Find the values of x and y that satisfies the equation.

5x + 3i = 15 + yi

See the attachment for a labeled graph.

_____

I find it convenient to use "technology" to draw the graph. A spreadsheet, graphing calculator, or on-line graphing program can do this for you.

[tex]\displaystyle x^{\frac{2}{3}}[/tex]

The rules of exponents tell you ...

... (a^b)(a^c) = a^(b+c) . . . . . . applies inside parentheses

... (a^b)^c = a^(b·c) . . . . . . . . applies to the overall expression

The Order of Operations tells you to evaluate inside parentheses first. Doing that, you have ...

... x^(4/3)·x^(2/3) = x^((4+2)/3) = x^2

Now, you have ...

... (x^2)^(1/3)

and the rule of exponents tells you to multiply the exponents.

... = x^(2·1/3) = x^(2/3)

Answer:

x^(2/3)

Step-by-step explanation:

(x^a.x^b)^c = x^[c*(a+b)]

using the above eqn, u can simplify the given expression to

x^[1/3*(4/3+2/3)]

=x^[1/3*(6/3)]

=x^(2/3)

ans is the 2nd choice

Answer:

[tex]AS=18\ units,\ DS=\dfrac{54}{7}\ units.[/tex]

Step-by-step explanation:

Consider trapezoid ABCD with bases BC=11 units and AD=18 units.  The lengths of legs are AB=7 units and CD=3 units. Point S is the point of intersection of the extensions of the legs AB and CD.

Let BS=x units and CS=y units.

Consider triangles BSC and ASD. By AAA theorem these triangles are similar (because ∠SAD≅∠SBC, ∠ADS≅∠BCS and ∠S is common).

Then

[tex]\dfrac{BS}{AS}=\dfrac{CS}{DS}=\dfrac{BC}{AD},\\ \\\dfrac{x}{x+7}=\dfrac{y}{y+3}=\dfrac{11}{18}.[/tex]

Therefore,

[tex]\dfrac{x}{x+7}=\dfrac{11}{18},\\ \\18x=11(x+7),\\ \\18x=11x+77,\\ \\7x=77,\\ \\x=11\ units.[/tex]

[tex]\dfrac{y}{y+3}=\dfrac{11}{18},\\ \\18y=11(y+3),\\ \\18y=11y+33,\\ \\7y=33,\\ \\y=\dfrac{33}{7}\ units.[/tex]

The lengths of segments between point S and the vertices of the greater base are

[tex]AS=AB+BS=7+11=18\ units,\ DS=DC+CS=3+\dfrac{33}{7}=\dfrac{54}{7}\ units.[/tex]

Answer:

216 cupcakes

Step-by-step explanation:

81 = 0.375 × order

81/0.375 = order = 216 . . . . . divide by the coefficient of the variable

_____

About percentages

% means /100

37.5% = 37.5/100 = 375/1000 = 0.375

9.9 years

A = P(1 + rt) . . . . account balance after time t at rate r starting with principal P

... 3000 = 2500(1 + 0.0202t) . . . . filling in the given numbers

... 1.2 = 1 + 0.0202t . . . . divide by 2500

... 0.2 = 0.0202t . . . . . . subtract 1

... 0.2/0.0202 = t ≈ 9.901

It will take about 9.9 years for the account balance to reach $3000.

Part A.

We're only told that the product of the mystery factor and the number shown is even. This is true when the number shown is odd as well as even, hence the mystery factor is even.

There are no other conditions on the product that might limit the value of the mystery factor (non-zero, or < 10, for example).

Hence, the mystery factor could be any single-digit even number. In our base-10 number system, there are 5 of them: {0, 2, 4, 6, 8}.

___

Part B.

There are only 5 even single-digit numbers in our base-10 number system, so by choosing 5, I know I have all of them.

The number of rooms booked to produce maximum revenue is required.

The number of rooms booked to produce the maximum revenue is 250.

The revenue function is

[tex]R(x)=200x-0.4x^2[/tex]

Differentiating with respect to x we get

[tex]R'(x)=200-0.8x[/tex]

Equating with zero

[tex]0=200-0.8x\\\Rightarrow x=\dfrac{-200}{-0.8}\\\Rightarrow x=250[/tex]

Double derivative of the function is

[tex]R''(x)=-0.8x[/tex]

Substituting the value of [tex]x=250[/tex]

[tex]R''(250)=-0.8\times 250=-200[/tex]

Since, it is negative the maximum value of x will be 250.

The number of rooms booked to produce the maximum revenue is 250.

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The number of rooms that produces the maximum revenue for Fred's Estates LLC is 250 rooms, calculated using the formula -b/2a, where a and b are the coefficients of the quadratic revenue function.

To calculate the maximum revenue for Fred's Estates LLC, we need to find the value of x that maximizes the function [tex]R(x)=200*-0.4x2.[/tex]

The maximum value of a quadratic function can be found using the formula -b/2a, where a and b are the coefficients of x² and x in the function. Here, a=-0.4 and b=200.

Using the formula, [tex]x=-b/2a = -200/(2*(-0.4)) = 250[/tex] rooms. Therefore, booking 250 rooms results in the maximum revenue for Fred's Estates LLC.

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Answer:

D. 6 hours

Step-by-step explanation:

There are 3 ft in 1 yd, so 400 yd = 1200 ft.

At 200 ft/h, Jim's travel time for 1200 ft will be

... time = distance/speed = (1200 ft)/(200 ft/h) = (1200/200) · ft · (h/ft)

... time = 6 h

(2.8×10¹)x⁶y⁴

Multiplication can be done in the usual way, then the number converted to scientific notation. Scientific notation does not apply to the variables.

[tex](7x^3y^2)(4x^3y^2)=(7\cdot 4)x^{3+3}y^{2+2}=28x^6y^4\\\\=2.8\cdot 10^1x^6y^4[/tex]

_____

If you write the numbers in scientific notation to start, then do the multiplication, the effect is virtually the same: an adjustment is needed in the product to get it back to scientific notation.

(7)(4) = (7×10⁰)(4×10⁰) = 7·4×10⁰⁺⁰ = 28×10⁰ = 2.8×10¹

Answer:

21d +25 ≤ 115

Step-by-step explanation:

Jim's cost will be ...

... 21.00·d + 0.10·250 = 21d +25

He wants his cost not to exceed his budget, so ...

... 21d +25 ≤ 115

_____

The solution is ...

... 21d ≤ 90 . . . . subtract 25

... d ≤ 90/21 ≈ 4.3

so Jim can rent the car a maximum of 4 days.

The inequality that can be used to express this scenario is

115 ≤  21*d+ 25

Given data

Charges = $21 per day

Cost per driven distance = $0.10

Distance he will travel = 250 miles

Amount her has to spend = $115

Let the maximum number of days Jim can rent a car with $115 be "m"

Hence

Total amount = 21*d+ 0.1*250

Substituting and Simplifying we have

115 ≤  21*d+ 25

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The volume of metal is the difference of the overall volume of the cylinder and the volume of the hole in it. The formula for the volume of a cylinder is ...

... V = π·r^2·h . . . . . radius r and height h

For the overall dimensions, the radius is half the diameter, so is 10 mm. The hole is said to have a radius of 6 mm. The overall "height" is 21 mm, so the volume in mm³ will be ...

... V_overall -V_hole = π(10)^2(21) -π(6)^2(21)

... = 21π·10^2 -21π·6^2 . . . . . . . matches the first selection

... = 2100π -756π . . . . . . . . . . . matches the third selection

... = 1344π . . . . . . . . . . . . . . . . doesnt' match any selection

The correct expressions for the volume of metal needed, in cubic millimeters, to make the pipe are,

⇒ 21π(10)² – 21π(6)²

⇒ 2,100π – 756π

To multiply means to add a number to itself a particular number of times. Multiplication can be viewed as a process of repeated addition.

Given that;

A cylindrical metal pipe has a diameter of 20 millimeters and a height of 21 millimeters.

And,  A cylindrical hole cut out of the center has a radius of 6 millimeters.

Hence, The formula for the volume of a cylinder is,

V = π·r²·h

Where, radius r and height h.

Now, For the overall dimensions, the radius is half the diameter, so is 10 mm. The hole is said to have a radius of 6 mm. The overall "height" is 21 mm,

so the volume in mm³ will be;

V (overall) -V (hole) = π(10)²(21) -π(6)²(21)

= 21π·10² -21π·6²

= 2100π -756π

= 1344π

Thus, The correct expressions for the volume of metal needed, in cubic millimeters, to make the pipe are,

⇒ 21π(10)² – 21π(6)²

⇒ 2,100π – 756π

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Answer:

$300

Step-by-step explanation:

Let x represent the side length of the square base in feet. Then the height of each side is ...

... h = (50 ft³)/(x ft)² = (50/x²) ft

The cost of the sides of the box is then ...

... (4 sides) × (x ft)(50/x² ft)/side × $2/ft² = $400/x

The cost of the bottom is ...

... (x ft)² × $25/ft² = $25x²

So, the total dollar cost is

... C = 400/x + 25x²

This will be a minimum where its derivative with respect to x is zero.

... 0 = -400/x² +50x

... 400/50 = 8 = x³ . . . . . add 400/x²; multiply by x²/50

... x = ∛8 = 2

For this value of x, the minimum cost is ...

... C = 400/2 + 25·2² = 300

The minimum possible cost is $300.

_____

Comments on the problem

1) Cardboard boxes are usually made of cardboard. They are rarely made of wood and alumninum.

2) The cost of the bottom is half the cost of the sides. When the dimensions are unconstrained, you will find (as here) the cost is shared equally between the bottom and pairs of opposite sides—each being 1/3 the total cost.

I like to look at a graph of the function to see where the zeros might be. Here, there are x-intercepts at x=-2 and x=3. These can be factored out using synthetic division to find the factorization to be ...

... (x +2)(x -3)(x² +x +1) = 0

By completing the square, using the quadratic formula, or by looking at the graph of it, the complex roots of the quadratic factor can be found to be ...

... x = (-1 ±i√3)/2

_____

The second attachment shows my synthetic division. The first division takes out the root x=3 to give a quotient of x³ +3x² +3x +2. The second division takes out the root -2 to give the quotient of x² +x +1. (You can see that I tried -1 as a root first.)

The graph shows both the quartic and the quadratic factor of it. The latter has a leading coefficient of 1 and a vertex at (-1/2, 3/4), so you know the complex roots are -1/2 ±i√(3/4).

_____

From the beginning

There is only a very complicated formula for the roots of a quartic equation, so these are usually solved by machine or by some form of trial and error (iteration). There are some helps, like Descarte's Rule of Signs, and the Rational Root theorem.

Here, the former looks at the one sign change in the coefficients to tell you there will be 1 positive real root. Changing the sign of the odd-degree terms makes there be 3 sign changes, so there will be 3 or 1 negative real roots. Thus, we're assured at least two real roots, one of each sign.

We can look at the constant term to find the y-intercept to be -6. We can add the coefficients to find the value of the function is -18 for x=1, so the positive real root is larger than 1.

The Rational Root theorem says any rational roots will be factors of 6, the constant term. Choices are 1, 2, 3, 6. We have already eliminated 1 as a possibility, and we consider it unlikely that 6 will be a root. (The 4th power overwhelms the other terms in the function.) We tried 2 and found it doesn't work (this was before we graphed the function). The attached division result shows that 3 is a root, as does the graph.

Once you get down to a quadratic, you can find the remaining roots in the usual way. Because it is so simple to read them from the graph, we decided to graph the quadratic factor.

_____

Comment on terminology

"root" and "zero" are essentially the same thing when the function is equated to zero, as here. The terms refer to the value(s) of x that make the polynomial function evaluate to zero.

To find the zeros of the equation x^4-6x^2-7x-6=0, we can apply the rational root theorem and synthetic division. By trying different factors of the constant term -6, we can determine that one of the roots is x = -2. After dividing the equation by (x + 2), we obtain a quadratic equation x^3 - 2x^2 - 11x + 3 = 0. We can then solve this equation by factoring or using a graphing calculator to find the remaining two roots.

To find the zeros of the equation x^4-6x^2-7x-6=0, we can apply the rational root theorem and synthetic division. By trying different factors of the constant term -6, we can determine that one of the roots is x = -2. After dividing the equation by (x + 2), we obtain a quadratic equation x^3 - 2x^2 - 11x + 3 = 0. We can then solve this equation by factoring or using a graphing calculator to find the remaining two roots.

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Answer:

Right part: [tex]\dfrac{4}{5}x+a,[/tex] where [tex]a\neq -8.[/tex]

Step-by-step explanation:

Linear equation have no solution if it is impossible for the equation to be true no matter what value we assign to the variable x.

The left part of the equation Tomas wrote is

[tex]\dfrac{4}{5}x-8.[/tex]

The right part of the equation should be of the form

[tex]\dfrac{4}{5}x+a,[/tex]

where [tex]a\neq -8.[/tex]

In this case, the equation will take look

[tex]\dfrac{4}{5}x-8=\dfrac{4}{5}x+a,\\ \\-8=a.[/tex]

Since [tex]a\neq -8,[/tex] this equality is always false.

Remark: 1) If a=8, the equation [tex]\dfrac{4}{5}x-8=\dfrac{4}{5}x+a[/tex] is equivalent to the equation

[tex]\dfrac{4}{5}x-8=\dfrac{4}{5}x-8,\\ \\0=0[/tex]

and the lest equation has infinitely many solutions.

2) When coefficient at x differs from [tex]\dfrac{4}{5},[/tex] the equation has unique solution.

Answer:

19

Step-by-step explanation:

b - 7 = 12

b = 7 + 12

b = 19

Hi there! :)

Answer:

b=19

*The answer must have a positive sign.*

Step-by-step explanation:

First, you add by 7 from both sides of an equation.

[tex]b-7+7=12+7[/tex]

Then, you add by the numbers from left to right.

[tex]12+7=19[/tex]

Final answer is b=19

I hope this helps you!

Have a nice day! :)

:D

-Charlie

Thank you so much! :)

Answer:

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Step-by-step explanation:

Answer:

31,250 ft²

Step-by-step explanation:

Let x represent the length of fence parallel to the river. Then the ends of the rectangle will have dimesion (500-x)/2, and the total area will be ...

... A = x(500-x)/2

This function describes a parabola with zeros at x=0 and x=500. The vertex (maximum) will be located halfway between these, at x=250.

The maximum size pen has area ...

... A = 250(500 -250)/2 = 31,250 . . . . sq ft

Answer:

x=0

Step-by-step explanation:

-0.2(x-20)=4-x

The first step is to distribute the -.2

-.2 x -.2 * -20 = 4-x

-.2x +4 = 4-x

Add x to each side

x-.2x +4 = 4-x+x

.8x +4 = 4

Subtract 4 from each side

.8x +4-4 = 4-4

.8x=0

Divide by .8

.8x/.8 = 0/.8

x =0

Answer:

206 Cornish hens

169 turkeys

Step-by-step explanation:

Let c represent the number of Cornish hens sold. Then c-37 is the number of turkeys sold. The total sales would be ...

... c + (c-37) = 375

... 2c = 412 . . . . collect terms, add 37

... c = 206 . . . . Cornish hens sold

... (c-37) = 169 . . . turkeys sold

_____

Comment on this type of problem

Note that the solution to this problem is that the larger number (number of cornish hens) is half the sum of the total and the difference: (375+37)/2 = 206. This is the general solution for this type of "sum and difference" problem.

The larger contributor is half the total plus half the difference; the smaller contributor is half the total minus half the difference.

Cornish hens = (1/2)(375 +37) = 206

Turkeys = (1/2)(375 -37) = 169

Answer:

10 tables are needed to seat 65 people.

Step-by-step explanation:

Given :

There are total 65 people.

7 people can be seated on each table .

To Find : No. of tables required to seat 65 people.

Solution :

Since 7 people can sit on no. of tables = 1

1 people sit on no. of tables = 1/7

65 people sit on no. of tables

= [tex]\frac{1}{7} *65[/tex]

⇒ [tex]9.28[/tex]

⇒ 9.28 ≈ 10 tables are needed to seat 65 people.

Thus 10 tables are needed to seat 65 people.

Answer:

see attached

Step-by-step explanation:

Each of the pie slices can be cut in half different ways. An easy way to do it and to understand it is to draw another cut from the center to the middle of the edge.

The result of cutting these slices is that instead of 8 equal pieces (of which 3 are colored), there will be 16, of which 6 are colored.

The new fraction is 6/16.

a ≤ 3

Undo what is done to the variable. The variable is multiplied by 3 and 4 is subtracted from that result. To undo these operations, you add 4, then divide by 3.

... 3a ≤ 9 . . . . 4 is added to both sides of the inequality

... a ≤ 3 . . . . . both sides of the inequality are divided by 3

_____

Comment on solving inequalities

These are the same steps you would use to solve the 2-step equation ...

... 3a -4 = 5

The only difference between solving equations and solving inequalities is that when you multiply or divide by a negative number, you reverse the sense of the comparison (> becomes <, and vice versa; ≥ becomes ≤, and vice versa). You can see this if you consider numbers: 2 > 1. Multiplying by -1 gives -2 < -1.

The first line.

If we let x represent "a number", then "a number divided by 3" is x/3. The denominator is missing on the first line.

Further, this "is 4 more than -11", so is = 4 + (-11). The constant 4 is on the wrong side of the equal sign in line 1.

_____

The solution should be ...

... x/3 = 4 + (-11)

... x = 3·(-7)

... x = -21

___

Check

-21 divided by 3 is -7. That is 4 more than -11.

Answer:

B. -9/4 -4i

Step-by-step explanation:

Collect terms the way you would with any algebraic expression.

= (-3 +2 -3)i + (3/4 -3)

= -4i +(3/4 -12/4)

= -9/4 -4i

The product of the complex numbers [tex]\( (3 - 2i) \)[/tex] and [tex]\( (1 + i) \)[/tex] is [tex]\( 5 - 4i \)[/tex].

To find the product of two complex numbers, we multiply them as we would with binomials, remembering that [tex]\( i^2 = -1 \)[/tex]. Let's perform the multiplication step by step:

Given complex numbers [tex]\( (3 - 2i) \)[/tex] and [tex]\( (1 + i) \)[/tex], we multiply them directly:

[tex]\[(3 - 2i) \cdot (1 + i) = 3 \cdot 1 + 3 \cdot i - 2i \cdot 1 - 2i \cdot i. \][/tex]

Now, we simplify the expression by combining like terms and using the fact that [tex]\( i^2 = -1 \)[/tex]:

[tex]\[ = 3 + 3i - 2i - 2i^2 = 3 + i - 2(-1) = 3 + i + 2. \][/tex]

Finally, we combine the real parts and the imaginary parts:

[tex]\[ = (3 + 2) + i = 5 - 4i. \][/tex]

Therefore, the product of the complex numbers [tex]\( (3 - 2i) \)[/tex] and [tex]\( (1 + i) \)[/tex] is[tex]\( 5 - 4i \)[/tex].

Answer:

The distance from he ball to the goal is 11.85 feet (Approx) .

Step-by-step explanation:

As given

The angle of elevation from a soccer ball on the ground to the top of the goal is 34° .

If the goal is 8 feet tall.

Now by using the trigonometric identity .

[tex]tan \theta = \frac{Perpendicular}{Base}[/tex]

As shown in the diagram given below

[tex]\theta = 34^{\circ}[/tex]

Perpendicular = AB = 8 feet

Base = BC

Put all the values in the identity .

[tex]tan\ 34^{\circ} = \frac{AB}{BC}[/tex]

[tex]tan\ 34^{\circ} = \frac{8}{BC}[/tex]

[tex]tan\ 34^{\circ} = 0.675\ (Approx)[/tex]

[tex]BC = \frac{8}{0.675}[/tex]

BC = 11.85 feet (Approx)

Therefore the distance from he ball to the goal is 11.85 feet (Approx) .

To calculate the distance from the soccer ball to the goal with an angle of elevation of 34 degrees and a goal height of 8 feet, use the tangent trigonometric ratio. The distance is found to be approximately 11.86 feet.

Given the angle of elevation is 34 degrees and the goal's height is 8 feet, we're looking to calculate the adjacent side (distance from the ball to the goal) in a right-angled triangle where the opposite side (goal's height) and the angle are known.

To calculate the distance (let's call it d), we use the tangent function:

So, d = 8/tan(34 degrees).

Calculating this, we find:

Therefore, the distance from the soccer ball to the goal is approximately 11.86 feet.

Answer:

561 because if you multiple 300 /8 times the number of the before populaiton u got it right Step-by-step explanation:

ANSWER:

Your answer is the 3rd one: y - 8 = -3/7(x - 5)

ABOUT POINT SLOPE FORM:

ABOUT PROBLEM:

y - Y1 = m (x - X1)

y - 8 = -3/7(x - 5) --- IN POINT SLOPE FORM

Hope this helps you!!! :)

The line with a slope -3/7 and passes through the point (5, 8) has an equation of y - 8 = (-3/7)(x - 5)

The equation of a straight line is given by:

y = mx + b;

where y,x are variables, m is the slope of the line and b is the y intercept.

Since the line has a slope -3/7 and passes through the point (5, 8), the equation of the line is:

[tex]y-y_1=m(x-x_1)\\\\y-8=-\frac{3}{7} (x-5)[/tex]

Hence a line with a slope -3/7 and passes through the point (5, 8) has an equation of y - 8 = (-3/7)(x - 5)

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Answer:

$6.40 because 32 ounces is the difference 128 and 96 hence 32 is 1/3 of 96 so divide $4.80 by 3 which is $1.60 then add $1.60 + $4.80 =$6.40

Step-by-step explanation: