Solve the mathematical puzzle.

Determine the digits of Y

from these clues.

The digits of Y

add to 16.
The first digit is 4 times the third digit.
The second digit is 3 times the third digit.

Y is a three digit number.

Answer:

[tex]\sf x = \dfrac{-1 + i\sqrt{3} }{2} \quad or \quad \dfrac{-1 - i\sqrt{3} }{2}[/tex]

Explanation:

[tex]\sf Given : x^2 + x + 1 = 0[/tex]

Solve it using quadratic formula

[tex]\sf x = \dfrac{ -b \pm \sqrt{b^2 - 4ac}}{2a} \quad \:when\: \: ax^2 + bx + c = 0[/tex]

So, here constants are: a = 1, b = 1, c = 1

When the value's are substituted inside the formula

[tex]\sf \rightarrow x = \dfrac{-1 \pm \sqrt{1^2 - 4(1)(1)} }{2(1)}[/tex]

[tex]\sf \rightarrow x = \dfrac{-1 \pm \sqrt{-3} }{2}[/tex]

[tex]\sf \rightarrow x = \dfrac{-1 \pm i\sqrt{3} }{2}[/tex]

[tex]\sf \rightarrow x = \dfrac{-1 + i\sqrt{3} }{2} \quad or \quad \dfrac{-1 - i\sqrt{3} }{2}[/tex]

The answers are :

This problem can be solved using the quadratic equation.

x = -1 ± √1² - 4(1)(1) / 2a

x = -1 ± √1 - 4 / 2a

x = -1 ± √-3 / 2a

x = -1 ± i√3 / 2a (∴ i = √-1)

The possible values of x are :

Answer:

[tex]108 m^{2}[/tex]

Step-by-step explanation:

[tex]27\times4=108[/tex]

The angle of elevation for two parasailing boats traveling at the same speed is equal because they have the same length of tow line.

Parasailing is also referred to as parascending and it can be defined as a recreational activity which involves a person wearing an open parachute (canopy wing) and gliding through the air, while being towed by a boat.

By critically observing the image shown below, we can infer and logically deduce that the reason why the angle of elevation for two parasailing boats traveling at the same speed is equal, is simply because they have the same length of tow line.

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Complete Question:

Why is the angle of elevation for two parasailing boats traveling at the same speed equal?

The balance in the Work in Process Inventory on January 1, if this was the only unfinished job for Ryan Company, is $18,200.

The work-in-process inventory is made up of the costs of direct materials, direct labor, and manufacturing overhead at the beginning or end of the period.

Work in Process Balance on January 1:

Direct materials $6,300

Direct labor $7,000

Manufacturing overhead $4,900

Total = $18,200

What was the balance in the Work in Process Inventory on January 1 if this was the only unfinished job?

Thus, the balance in the Work in Process Inventory on January 1, if this was the only unfinished job for Ryan Company, is $18,200.

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Answer:

94 hamburgers

Step-by-step explanation:

Let's solve this question through the use of equations.

Start by defining the variables used.

Let the number of hamburgers and cheeseburgers sold on Friday be h and c respectively.

Form 2 equations using the given information.

Given that the total sold is 376,

h +c= 376 -----(1)

The number of cheeseburgers sold was thrice the number of hamburgers sold.

c= 3h -----(2)

Solving by substitution:

Substitute (2) into (1):

h +3h= 376

Now that we have an equation expressed only in terms of h, we can find the value of h.

4h= 376

Divide both sides by 4:

h= 376 ÷4

h= 94

Thus, 94 hamburgers were sold on Friday.

There is no relationship between the number of hours cycled and the number of hours worked (Option D).

A scatterplot is a graph that is used to show the association between two variables. The relationship between the variables is shown by the use of  line of best fit.

From the scatterplot shown, we can conclude that there is no relationship between the number of hours cycled and the number of hours worked (Option D).

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Answer:

3

Step-by-step explanation:

[tex]\frac{y+x}{xy} =3\\x+y=3xy\\xy+x+y=4\\xy+3xy=4\\4xy=4\\xy=1\\x^2y+xy^2=xy(x+y)=xy(3xy)=3(xy)^2=3(1)^2=3[/tex]

Answer:

3

Step-by-step explanation:

So we're given: [tex]\frac{1}{x} + \frac{1}{y} = 3[/tex] and that: [tex]xy+x+y=4[/tex]. And now we need to solve for: [tex]x^2y+xy^2[/tex].

Original equation:

[tex]\frac{1}{x} + \frac{1}{y} = 3[/tex]

Multiply both sides by xy

[tex]y+x=3xy[/tex]

Now take this and plug it as x+y into the second equation:

Original equation:

[tex]xy+x+y=4[/tex]

Substitute 3xy as x+y

[tex]xy + 3xy = 4[/tex]

Combine like terms:

[tex]4xy = 4[/tex]

Divide both sides by 4

[tex]xy=1[/tex]

Divide both sides by x:

[tex]y=\frac{1}{x}[/tex]

Original equation:

[tex]x^2y+xy^2[/tex]

Substitute 1/x as y

[tex]x^2(\frac{1}{x})+x(\frac{1}{x})^2[/tex]

Multiply values:

[tex]\frac{x^2}{x}+\frac{x}{x^2}[/tex]

Simplify:

[tex]x+\frac{1}{x}[/tex]

Substitute y as 1/x back into the equation:

[tex]x+y[/tex]

so now we just need to solve for x+y

Look back in steps to see how I got this:

[tex]y+x=3xy[/tex]

Divide both sides by 3

[tex]\frac{x+y}{3}=xy[/tex]

Original equation:

[tex]xy+x+y=4[/tex]

Substitute

[tex]\frac{x+y}{3}+x+y=4[/tex]

Multiply both sides by 3

[tex]x+y+3x+3y=12[/tex]

Combine like terms:

[tex]4x+4y=12[/tex]

Divide both sides by 4

[tex]x+y=3[/tex]

So now we finally arrive to our solution 3!!!!! I swear I felt like I was going in circles, and I was about to just stop trying to solve, because I had no idea what I was doing, sorry if I made some unnecessary intermediate steps.

Answer:

  (d) Gauss-Jordan Elimination

  80 carnations; 50 roses; 70 daisies

Step-by-step explanation:

The given relations can be written as equations, which can be expressed as one matrix equation. Any of the methods listed can be used to solve the matrix equation.

If we let c, r, d represent numbers of carnations, roses, and daisies ordered, respectively, then the given relations can be written as ...

  c + r + d = 200 . . . . . . 200 flowers were ordered

  0c -r +d = 20 . . . . . . . . . . . . . 20 more daisies than roses were ordered

  1.50c +5.75r +2.60d = 589.50 . . . . . the total value of the order

Written as a matrix equation, it will be of the form ...

  AX = B

where A is the square matrix of variable coefficients, X is the column vector of variables, and B is the column vector of equation right-side constants. This is the matrix equation:

  [tex]\left[\begin{array}{ccc}1&1&1\\0&-1&1\\1.50&5.75&2.60\end{array}\right] \left[\begin{array}{c}c\\r\\d\end{array}\right] =\left[\begin{array}{c}200\\20\\589.50\end{array}\right][/tex]

The mathematical operations required to find the equation solution can be briefly described as ...

Inverse Matrices

The coefficient matrix is inverted and multiplied by the constant column vector:

  [tex]X=A^{-1}B[/tex]

The inversion operation requires computation of 10 determinants, of which 9 are of 2×2 matrices. That's a total of about 39 multiplications, 9 divisions, and 20 additions.

Cramer's Rule

Using Cramer's rule requires computation of 4 determinants of 3×3 matrices. The total number of operations comes to about 48 multiplications, 3 divisions, and 20 additions.

Gaussian Elimination

To obtain the upper triangular matrix that results from Gaussian Elimination requires about 11 multiplications, 11 additions, and 2 divisions. This finds the value of one variable, but the others must be found by substitution into the remaining two equations, requiring an additional 3 multiplications and 3 additions.

Gauss-Jordan Elimination

This method starts with an augmented matrix that appends column vector B to the square matrix A. The result of this is shown in the attachment. It is a diagonal matrix with the variable values a direct result of the matrix operations. The calculator's RREF( ) function performs matrix row operations to transform the augmented matrix to this Reduced Row-Echelon Form. About 6 multiplications, 6 additions, and 4 divisions are required.

Clearly, Gauss-Jordan Elimination is the method that requires the least computational work, so it would probably be used for this question.

The attachment shows the order to be ...

__

Additional comment

The estimates of computational load presented by each of the solution methods are not intended to be exact counts. For this specific problem, some of the operations can be avoided due to the fact that some coefficients are already 1. Also, some computations are not needed simply because they are intended to produce an outcome that is already known. The intention is to give an idea of the relative difficulty of using these different methods.

In some cases, computationally less-efficient methods may be preferred because they are simpler to describe.

Given :

Midpoint formula :

[tex]\boxed {M = (\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})}[/tex]

Let's input the values !

⇒ M = (4 - 4 / 2, -5 + 5 / 2)

⇒ [tex]\boxed {M = (0,0)}[/tex]

The midpoint between the points is (0, 0).

Answer:

1/36

Step-by-step explanation:

There is an 1/6 chance of rolling 1 the first time and 1/6 chance of rolling a 4 the second time. 1/6 * 1/6 = 1/36 chance of rolling a 1 then 4.

The end behavior of the function is the one in option C.

"One end increases and one end decreases"

Here we have the function:

f(x) = -ln(2x) + 4.

Remember that for the natural logarithm, as x tends to zero the function tends to negative infinity.

And as x tends to infinity, the natural logarithm tends to infinity.

So, for our function where we have a negative sign before the logarithm, as x tends to zero the function tends to infinity and as x tends to infinity the function tends to zero.

Then the correct option is C:

"One end increases and one end decreases"

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Answer:

[tex]967.6cm^{2} (1 d.p)[/tex]

Step-by-step explanation:

Total Surface Area of Cylinder = [tex]2\pi rh+2\pi r^{2}[/tex]

Given from the question, r = 7 cm and h = 15cm

Lets Substitute r and h into the formula to find the Total Surface Area of the cylinder.

Total Surface Area of Cylinder = [tex]2\pi (7)(15)+2\pi (7)^{2} \\=210\pi +98\pi \\=308\pi cm^{2} \\=967.6cm^{2} (1 d.p)[/tex]

The value of x is 3 units.

We know that a portion of a line with two endpoints is referred to as a line segment. A line segment, in contrast to a line, has a known length. A line segment's length can be calculated using either metric measurements like millimeters or centimeters, or conventional measurements like feet or inches.

Given that FH is a line segment and FH = 18 units. G is a point on FH such that FG = 4x units and GH = 2x units.

So, we can write

4x + 2x = 18

i.e. 6x = 18

i.e. x = 18/6 = 3

Then, the value of x is 3 units. So, the first option is correct.

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Answer: A / 3

Step-by-step explanation: edge 2022

The slope is 4 and the y-intercept is 6.

The slope-intercept form is [tex]y=mx+b[/tex], where [tex]m[/tex] is the slope and [tex]b[/tex] is the y-intercept.

Attached is an image of the graph.

Answer:

27π cm² or 84.8 cm²

Explanation:

[tex]\sf Formula \ for \ area \ of \ sector = \dfrac{\theta}{360 } \ x \ \pi (radius)^2[/tex]

Shaded region angle: 360° - 90° = 270°

 Given radius: 6 cm

Applying formula:

[tex]\rightarrow \sf \dfrac{270}{360} \ x \ \pi (6)^2[/tex]

[tex]\rightarrow \sf 27\pi[/tex]

[tex]\rightarrow \sf 84.823 \ cm^2[/tex]

[tex]\rightarrow \sf 84.8 \ cm^2 \quad (rounded \ to \ nearest \ tenth)[/tex]

The equivalent expression of [tex]\sqrt{-12}[/tex] is [tex]2\sqrt{-3}[/tex]

The expression is given as:

[tex]\sqrt{-12[/tex]

Express -12 as 4 * -3

[tex]\sqrt{-12} = \sqrt{4 * -3}[/tex]

Expand the expression

[tex]\sqrt{-12} = \sqrt{4} * \sqrt{-3}[/tex]

Take the positive square root of 4

[tex]\sqrt{-12} = 2 * \sqrt{-3}[/tex]

Evaluate the product

[tex]\sqrt{-12} = 2\sqrt{-3}[/tex]

Hence, the equivalent expression of [tex]\sqrt{-12}[/tex] is [tex]2\sqrt{-3}[/tex]

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A bisector is a line that divides either a given line or an angle into two equal parts. The answer to the given question is in the attachments to this answer.

The process of bisection implies dividing a given angle or line into two equal parts. Thus a bisector should be constructed.

The construction required is as given below:

For figure 1:

For figure 2:

For figure 3:

The required construction is as shown in the attachments to this answer.

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Answer:

6 yds^3

Step-by-step explanation:

I find these easiest if you put all of the dimensions into yards and then multiply

18 ft =  6 yds

36 ft = 12 yds

3 in = 3/36 yds    = 1/12 yds     now multiply them together

6 x 12 x 1/12 = 6 yds^3

2t + 4 < -8

2t < -12

t < -6

For the inequality to be true, t must be a number that is less than -6.

Hope this helps!

The time it will take runner B to finish the race is 7/15 of a minutes

Time taken by runner B = 21/25 × 5/12 minutes

= (21 × 5) / (25 × 12)

= 105 / 300

= 7/15 minutes

Therefore, it takes runner B 7/15 of a minutes to run the same race as runner A.

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It's easier to sketch if you simplify [tex]f(x)[/tex]. Factorize the numerator and denominator - simple to do with a difference of cubes or squares.

[tex]\dfrac{x^3 - 1}{x^2 - 1} = \dfrac{(x - 1)(x^2 + x + 1)}{(x - 1) (x + 1)}[/tex]

If [tex]x\neq1[/tex], we can cancel the factors of [tex]x-1[/tex]. Note that [tex]f(1)[/tex] itself is undefined. For all intents and purposes, aside from the singularity at [tex]x=1[/tex], the graph of [tex]f(x)[/tex] will look exactly like the graph of

[tex]\dfrac{x^2 + x + 1}{x + 1}[/tex]

Now, rewrite this as

[tex]\dfrac{x(x + 1) + 1}{x + 1}[/tex]

Then when [tex]x\neq-1[/tex] (note that [tex]f(-1)[/tex] is also undefined), we can cancel [tex]x+1[/tex] to reduce this to

[tex]x + \dfrac1{x+1}[/tex]

On its own, the graph of [tex]x[/tex] is a line through the origin. When [tex]x[/tex] is a large number [tex]\frac1{x+1}[/tex] is small. But as [tex]x[/tex] gets closer to -1, the rational term blows up. Effectively, this means the graph of [tex]f(x)[/tex] looks like [tex]\frac1{x+1}[/tex] around [tex]x=-1[/tex], and far enough away it looks like [tex]x[/tex].

See the attached plot for a sketch of these details.

the intersection points are (1, 7) and (4, 13), So the correct option is A.

To see that, we need to see when the two equations:

y = -x²+7x+1

y = 2x + 5

Can be solved simultaneously for a point (x, y). So we need to solve a system of equations.

y = -x²+7x+1

y = 2x + 5

We can rewrite:

-x²+7x+1 = y =  2x + 5

Then we can solve this for x:

-x²+7x+1 =  2x + 5

x² - 5x + 4 = 0

This is just a quadratic equation, the solutions are:

[tex]x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4*(1)*(4)} }{2} \\\\x = \frac{5 \pm 3 }{2}[/tex]

So we have two values of x:

The correspondent values of y are:

y = 2*(4) + 5 = 13

y = 2*(1) + 5 = 7

Then the intersection points are (1, 7) and (4, 13)

Then the correct option is A. The dog will catch the frisbee.

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A circle is a curve sketched out by a point moving in a plane. If the distance from A to C is 6 units, then the distance from C to B is 6 units.

A circle is a curve sketched out by a point moving in a plane so that its distance from a given point is constant; alternatively, it is the shape formed by all points in a plane that are at a set distance from a given point, the centre.

If the distance from A to C is 6 units, then the distance from C to B is 6 units. This is because AC and BC are the radii of the circle, therefore, both are equal to each other.

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Answer: c and 6

Step-by-step explanation:

Factorial of a positive integer is the result of the repeated multiplication of all the integers from 1 to that considered integer. The number of ways Mrs Evans can display the 5 baskets is 120.

Factorial of a positive integer is the result of the repeated multiplication of all the integers from 1 to that considered integer.

Thus, if we want to take the factorial of 'n',(n being a positive integer), then:

[tex]n! = 1 \times 2 \times \cdots \times (n-1) \times n[/tex]

We usually write it in reverse order because, in calculations, its cancellation comes useful. Thus, we have:

[tex]n! = n \times (n-1) \times \cdots \times 2 \times 1[/tex]

Mrs. evans has 5 spaces in which to display 5 baskets. Therefore, the number of ways Mrs Evans can display the 5 baskets are,

Number of ways = 5! = 5×4×3×2×1 = 120

Hence, the number of ways Mrs Evans can display the 5 baskets is 120.

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The amount that the designer will have to pay in FICA taxes is:$3,978.

FICA taxes comprises of:

Social Security tax= 6.2%

Medicare tax= 1.45%

Hence:

FICA taxes=($52,000×6.2%)+($52,000×1.45%)

FICA taxes=$3,224+$754

FICA Taxes=$3,978

Therefore the amount that the designer will have to pay in FICA taxes is:$3,978.

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Answer:

The volume of the suitcase will be 199.5 [tex]ft^{3}[/tex]

Step-by-step explanation:

To find the volume of a 3D shape (in question, the suitcase) we use the formula length x width x height. With this in mind, we multiply 19 x 7 = 133. Multiply 133 by 1.5, which equals 199.5. With a final answer of 199.5, the volume of the suitcase will be 199.5 [tex]ft^{3}[/tex].

The value of y when x= 2401 is -10

An inverse variation is represented as:

[tex]y = \frac{k}{\sqrt x}[/tex]

Where k is the proportionality constant

When y = -70, x = 49

So, we have:

[tex]-70 = \frac{k}{\sqrt {49}}[/tex]

Take the square root of 49

[tex]-70 = \frac{k}{7}[/tex]

Multiply through by 7

k = -490

Substitute k = -490 in [tex]y = \frac{k}{\sqrt x}[/tex]

[tex]y = -\frac{490}{\sqrt x}[/tex]

When x =2401, we have

[tex]y = -\frac{490}{\sqrt {2401}}[/tex]

Evaluate the square root

[tex]y = -\frac{490}{49}[/tex]

Divide

y = -10

Hence, the value of y when x= 2401 is -10

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The type of solution set of the given equation is; "No Solution"

Let's solve for x in the given equation;

8(3x - 2) - 4 = 8x + 4(4x - 3)

Expanding the bracket gives;

8(3x) + 8(-2) - 4 = 8x + 4(4x) + 4(-3)

24x - 16 - 4 = 8x + 16x - 12

24x - 20 = 24x - 12

Subtract 24x from both sides to get;

-20 = -12

This is a false equation as both sides are not the same number and as such the type of solution set is "No Solution"

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The required mass of the solid is 8.52g. Hence option B is correct.

We have,
mb = mass/89.0, ΔTb = 2.42° C and Kb = 2.53°C•kg/mol.

Mass of the object is defined as the space occupied by an object.

Given equation, ΔTb=Kb x mb
2.42 = 2.53 x mb
mb =0.95
mass / 89.0 x 0.1 = 0.95
mass = 8.52 g

Thus, the required mass of the solid is 8.52g.

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