dashrightarrow \: volume = length \sdot width \sdot height[/tex]

[tex]\qquad \sf  \dashrightarrow \: v = (24) \sdot(15) \sdot(2)[/tex]

[tex]\qquad \sf  \dashrightarrow \: v = 720 \: \: ft {}^{3} [/tex]

now, we know :

[tex]\qquad \sf  \dashrightarrow \: 1 \: \: yard {}^{3} = 27 \: \: ft {}^{3} [/tex]

[tex]\qquad \sf  \dashrightarrow \: 1 \: \: ft {}^{3} = \cfrac{1}{27} \: \: yard {}^{3} [/tex]

[tex]\qquad \sf  \dashrightarrow \: 720 \: \: ft {}^{3} = \cfrac{1}{27} \times 720 \: \: yard {}^{3} [/tex]

[tex]\qquad \sf  \approx\: 26.67 \: \: yard {}^{3} [/tex]

now, total cost of ingredients used is :

[tex]\qquad \sf  \dashrightarrow \: 26.67 \times 12.75[/tex]

[tex]\qquad \sf  \dashrightarrow \: 340.04[/tex]

so, total cost is $ 340.04

The number of people who can go to the game only on Sunday by applying the concept of set theory is equal to 5.

Total number of people for a trip to a baseball game = 15

Number of people planning to go on Saturday = 10

Number of people planning to go Sunday = 12

Let x be the number of people who can go on both Saturday and Sunday.

Then, the number of people who can go only on Saturday is 10 - x.

And the number of people who can go only on Sunday is 12 - x.

Total of 15 people are going to the baseball game.

Using set theory we have,

Number of people going on Saturday only + number of people going on Sunday only + number of people going on both = 15

Substitute the value we have,

⇒(10 - x) + (12 - x) + x = 15

Simplifying the above equation, we get,

⇒22 - x = 15

⇒ x = 7

The number of people who can only go to the game on Sunday is

= 12 - x

= 12 - 7

= 5

Therefore, using set theory number of people can only go to the game on Sunday is equal to 5 .

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Answer:

C

Step-by-step explanation:

y = (x - 2)² = x² - 4x + 4 → (1)

x + y - 2 = 0 → (2)

substitute y = x² - 4x + 4 into (2)

x + x² - 4x + 4 - 2 = 0

x² - 3x + 2 = 0

(x - 1)(x - 2) = 0

equate each factor to zero and solve for x

x - 1 = 0 ⇒ x = 1

x - 2 = 0 ⇒ x = 2

substitute these values into (2) and solve for y

x = 1 : 1 + y - 2 = 0 ⇒ y - 1 = 0 ⇒ y = 1 ⇒ (1, 1 )

x = 2 : 2 + y - 2 = 0 ⇒ y + 0 = 0 ⇒ y = 0 ⇒ (2, 0 )

the line intersects curve C at 2 points (1, 1 ) and (2, 0 )

Answer:

0.375 second and 3.5 second

Step-by-step explanation:

The position can be modeled by a quadratic function [tex]\displaystyle{h=-16t^2+62t+6}[/tex]. We are tasked to find the time when a ball reaches a height of 27 feet. Therefore, let h = 27:

[tex]\displaystyle{27=-16t^2+62t+6}[/tex]

Solve for t:

[tex]\displaystyle{27-6=-16t^2+62t}\\\\\displaystyle{21=-16t^2+62t}\\\\\displaystyle{16t^2-62t+21=0}[/tex]

Since the equation is quite complicated and more time-consuming to solve, i'll skip the factoring or quadratic part:

[tex]\displaystyle{t=0.375, 3.5}[/tex]

After done solving the equation, you'll get t = 0.375 and 3.5 seconds. These solutions are valid since both are positive values and time can only be positive.

Hence, it'll take 0.375 and 3.5 seconds for a ball to reach 27 feet.

Option b) is correct

In statistics, a histogram is a graphical representation of the distribution of data. A histogram is represented by a set of rectangles, adjacent to each other, where each column represents a certain type of data. Statistics is a stream of mathematics that is used in various fields. When figures are repeated in statistical data, this repetition is known as frequency and can be written in the form of a table, called a frequency distribution. Frequency distributions can be displayed graphically using different types of graphs and one of them is a histogram.

A histogram chart is used under certain conditions. They are:

Data should be numeric.

A histogram is used to check the shape of the data distribution.

It is used to check if the process changes from one period to another.

It is used to determine whether the output differs when it involves two or more processes.

It is used to analyze whether the given process meets the customer's requirements.

The data 1,2,12,14,15,16,18 is given

We need to find the histogram which best fits to the given data

Total number that falls between 0-9 is 2

Total Number that falls between 10-19 is 5

Total Number that falls between 20-29 is 1

Total Number that falls between 30-39 is 5

Total Number that falls between 40-49 is 3

Total Number that falls between 50-59 is 1

Hence the histogram is obtained

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Answer:

40. 398nine

41: 432six

42: 14714eight

Step-by-step explanation:

765-367=398

The 'nine' is probably confusing you

3x144=432

The 'six' is there to confuse you.

2x7357=14714

The 'eight' is again here to confuse you.

Ignore the 'six' and 'eight' and 'nine' they are making it harder for no reason

Answer: 178 ft^3

Step-by-step explanation:

A cylinder has the formula V=pi radius ^2 height

A cone has the formula V= 1/3 pi radius ^2 height

So 1/3 of the volume of the cylinder is the volume of a cone

1/3 of 534 = 178 ft^3

Answer:

The first one, you cannot factor because the two values do not share any common factors. Hence, the answer would just be:

a. 51x-25

The other one, we can factorise to:

b. 3x(4-x)

Step-by-step explanation:

For the second option:

Common factor between 12x and 3x^2 is 3x.

Take this out to the front, we are left with:

3x(4-x)

A website I recommend you use (math calculator with some instructions) is Symbolab.

https://www.symbolab.com/solver/factor-calculator/factor%2012x-3x%5E%7B2%7D?or=input

Considering the definition of an inequality, the inequality that can be used to determine x is 20x + 10≤ 80

An inequality is the existing inequality between two algebraic expressions, connected through the signs:

An inequality contains one or more unknown values ​​called unknowns, in addition to certain known data.

Solving an inequality consists of finding all the values ​​of the unknown for which the inequality relation holds.

A rental car company charges $20 per day to rent a car and $0.10 for every mile driven.

Addison wants to rent a car, knowing that:

On the other hand, x represents the maximum number of days Alyssa can afford to rent for while staying within her budget.

Then, the inequality that can be used to determine x is:

20x + 0.1×100≤ 80

So:

20x + 10≤ 80

In summary, the inequality that can be used to determine x is 20x + 10≤ 80

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Answer: (-2,2)

Step-by-step explanation:

The solution to a system of equations represented graphically is the point where the graphs intersect.

The equation that shows the proportional relationship between his defense and offense cards is y = 2x

An equation is an expression that shows the relationship between two or more numbers and variables.

Let y represent the defense cards and x represent the offense cards.

For every 2 offense cards, he owns 4 defense cards. Hence:

4 = 2m

m = 2

y = 2x

The equation that shows the proportional relationship between his defense and offense cards is y = 2x

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By reduction to the absurd, the two lines do not intercept each other because of the following absurd: 5 = - 4.

From statement we have two lines in space, whose vector forms are described below:

(x, y, z) = (1, - 9, 1) + t · (4, 1, 2)     (1)

(x, y, z) = (1, 0, - 9) + s · (5, - 1, 5)     (2)

The point of intersection satisfies the following condition:

(1, - 9, 1) + t · (4, 1, 2) = (1, 0, - 9) + s · (5, - 1, 5)

(1, - 9, 1) - (1, 0, - 9) = s · (5, - 1, 5) + t · (- 4, - 1, - 2)

s · (5, - 1, 5) + t · (- 4, - 1, - 2) = (0, - 9, - 8)

There is a point of intersection if the resulting system of linear equations has at least a solution:

5 · s - 4 · t = 0      (2)

- s - t = - 9      (3)

5 · s - 2 · t = - 8     (4)

By (2):

s = 4 · t/5

(2) in (3):

- 4 · t/5 - t = - 9

- 9 · t/5 = - 9

t/5 = 1

t = 5

(2) in (4):

5 · (4 · t/5) - 2 · t = - 8

4 · t - 2 · t = - 8

2 · t = - 8

t = - 4

By reduction to the absurd, the two lines do not intercept each other because of the following absurd: 5 = - 4.

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The definition of the piece-wise function is:

A piece-wise function is a function that has different definitions, depending on the input.

In this problem, for inputs between -2 and 2, the function is a line that goes through (-2,-4) and (2,6). The slope is:

m = (6 - (-4))/(2 - (-2)) = 2.5.

The y-intercept is of b = 1, hence the rule is:

[tex]f(x) = 2.5x + 1, -2 \leq x \leq 2[/tex]

For x greater than 2 and less than 6, the function is constant at -3, hence the rule is:

[tex]f(x) = -3, 2 < x < 6[/tex].

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Answer:

Absolute value parent function

Explanation:

What is an Absolute value function?

It is a function in Algebra in where the variable is inside the absolute value bars (| |)

f(x)=|x|

y=x on the right side and y=-x on the left side of the graph

Hope it helps!

Step-by-step explanation:

The formula for the angle between vectors

[tex] \alpha = \cos {}^{ - 1} ( \frac{uv}{ |u| |v| } ) [/tex]

To multiply vectors, multiply the first component and multiply the second component and add them.

(-4*-1) + (-3*5)= 4-15=-11.

To find magnitude of vectors, use the Pythagorean theorem

[tex]u = \sqrt{ { - 4}^{2} + { - 3}^{2} } = 5[/tex]

[tex]v = \sqrt{ - 1 {}^{2} + 5 {}^{2} } = \sqrt{26} [/tex]

so

[tex] |u| |v| = 5 \sqrt{26} [/tex]

Know we have,

[tex] \alpha = \cos {}^{ - 1} ( \frac{7}{5 \sqrt{26} } ) [/tex]

[tex] \alpha = 105.94[/tex]

in degrees,

[tex] \alpha = 1.849[/tex]

in radians

Answer:

115.6° (1 d.p.)

Step-by-step explanation:

To find the angle between two vectors:

**Please see attached for the triangle diagram**

Given vectors:

[tex]\textbf{u}=-4\textbf{i}-3\textbf{j}[/tex]

[tex]\textbf{v}=-\textbf{i}+5\textbf{j}[/tex]

Use Pythagoras Theorem to find the magnitude of each vector:

[tex]\implies |\textbf{u}|=\sqrt{(-4)^2+(-3)^2}=5[/tex]

[tex]\implies |\textbf{v}|=\sqrt{(-1)^2+5^2}=\sqrt{26}[/tex]

[tex]\overrightarrow{\text{UV}}=\textbf{v}-\textbf{u}=(-\textbf{i}+5\textbf{j})-(-4\textbf{i}-3\textbf{j})=3\textbf{i}+8\textbf{j}[/tex]

[tex]|\overrightarrow{\text{UV}}|=\sqrt{3^2+8^2}=\sqrt{73}[/tex]

Cosine Rule (for finding angles)

[tex]\sf \cos(C)=\dfrac{a^2+b^2-c^2}{2ab}[/tex]

where:

Find angle θ using the cosine rule:

[tex]\implies \cos(\theta)=\dfrac{|\textbf{u}|^2+|\textbf{v}|^2-|\overrightarrow{\text{UV}}|^2}{2|\textbf{u}||\textbf{v}|}[/tex]

[tex]\implies \cos(\theta)=\dfrac{5^2+\left(\sqrt{26}\right)^2-\left(\sqrt{73}\right)^2}{2(5)\left(\sqrt{26}\right)}[/tex]

[tex]\implies \cos(\theta)=\dfrac{-22}{10\sqrt{26}}[/tex]

[tex]\implies \theta=\cos^{-1}\left(\dfrac{-22}{10\sqrt{26}}\right)[/tex]

[tex]\implies \theta=115.5599652...^{\circ}[/tex]

Therefore, the angle between the vectors is 115.6° (1 d.p.).

Bearing is a topic that can be used to locate the position of an object at a given time. Therefore, the blimp is 80 m far away from its starting point. And in the North-East direction.

The speed of an object relates the distance it covers to the time taken to cover the said distance.

i.e speed = [tex]\frac{distance}{time taken}[/tex]

⇒ distance = speed x time taken

In the given question, the initial distance covered can be determined as;

distance = 40 mph x 3 h

              = 120 m

The final distance covered can be determined as;

distance = 40 mph x 1 h

              = 40 m

Let the distance from its starting point to its final location be represented by x.

Thus applying the Cosine rule, we have:

[tex]c^{2}[/tex] = [tex]a^{2}[/tex] + [tex]b^{2}[/tex] - 2abCos C

[tex]x^{2}[/tex] = [tex]120^{2}[/tex] + [tex]40^{2}[/tex] - 2(120 x 40) Cos 90

   = 14400 + 1600 - 9600

[tex]x^{2}[/tex] = 16000 - 9600

   = 6400

x = [tex]\sqrt{6400}[/tex]

 = 80

x = 80 m

Therefore, the distance of the blimp from its starting point to its final destination is 80 m.

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Answer: [tex]\frac{3x+1}{x^{2}-6}[/tex]

Step-by-step explanation:

(f/g) represents the division of function f by function g.