Absolute Value Functions:
Determine the value of y, if x is -7

To minimize our double integral, we want to find the region over which the function we are integrating has negative values

x^2 +y^2 − 9</= 0

Solving the inequality

x^2 +y^2 </= 9

We can verify this solution by graphing. f(x,y) = x^2 +y^2 − 9

Our function is a paraboloid. Only the negative values of f(x, y) are graphed here.

We can see that f(x, y) is negative below the circle in the x-y plane given by the formula x^2 + y^2 <= 9. If we were to include any points outside of this region in our integral, we would add positive values of f(x, y) to our double integral and the integral wouldn't be minimized

therefore, R : x^2 +y^2 </= 9

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Answer:

519.2 in²

Step-by-step explanation:

First find the slant height  (s) of the triangle that makes up 1 side of the pyramid:

s² = 18² + (24/2)² = 468

s= √468 = 21.63

Now find the area of one side of the pyramid:

A = 1/2bh = 1/2(24)(21.63) = 519.2 / 2 = 259.56

Total area (all 4 sides) of the pyramid = 4(259.56) = 1038.2

1/2 the surface area is yellow:

1038.2 / 2 = 519.2 in²

Answer: 25.13cm

Step-by-step explanation:

Answer:

31.4 cm

Step-by-step explanation:

C=2πr

Plug in numbers: C= 2 * 3.14 * 5

The annual percentage rate, APR, for the loan is about 5.486%

The annual percentage rate is the cost of a loan or interest on an investment per year during the loan term or period.

Price of the car = $35,000.00

The down payment = 20% of $35,000 = $7,000

The installment loan details include;

The number of installment = 48 monthly installment

The amount paid each month = $651

The total payment, A= 48 × 651 = 31248

The equal monthly installment (EMI) formula is presented as follows;

[tex]EMI = P\cdot \frac{r\cdot (1+r)^n}{(1+r)^n -1}[/tex]

Where;

EMI = The equal monthly installment paid = $651

P = The principal amount of the loan

r = The interest rate

n = Number of payments to be made = 48

The principal is therefore;

P = $35,000.00 - $7,000.00 = $28,000.00

Therefore;

The

[tex]651 = 28000\times \frac{r\cdot (1+r)^{48}}{(1+r)^{48} -1}[/tex]

Using an online calculator and by trial and error, or by graphing we get;

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The integer that represents the score in points is -42

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

Thumbs down = 42

A thumbs down indicates a negative rating

So, we have

Point = -42

Hence, the score is -42

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Answer:

n = [tex]\frac{1}{3}[/tex]

Step-by-step explanation:

for the equation to be true for all values of x , then both sides have to be the same.

expand both sides and equate like terms

left side

12(3n - [tex]\frac{1}{2}[/tex] x) ← distribute each term in the parenthesis by 12

= 36n - 6x

right side

- [tex]\frac{2}{3}[/tex] (9x - 18) ← distribute each term in the parenthesis by - [tex]\frac{2}{3}[/tex]

= - 6x + 12

we require

36n - 6x = - 6x + 12

that is

36n = 12 ( divide both sides by 36 )

n = [tex]\frac{12}{36}[/tex] = [tex]\frac{1}{3}[/tex]

for the equation to be true for all x then n = [tex]\frac{1}{3}[/tex]

Answer:

4357

Step-by-step explanation:

The required measure of the segment CF in the given triangle is 7.5 units.

The triangle is a geometric shape that includes 3 sides and the sum of the interior angle should not be greater than 180°.

Here,

The median of the triangle is the line joining the midpoint of the side to the opposite vertex, so if f is the midpoint then,
CF = EC/2
CF = 15 / 2
CF  = 7.5 units

Thus, the required measure of the segment CF in the given triangle is 7.5 units.

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Answer:

[tex]\boxed{x^4+4x^3y+6x^2y^2+4xy^3+y^4\\}[/tex]

Step-by-step explanation:

Pascal’s triangle defines the coefficients which appear in binomial expansions. That means the nth row of Pascal’s triangle comprises the coefficients of the expanded expression of the polynomial (a + b)ⁿ

To create Pascal's triangle for any n in (a + b)ⁿ ,

Here is Pascal's triangle for the first 5 rows. The first row is row 0

Row #                                      

0                  1              

1               1    1            

2            1    2    1        

3         1    3    3    1      

4      1    4    6    4    1

[tex]\textrm{ Here row 0 represents $(a + b)^0$ }[/tex] =   [tex]\boxed{\textbf{1}}[/tex]

[tex]\textrm{Row 1 corresponds to $(a + b)^1$} =[/tex]  [tex]\boxed{1}\;a + \boxed{1}\;b[/tex]

[tex]\textrm{Row 2 corresponds to $(a + b)^2$} = \boxed{1}\;a^2 + \boxed{2} \;ab +\boxed{1}\;b^2[/tex]

[tex]\textrm{Row 3 corresponds to $(a + b)^3$} = \boxed{1}\;a^3 + \boxed{3}\;a^2b + \boxed{3}\;ab^2 + \boxed{1}\;b^3[/tex]

[tex]\textrm{Row 4 corresponds to $(a + b)^4$} = \boxed{1}\;b^4 + \boxed{4} \;a^3b +\boxed{6}\;a^2b^2 + \boxed{4} \;ab^3 + \boxed{1} \;b^4[/tex]

For the specific problem, (4x + 3y)³ use coefficients of row 3 and use 4x instead of a and 3y instead of b

[tex]\mbox{\normal(4x + 3y)^3= \boxed{1}\;(4x)^3 + \boxed{3}\;(4x)^2(3y) + \boxed{3}\;4x(3y)^2 + \boxed{1}\;(3y)^3}[/tex]

[tex]= \boxed{x^4+4x^3y+6x^2y^2+4xy^3+y^4\\} \;\;\; \textrm{ANSWER}[/tex]

Step-by-step explanation:

l don't know how to solve

it

The sine function for this problem is defined as follows:

y = 2sin(2x).

The sine function for this problem is defined as follows:

y = Asin(B(x - C)) + D.

The parameters are described as follows:

The function has a minimum value of -2 and a maximum value of 2, for a difference of 4, hence the amplitude is obtained as follows:

2A = 4

A = 2.

The period is of π, hence the coefficient B is given as follows:

2π/B = π

B = 2.

The function does not have a phase shift (when x = 0, y = 0), neither a vertical shift (between 2 x -1 = -2 and 2 x 1 = 2), hence it is defined as follows:

y = 2sin(2x).

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The correct answer is option 3. [tex]4x^2 + 17[/tex] expressions represent the total perimeter of her sandwich, and if x = 1.2, the approximate length of the crust is 21.8 centimeters.

The first expression, [tex]4x^2 + 17[/tex] , represents the total perimeter of the sandwich. When x = 1.2, the expression evaluates to[tex]4 * 1.2^2 + 17 = 21.8[/tex].

So the total perimeter of the sandwich is 21.8 cm. However, this length is not the length of the crust, since the length of the crust would only be a portion of the total perimeter.

The expression [tex]4x^2 + 17[/tex] represents a mathematical formula for the total perimeter of the sandwich. The variable x represents the thickness of each slice of bread. The expression [tex]4x^2[/tex] represents the total length of the four edges of the bread slices, and the 17 represents the total length of the filling. When x = 1.2, the expression evaluates to 21.8, which represents the total perimeter of the sandwich.

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Okun's law states that changes in unemployment and production are inversely related, with a coordinate of -2.

(a) When the unemployment rate stays the same in a given year, Okun's law's number 3.5 indicates the anticipated percent rise in yearly US national production.

(b) In accordance with Okun's law, a year in which unemployment increases from 5% to 8% results in a 1% decrease in national production.

(c) In accordance with Okun's law, a year in which production is equal to the previous year results in a decline of 1.75% in the unemployment rate.

(d) The inverse link between the unemployment rate and national output is represented by the coefficient -2 in Okun's law, which states that a rise in the unemployment rate would result in a fall in national production.

According to Okun's Law, the annual US national production's percent change (y) and the unemployment rate's percent change are connected (u). When the unemployment rate stays constant in a given year, the projected percent increase in national production is represented by the constant 3.5 in Okun's law. The inverse link between the unemployment rate and national output, represented by the coefficient -2 in Okun's law, states that a rise in the unemployment rate will result in a fall in national production. As a result, if unemployment increases from 5% to 8%, national production will decline by 1%. The change in the unemployment rate would need to be a decrease of 1.75 percent for national production to be the same as it was the previous year.

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The calculation is performed in three stages, and the result represents the volume of the object in cubic units.

The expression ∭ e ⎛ ⎝y ln x z ⎞ ⎠dv represents a triple integral, which is a type of mathematical operation that calculates the volume of a 3-dimensional object bounded by certain conditions.

In this particular case, the volume is defined by the limits of integration (1 ≤ x ≤ e, 0 ≤ y ≤ ln x, 0 ≤ z ≤ 1), meaning that it considers only the points in the space that satisfy these conditions.

To understand the triple integral, it is important to understand the concept of single and double integrals.

The calculation of a triple integral is performed in three stages.

First, the integral with respect to x is calculated, then the integral with respect to y, and finally the integral with respect to z.

The expression e ⎛ ⎝y ln x z ⎞ ⎠ represents the integrand, or the function that we are integrating. In this case, it is a product of three functions of x, y, and z.

The ∭ symbol represents the triple integral and the dv represents the differential of the volume, which is used to calculate the volume in small increments.

In conclusion, the triple integral

=> ∭ e ⎛ ⎝y ln x z ⎞ ⎠dv

calculates the volume of a 3-dimensional object defined by the limits of integration.

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Answer:

Step-by-step explanation:

The diagonals of a quadrilateral divide it into two congruent triangles. In this case, the diagonals of quadrilateral ABCD with vertices A (-4,9), B (3,9), C(2,3), D(-5,3) are AC and BD.

To find the intersection point of the two diagonals, we can use the equations of the two lines.

The equation of line AC can be represented as y = mx + b, where m is the slope of the line and b is the y-intercept.

m = (y2-y1)/(x2-x1)

m = (3-9)/(2-(-4)) = -6/6 = -1

b = y1 - mx1

b = 9 - (-1)(-4) = 9+4 = 13

The equation of the line is y = -x + 13

Similarly, the equation of line BD can be represented as y = nx + c, where n is the slope of the line and c is the y-intercept.

n = (y4-y3)/(x4-x3) = (3-9)/(-5-3) = -6/8 = -3/4

c = y3 - nx3

c = 3 - (-3/4)(-5) = 3 + 15/4 = 27/4

The equation of the line is y = -3/4 x + 27/4

Now we can find the point of intersection (x,y) by solving the following system of equations

-x + 13 = -3/4 x + 27/4

Multiply the equation 1 by 4

-4x + 52 = -3x + 27

Add 3x to both sides

-x + 52 = 27

Subtract 27 from both sides

-x = -25

x = 25

Now we can substitute this value of x in any of the equation to find the value of y

y = -x + 13

y = -25 + 13

y = -12

So, the coordinates of the intersection of the diagonals of quadrilateral ABCD are (25,-12).

The motion of the train, obtained from the velocity time graph, can be described as follows;

a) In the first 15 minutes, the train is accelerating at 200 km/h²

b) The distance between the two cities is 125 kilometers

c) The average speed is 62.5 km/h

The velocity of the train is the rate at which the displacement of the train changes with time of motion.

The graph indicates that 8 units = 1 hour

1 hour = 60 minutes

Therefore; 8 units = 60 minutes

1 unit = 60/8 minute/unit = 7.5 minutes

a) Acceleration = Change in velocity/Time = Δv/Δt

Therefore, acceleration of the train in the first 15 minutes = The change in the velocity in the first 15 minutes, divided by the time

The velocity of the train after 15 minutes = 50 km/h

The initial velocity of the train = 0 km/h

The change in time, Δt = 15 minutes = 0.25 hours, therefore;

Acceleration = (50 - 0)/0.25 = 200

(b) The distance is the area under the velocity time graph, therefore, the distance between the two cities is found as follows;

d = 0.5 × 0.25 × 50 + 6 × 0.25 × 50 + (1 + 0.5) × 0.5 × 50 + 0.5 × 0.25 × 50 = 125

(c) The average speed = Total distance ÷ Time

The time it takes the train to complete the journey = 2 hours

The average speed = 125 km/(2 hrs) = 62.5 km/h

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The system will have solution when an arbitrary constant have infinitely many solutions when a-2b+c =0

What is Linera Alegebra?

The study of the planes and lines, vector spaces, and mappings needed for linear transforms is known as linear algebra. It was first defined in the 1800s to help solve systems of linear equations and is a relatively new subject of research.

Given,

x + y - z = 2x + 2y + z = 3x + y + (k^2-5)z

[tex]\left[\begin{array}{ccc}1&1&-1\\1&2&1\\1&1&R^{2}-5 \end{array}\right] \left[\begin{array}{ccc}2\\3\\R\end{array}\right] = (A:B)[/tex]

Apply [tex]R_{2}[/tex] ⇒ R2 -R1, R3⇒R3-R1

[tex]\left[\begin{array}{ccc}1&1&-1\\0&1&2\\0&0&R^{2}-4 \end{array}\right]\left[\begin{array}{ccc}2\\1\\R-2\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc}1&1&-1\\0&1&2\\0&0&(K-2)(R+2)\end{array}\right] \left[\begin{array}{ccc}2\\1\\R-2\end{array}\right][/tex]

(1) For K = -2 the system has no solution as we get

[tex]\left[\begin{array}{ccc}1&1&-1\\0&1&2\\0&0&0\end{array}\right] \left[\begin{array}{ccc}2\\1\\-4\end{array}\right][/tex]

(2)For k =2 the system has infinite solution as we get

[tex]\left[\begin{array}{ccc}1&1&-1\\0&1&2\\-0&0&0\end{array}\right] \left[\begin{array}{ccc}2\\1\\0\end{array}\right][/tex]

(3) for K ∈ R, R∉ (-2,2) Here the system will have unique solution as doe above here we have

[tex]\left[\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right] \left[\begin{array}{ccc}A\\B\\C\end{array}\right][/tex]

Apply R2 = R2- 4R1

R3 = R3 -7R1

R1 To get

[tex]\left[\begin{array}{ccc}1&2&3\\0&-3&-6\\0&-6&-12\end{array}\right] \left[\begin{array}{ccc}A\\B-4A\\C-7A\end{array}\right][/tex]

Apply R2= R2 /3 To get

[tex]\left[\begin{array}{ccc}1&2&3\\0&1&2\\0&-6&-12\end{array}\right] \left[\begin{array}{ccc}A\\4A-B/3\\C-7A\end{array}\right][/tex]

Apply R1 = R1 - 2(R2)

[tex]\left[\begin{array}{ccc}1&0&-1\\0&1&2\\0&0&0\end{array}\right] \left[\begin{array}{ccc}-5A+2B/3\\4A-B/3\\A-2B+C\end{array}\right][/tex]

The system will have solution when an arbitrary constant have infinitely many solutions when a-2b+c =0

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Answer:

4/5

Step-by-step explanation:

The constant of proportionality is the number that relates the two quantities in a proportion. In the proportion (2/5) and (1/2), the constant of proportionality is the same for both fractions. To find the constant of proportionality, you can divide the two fractions and compare the results.

For example:

(2/5) / (1/2) = (2/5) * (2/1) = 4/5

So the constant of proportionality is 4/5.

The fish pulled the fishing line at a distance of 402.92 cm ≈ 402 cm.

We can use the formula for the circumference of a circle to find the distance the fishing line was pulled:

C = 2πr

Where C is the circumference, π (pi) is approximately equal to 3.14, and r is the radius of the spool.

The circumference of the spool is:

C = 2 × 3.14 × 4 = 25.12 cm

So, the distance the fishing line was pulled is:

25.12 × 16 = 402.92 cm

So, the fish pulled the fishing line at a distance of 402.92 cm ≈ 402 cm.

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--The given question is incorrect; the correct question is

"Bilal's fishing line was on a spool with a radius of 4cm. Suddenly, a fish pulled on the line, and the spool spun 16 times before Bilal began to reel in the fish. What is the distance the fish pulled the fishing line?"--

The number of minutes that Greg rode on the scooter is; 69 minutes

Algebraic word problems are defined as questions that require translating sentences to equations, then solving those equations. and a single variable.

The general form for the equation of a line in slope intercept form is;

y = mx + c

where;

m is slope

c is y-intercept

Initial fee paid is $2. Thus, y-intercept = $2.

Cost per minute = 12 cents of $0.12 and as such the slope is $0.12.

Thus, equation of line here is;

y = 0.12x + 2

where x is number of minutes

If total cost was $10.28, then;

10.28 = 0.12x + 2

0.12x = 8.28

x = 8.28/0.12

x = 69 minutes

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Answer:

y = 4

x= -6

Step-by-step explanation:

Substitution; so pretty much finding one equation and substituting it into the second

Let's find the equation of the top one... we'll find X (you can find either x or y though)

2x - 3y = -24

We'll add 3y to both sides

2x = 3y - 24

Dividing by 2 on both sides

x = (3y-24)/2... Simplifying it to x = 1.5y - 12

Let's put it into x + 6y = 18. Since we found X

1.5y - 12 + 6y = 18

Let's add 12

1.5y + 6y = 30

Combining like terms

7.5y = 30

Lastly, we'll divide by 7.5

y = 4

So now that we finally found Y, we can use either equation to solve for X. I'll do both to show both equations end up with the same X

2x-3y =-24

2x-3(4)=-24

2x-12=-24

2x=-12

x=-6

---------

x+6y=18

x+6(4)=18

x+24=18

x=-6

The number of items that should be tested so that the probability that one or more items are found defective is at least 99% is:

234 items.

To determine the number of items that need to be tested to have a 99% probability that one or more items are found defective, we need to use the binomial distribution and its cumulative distribution function (CDF).

The binomial distribution is used to model the number of successful outcomes in a fixed number of independent trials, where each trial has a probability of success p.

Let's assume that the probability of finding a defective item is p.

To find the minimum number of items that need to be tested, we need to solve the equation:

Where n is the number of items tested and (1 - p)^n is the probability that none of the items are defective.

To solve this equation, we can use a numerical method or a spreadsheet software, or we can use an approximate formula:

For a typical value of p = 0.01 (1% defect rate), the minimum number of items that need to be tested is approximately:

So, in this case, it would be advisable to test at least 234 items to have a 99% probability that one or more items are found defective.

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Answer:

[tex]\left(-\frac{3}{5},\frac{4}{5} \right)[/tex]

Step-by-step explanation:

We have to find [tex]n\in\mathbb{N}[/tex] such that [tex]\frac{1-n^2}{1+n^2} \neq 0[/tex] and [tex]\frac{2n}{1+n^2} \neq 0[/tex].

If we choose [tex]1\neq n\in\mathbb{N}[/tex]   [tex]\left(\frac{1-n^2}{1+n^2},\frac{2n}{1+n^2} \right)[/tex] is a rational point not on the x-axis or y-axis but on the unit circle.

Let [tex]n=2[/tex]

[tex]\left(\frac{1-2^2}{1+2^2},\frac{2*2}{1+2^2} \right)[/tex]

[tex]\left(\frac{1-4}{1+4},\frac{4}{1+4} \right)[/tex]

[tex]\left(-\frac{3}{5},\frac{4}{5} \right)[/tex]

The length of the side AC of the right-angle triangle will be 117 units.

The Pythagoras theorem states that the sum of two squares equals the squared of the longest side.

The Pythagoras theorem formula is given as

H² = P² + B²

The tangent and radius of the circle intersect at the right angle. Then the value of 'x' is given as,

AC² = AB² + B²

x² = 45² + 108²

x² = 2025 + 11664

x² = 13689

x = 117 units

The length of the side AC of the right-angle triangle will be 117 units.

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Φ(x) = x^m is a solution for the equation a , when m = -9 or m = 1

for function

Φ(x) = x^m

then

dΦ/dx (x) = m*x^(m-1)

d²Φ/dx² (x) = m*(m-1)*x^(m-2)

then

for a expression

3x^2 (d^2y/dx^2) + 11x(dy/dx) - 3y = 0

3x^2*m*(m-1)*x^(m-2) + 11*x* m*x^(m-1) - 3*x^m = 0

3*m*(m-1)*x^m + 11*m*x^m- 3*x^m = 0

dividing by x^m

3*m*(m-1) + 11*m - 3 =0

3*m² + 8 m - 3 =0

m= [-8 ± √(64 + 4*3*3)]/2 = (-8±10)/2  

m₁ = -9 , m₂= 1

then Φ(x) = x^m is a solution for the equation a , when m = -9 or m = 1

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The measure of angle POQ is 118°.

Having three edges and three vertices, a triangle is a three-sided polygon. The fact that a triangle's internal angles add up to 180 degrees is its most crucial characteristic. This characteristic is known as the triangle's angle sum property.

Given ΔPQR

∠R = 62°

QS and PT are altitudes,

Take ΔQRS

∠R = 62°

∠QSR = 90°

so, ∠SQR + ∠QSR + ∠R = 180°

∠SQR = 180 - 90 - 62

∠SQR = 28°

Taking ΔPRT

∠R = 62°

∠PTR = 90°

so, ∠RPT + ∠PTR + ∠R = 180°

∠RPT = 180 - 90 - 62

∠RPT = 28°

now let  ∠TPQ = x° and ∠SQP = y°

in ΔPQS,

∠SPQ = 28 + x

∠PQS = y°

∠QSP = 90°

∠QSP + ∠PQS + ∠SPQ = 180°

90 + y + 28 + x = 180

x + y = 180 - 90 - 28

x + y = 62°

now in  ΔPOQ

∠OPQ = x°, ∠OQP = y°

∠POQ + ∠OPQ + ∠OQP = 180°

∠POQ + x + y = 180

sunstitute value of x + y =  62°

∠POQ = 180 - 62

∠POQ = 118°

Hence the value of angle POQ is 118°.

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The system of equations that represents the given situation is:

q + d = 35

q*0.25 + d*0.10 = 4.90

Here we want to write a system of equations that represents the given situation.

First, let's define the two variables:

q = number of quarters.

d = number of dimes.

There are 35 coins in total, so the first equation is:

q + d = 35

And the value of these coins is $4.90, then the second equation is:

q*0.25 + d*0.10 = 4.90

Then the system of equations is:

q + d = 35

q*0.25 + d*0.10 = 4.90

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