How many solutions does the equation −2y + 2y + 3 = 3 have? One Zero Infinitely many Three

The student's question relates to the point P with the polar coordinates (3, -π/3). Polar coordinates are not unique, so we can find all coordinates of point P by adding multiples of 2π to the angle part of the coordinate, that is, (3, -π/3 + 2πn) where n is an integer.

The polar coordinates system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from the origin (point O) and an angle measured anti-clockwise from an arbitrary direction, usually the x-axis.

Each point is represented by the ordered pair (r, θ). Our point P has the polar coordinates (3, -π/3). However, polar coordinates are not unique for a given point. To find all polar coordinate pairs for point P, we add multiples of 2π to the angle part of the coordinate pair, as a complete revolution is 2π in radians. Therefore, alternative polar coordinate pairs for point P would include (3, -π/3 + 2πn) where n is an integer.

Examples include:

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Answer:

[tex]\frac{\tan 60\degree}{\cos45 \degree}= \sqrt{6}[/tex]

Step-by-step explanation:

We want to evaluate

[tex]\frac{\tan 60\degree}{\cos45 \degree}[/tex]

We use special angles or the unit circle to obtain;

[tex]\frac{\tan 60\degree}{\cos45 \degree}=\frac{\sqrt{3}}{\frac{\sqrt{2}}{2}}[/tex]

This implies that;

[tex]\frac{\tan 60\degree}{\cos45 \degree}=\sqrt{3}\div \frac{\sqrt{2}}{2}[/tex]

[tex]\frac{\tan 60\degree}{\cos45 \degree}=\sqrt{3}\times \sqrt{2}[/tex]

[tex]\frac{\tan 60\degree}{\cos45 \degree}= \sqrt{6}[/tex]

Answer:

[tex]\sqrt{6}[/tex].

Step-by-step explanation:

[tex]\frac{tan(60)}{cos(45)}[/tex]

[tex]= \frac{\frac{sin(60)}{cos(60)}}{cos(45)}[/tex]

[tex]= \frac{sin(60)}{cos(60)*cos(45)}[/tex]

[tex]= \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}*\frac{\sqrt{2}}{2}}[/tex]

[tex]= \frac{\frac{\sqrt{3}}{2}}{\frac{\sqrt{2}}{4}}[/tex]

[tex]= \frac{4\sqrt{3}}{2\sqrt{2}}[/tex]

[tex]= \frac{2\sqrt{3}}{\sqrt{2}}[/tex]

[tex]= \frac{2\sqrt{3}\sqrt{2}}{2}[/tex]

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

[tex]=\sqrt{6}[/tex].

Answer:

see explanation

Step-by-step explanation:

Note that the sum of the coefficients of g(x)

1 + 2 - 3 = 0

Hence x = 1 is a root of g(x) and (x - 1) is a factor

Note the sum of the coefficients of f(x)

1 - 3 - 13 + 15 = 0

hence x = 1 is a root of f(x) and (x - 1) is a factor

Since (x - 1) is a factor of both

Then f(x) is also divisible by x² + 2x - 3

Answer:

2 miles.

Step-by-step explanation:

Answer:

14

Step-by-step explanation:

2k - 1 is linear, so this is an arithmetic series.  The sum of an arithmetic series is:

S = (n/2) (a₁ + an)

Here:

S = 196

a₁ = 2(1) - 1 = 1

an = 2n - 1

Solving:

196 = (n/2) (1 + 2n - 1)

196 = (n/2) (2n)

196 = n²

n = 14

Answer:

Population will be approx 1606300.

Step-by-step explanation:

Given that a city’s population is about 763,000 and is increasing at an annual rate of 1.5%. Now we need to predict the population of the city in 50 years.

We can use growth formula

[tex]A=P\left(1+r\right)^t[/tex]

Where P=763000

rate r=1.5% = 0.015

time t = 50 years

Plug these values into above formula

[tex]A=763000\left(1+0.015\right)^{50}[/tex]

[tex]A=763000\left(1.015\right)^{50}[/tex]

[tex]A=763000\left(2.10524242061\right)[/tex]

[tex]A=1606299.96692[/tex]

Hence population will be approx 1606300.

Therefore, the predicted population of the city in 50 years is approximately [tex]$1,586,997$[/tex].

To predict the population of the city in 50 years, we can use the formula for compound interest, where the principal is the initial population, the interest rate is the annual growth rate, and the time is 50 years.

Given:

- Initial population = 763,000

- Annual growth rate = 1.5% = 0.015

Step 1: Calculate the total growth factor after 50 years using the compound interest formula.

Growth factor = [tex](1 + r)^_t[/tex]

Growth factor =[tex](1 + 0.015)^_{50}[/tex]

Growth factor = [tex]$1.015^{50} = 2.079$[/tex]

Step 2: Calculate the final population by multiplying the initial population with the growth factor.

Final population = Initial population × Growth factor

Final population = 763,000 × 2.079

Final population = [tex]$1,586,997$[/tex]

Answer:

The length of the hypotenuse is [tex]h = 6.71\ units[/tex]

Step-by-step explanation:

For a straight triangle it is true that

[tex]h = \sqrt{a ^ 2 + b ^ 2}[/tex]

Where has is the hypotenuse of the right triangle and a and b are the lengths of the other two sides.

In this case we know that:[tex]a = 3\\b = 6[/tex]

So the hypotenuse is:

[tex]h = \sqrt{3 ^ 2 + 6 ^ 2}[/tex]

[tex]h = \sqrt{3 ^ 2 + 6 ^ 2}[/tex]

[tex]h = 3*\sqrt{5}[/tex]

[tex]h = 3*\sqrt{5}[/tex]

[tex]h = 6.71[/tex]

ANSWER

The hypotenuse is 3√5 units.

EXPLANATION

We use the Pythagoras Theorem.

Let h be the hypotenuse.

The Pythagoras Theorem says that, the hypotenuse square is equal to the sum of the squares of the two shorter legs.

[tex] {h}^{2} = {3}^{2} + {6}^{2} [/tex]

[tex]{h}^{2} = 9+ 36[/tex]

[tex]{h}^{2} = 45[/tex]

Take positive square root.

[tex]h = \sqrt{45} [/tex]

[tex]h = 3 \sqrt{5} units[/tex]

Answer: [tex]V=11.775\ in^3[/tex]

Step-by-step explanation:

You need to use the formula for calculate the volume of a cone. This is:

[tex]V=\frac{1}{3}\pi r^2h[/tex]

Where "r" is the radius and "h" is the height.

You know that the diameter of the cone-shaped paperweight is 3 inches. Then, you need to divide the diameter by 2 to find the radius:

[tex]r=\frac{3in}{2}\\\\r=1.5\ in[/tex]

Now you know that:

[tex]r=1.5\ in\\h=5\ in\\\pi=3.14[/tex]

Substituting these values into the formula [tex]V=\frac{1}{3}\pi r^2h[/tex], you get that the volume of the  paperweight is:

[tex]V=\frac{1}{3}(3.14)(1.5\ in)^2(5\ in)[/tex]

[tex]V=11.775\ in^3[/tex]

The volume of the paperweight will be 11.76 cubic inches.

Let d be the diameter of the base circle and h be the height of the cone.

Then the volume of the cone will be

V = 1/12 x πd² x h

The cone-shaped paperweight has a diameter of 3 inches and a height of 5 inches.

Then the volume of the paperweight will be

V = 1/12 x π(3)² x 5

V = 1/12 x 3.14 x 9 x 5

V = 11.76 cubic inches

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Answer:

x = 14

Step-by-step explanation:

3(x + 3) - 2x = 23

3x + 9 - 2x = 23

x + 9 = 23

x = 14

Answer: SECOND OPTION

Step-by-step explanation:

The area of a parallelogram can be calculated with this formula:

[tex]A=bh[/tex]

Where "b" is the of one base and "h" is the height.

You can observe in the figure that "b" and "h" are:

[tex]b=AB=4\sqrt{2}ft\\\\h=AX=3ft[/tex]

Then, substituting these values into the formula, you get that the area of the given parallelogram is:

[tex]A=(4\sqrt{2}ft)(3ft)\\\\A=12\sqrt{2}ft^2[/tex]

This matches with the second option.

Answer:

12√2 sq. ft.

Step-by-step explanation:

Hope this helps.

Answer:

Options B,C,E are true.

Step-by-step explanation:

It has been given in the question ∠ACD is supplementary to ∠ACE and ∠BCD

∠ACD ≅ ∠BCE

Option A. ∠ACE is supplementary to ∠BCD

False. They are not lying on a point of straight line.

∠ACE + ∠BCD ≠180°

Option B. ∠BCE is supplementary to ∠ACE

Since ∠ACD + ∠ACE = 180° [Given]

and ∠ACD ≅ ∠BCE [Given]

Therefore, ∠BCE + ∠ACE = 180°

TRUE.

Option C. ∠BCD is supplementary to ∠BCE

Since ∠ACD + ∠BCD = 180°[Given]

and ∠ACD ≅ ∠BCE [Given]

So ∠BCE + ∠BCD = 180°

TRUE.

Option D. ∠ACE ≅ ∠BCE

Since ∠ACD + ∠ACE = 180°

And ∠ACD + ∠BCD = 180°

This is clear from these equations that ∠ACE is supplementary to ∠BCD.

FALSE :

Option E. ∠BCD is congruent to ∠ACE

As we have already proved in option D. ∠BCD ≅ ∠ACE

TRUE

Therefore, Options B, C, E are TRUE

Should be 160 chairs

To determine the total number of chairs in the restaurant, multiply the count of tables with 2 chairs (20 tables) by 2, and those with 4 chairs (30 tables) by 4, and sum the two products to get a total of 160 chairs.

To calculate the total number of chairs in the restaurant, we first need to find out how many tables are there with 2 chairs and how many with 4 chairs. Since 40% of the tables have 2 chairs, we multiply 40% (or 0.4) by the total number of tables (50) to find the number of tables with 2 chairs, which equals to 20 tables.

The remaining 60% of the tables have 4 chairs, so we multiply 60% (or 0.6) by the total number of tables (50) to find the number of tables with 4 chairs, which equals to 30 tables.

To find the total number of chairs, we multiply the number of tables with 2 chairs (20) by 2 and the number of tables with 4 chairs (30) by 4, then we add the two results together:

Tables with 2 chairs: 20 tables imes 2 chairs/table = 40 chairs

Tables with 4 chairs: 30 tables imes 4 chairs/table = 120 chairs

So, the total number of chairs in the restaurant is:

40 chairs + 120 chairs = 160 chairs

Answer:

[tex]\frac{1}{4}[/tex] of the books in the book store are mystery fiction books.

Step-by-step explanation:

Let x represent all the books in the books store.

Then, the fraction of books that are fiction books is [tex]\frac{3}{4}x[/tex]

We have that; [tex]\frac{1}{3}[/tex] of the fiction books are mystery books.

The fraction of the books at the  bookstore that are mystery fiction  books is [tex]\frac{1}{3}\times \frac{3}{4}x=\frac{1}{4}x[/tex].

Therefore [tex]\frac{1}{4}[/tex] of the books in the bookstore are mystery fiction books.

Answer:

1/4

Step-by-step explanation:

Answer:

2 reals and 2 rationals.

Step-by-step explanation:

The discriminate gives

b^2 - 4*a*c

b = -7

a = 5

c = 2

(-7)^2 - 4(5)(2)

49 - 40

9

Taking the square root gives you +/-3

The discriminate tells you that there are 2 roots, both real and both rational

x = [(-7) +/- 3  ]/2*5

x = (- 7 - 3)/10 = - 1

x = (- 7 + 3)/10 = -0.4

Answer:

x = 34; y = 14

Step-by-step explanation:

Step 1: Make the equations

x + y = 48

x - y = 20

Step 2: Solve the equations

x + y = 48

x - y = 20

2x = 68

x = 34

34 + y = 48

y = 14

Answer:

The volume of the smaller solid is [tex]339\ yd^{3}[/tex]

Step-by-step explanation:

step 1

Find the scale factor

we know that

If two figures are similar, then the ratio of its surface areas is equal to the scale factor squared

Let

z----> the scale factor

x----> surface area of the larger solid

y----> surface area of the smaller solid

[tex]z^{2}=\frac{x}{y}[/tex]

we have

[tex]x=361\ yd^{2}[/tex]

[tex]y=121\ yd^{2}[/tex]

substitute

[tex]z^{2}=\frac{361}{121}[/tex]

[tex]z=\frac{19}{11}[/tex]

step 2

Find the volume of the smaller solid

we know that

If two figures are similar, then the ratio of its volumes is equal to the scale factor elevated to the cube

Let

z----> the scale factor

x----> volume of the larger solid

y----> volume of the smaller solid

[tex]z^{3}=\frac{x}{y}[/tex]

we have

[tex]z=\frac{19}{11}[/tex]

[tex]x=1,747\ yd^{3}[/tex]

substitute

[tex](\frac{19}{11})^{3}=\frac{1,747}{y}\\ \\(\frac{6,859}{1,331})=\frac{1,747}{y} \\ \\y=1,331*1,747/6,859\\ \\y=339\ yd^{3}[/tex]

Answer:divide the diameter by 2 and plug the values for volume, pi, and radius into the formula for volume of a cylinder. Next, square the radius and multiply the values together. Then, divide both sides by 200.96 for the answer, remembering to include the appropriate unit of measurement

Step-by-step explanation:

Finding A Diameter Of A Cylinder Is Easy.

If You Know The Radius, Multiply The Radius By 2 To get Your Diameter,

Have A Great Day!

For this case we have that by definition, a polynomial in its standard form is given by:

[tex]P (x) = ax ^ {n} + bx ^ {n-1} + ... + cx ^ 3 + dx ^ 2 + ex + f[/tex]

Where:

a, b, c, d, e, f: They are the coefficients

n, n-1,3,2,1: They are the exponents. The degree of the polynomial is "n" because it is the largest exponent.

x: It is the variable

The given polynomial is:

[tex]4m-2m ^ 4-6m ^ 2 + 9[/tex]

Rewriting it in its standard form:

[tex]P (x) = - 2m ^ 4-6m ^ 2 + 4m + 9[/tex]

It is a polynomial of degree 4

ANswer:

[tex]P (x) = - 2m ^ 4-6m ^ 2 + 4m + 9[/tex]

Answer:

For this case we have that by definition, a polynomial in its standard form is given by:Where:a, b, c, d, e, f: They are the coefficientsn, n-1,3,2,1: They are the exponents. The degree of the polynomial is "n" because it is the largest exponent.x: It is the variableThe given polynomial is:Rewriting it in its standard form:It is a polynomial of degree 4ANswer:

Step-by-step explanation:

Answer:

The volume of the prism = 300 units³

Step-by-step explanation:

* Lets study the triangular prism

- The triangular prism has 6 faces

- Two right triangular bases

- Four rectangular side faces

- The volume of the prism = area of its base × its height (altitude)

* Now lets solve the problem

∵ The base is a right triangle with a hypotenuse of 13 units and

   a leg of 12 units

∵ The area of the right triangle = 1/2 × leg1 × leg2

- You can find the length of other leg by using Pythagoras theorem

∵ (hypotenuse)² = (leg1)² + (leg2)²

∵ hypotenuse = 13 units

∵ leg1 = 12 units

∴ (13)² = (12)² + (leg2)²

∴ 169 = 144 +  (leg2)² ⇒ subtract 144 from both sides

∴ 25 =  (leg2)² ⇒ take √ for both sides

∴ leg2 = 5 units

- The area of the right triangle = 1/2 × leg1 × leg2

∴ The area of the base = 1/2 × 12 × 5 = 30 units²

∴ The volume of the prism = 30 × 10 = 300 units³

In the equation: -6x - 19 - 4x replace x with -2

-6(-2) - 19 - 4(-2)

Use your rules of PEMDAS to evaluate

12 - 19 + 8

-7 + 8

1

Hope this helped!

~Just a girl in love with Shawn Mendes

Answer

1

Step-by-step explanation:

Step 1: Replace all x's with -2

-6(-2)-19-4(-2)

Step 2: Multiply

12-19+8

Tip: Remember that a double negative is a positive.

Step 3: Add and Subtract

12-19+8

-7 + 8

1

Answer:

1

Step-by-step explanation:

In each set, there is a difference of 1 between the two x's.

I just did it its 2 4 and 8 i think

Answer:

4

Step-by-step explanation:

-5+9=4

Answer:

a(n) = 3*(2)^(n-1)

Step-by-step explanation:

Each new term is found by multiplying the previous term by 2.  The first term is 3.  Using the standard explicit formula for a geometric series, we get:

a(n) = a(1)*r^(n-1).  In this particular case we have a(n) = 3*(2)^(n-1).

ANSWER

The discriminant is 49.

EXPLANATION

The given quadratic equation is:

[tex]0 =2{x}^{2} + 3x - 5[/tex]

We can rewrite this as

[tex]2{x}^{2} + 3x - 5 = 0[/tex]

Comparing this to

[tex]a{x}^{2} + bx + c = 0[/tex]

We have a=2,b=3, c=-5.

The discriminant is given by:

[tex]D = {b}^{2} - 4ac[/tex]

We plug in the values to get:

[tex]D = {3}^{2} - 4(2)( - 5)[/tex]

[tex]D =9 + 40[/tex]

[tex]D = 49[/tex]

Answer:

D IS THE ANSWER (49)

Step-by-step explanation:

The answer is -12 and 9. You have to factor the equation first. Once you do this, than you can set each problem equal to zero and solve. I hope this helps.

Answer:

A. 0.5

B. 4.5

C. 1.5

D. 0.5

Step-by-step explanation:

y varies inversely with x can be written as:

y = k/x

where k is constant of variation.

1. value of A

x=A, y = 9 and k = 4.5 (given)

y = k/x

9 = 4.5/A

=> A = 4.5/9

=> A=0.5

2. Value of B

x =1, y= B, k = 4.5

y = k/x

B = 4.5/1

B= 4.5

3. Value of C

x=C, y=3. k=4.5

y = k/x

3 = 4.5/C

3C = 4.5

C = 4.5/3

C = 1.5

4. Value of D

x= 9, y=D, k=4.5

y = k/x

D = 4.5/9

D = 0.5

Answer:

[tex]A=0.5[/tex]

[tex]B=4.5[/tex]

[tex]C=1.5[/tex]

[tex]D=0.5[/tex]

Step-by-step explanation:

The form an the equation of inverse variation is:

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

Being "k" the constant of variation.

Since we know "k" and we have the values given in the table, we can find the missing values:

To find A we need to substitute the [tex]y=9[/tex], the value of "k" and [tex]x=A[/tex] into the equation and solve for "A":

[tex]9=\frac{4.5}{A}[/tex]

[tex]A=\frac{4.5}{9}=0.5[/tex]

To find B we need to substitute the [tex]x=1[/tex], the value of "k" and [tex]y=B[/tex] into the equation:

[tex]B=\frac{4.5}{1}=4.5[/tex]

To find C we need to substitute the [tex]y=3[/tex], the value of "k" and [tex]x=C[/tex] into the equation and solve for "C":

[tex]3=\frac{4.5}{C}[/tex]

[tex]C=\frac{4.5}{3}=1.5[/tex]

To find D we need to substitute the [tex]x=9[/tex], the value of "k" and [tex]y=D[/tex] into the equation:

[tex]D=\frac{4.5}{9}=0.5[/tex]

186-140=46/2=23 weeks -2/23=slope

Answer:

Equation is W = -2t+186

It will take 23 weeks to achieve the goal.

Slope = -2 pounds per week

Step-by-step explanation:

Here  current Weight  is 186 pounds

and her goal is to weigh 140 pounds

she can safely lose 2 pounds per week which is the slope

for t =0 , W =186 pounds  (Y intercept )

    slope = 2 pounds per week

since its decreasing therefore its negative

W = -2t+186 is the equation

To achieve the desired goal W = 140

plugging W = 140 and solving for t

140 = -2t+186

2t = 186-140

2t = 46

t = 23

It will take 23 weeks to achieve the goal.

Slope = -2 pounds per week

Definition of average rate of change a function g(x) over an interval [a,b]:

[tex]A = \dfrac{g(b)-g(a)}{b-a}[/tex]

Substitute your function and your interval:

[tex]A = \dfrac{(3^6-6)-(3^0-6)}{3-0} = \dfrac{3^6-6-3^0+6}{3} = \dfrac{3^6-1}{3} = \dfrac{728}{3}[/tex]

The average rate of change of the function gx) = 3(2x) - 6 over the interval [0,3] is calculated as (g(3) - g(0)) / (3 - 0) which equals to 6.

The average rate of change of a function over an interval [a,b] is defined as:

(g(b) - g(a)) / (b - a)

Here, the function g(x) = 3(2x) - 6, and the interval is [0,3]. Let's calculate g(3) and g(0).

g(3) = 3(2*3) - 6 = 12

g(0) = 3(2*0) - 6 = -6

Now, apply these values to the average rate of change formula:

(g(3) - g(0)) / (3 - 0) = (12 - (-6)) / 3 = 18 / 3 = 6.

So, the average rate of change of the function g(x) over the interval [0,3] is 6.

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[tex]\bf ~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ (\stackrel{x_1}{5}~,~\stackrel{y_1}{6})\qquad (\stackrel{x_2}{8}~,~\stackrel{y_2}{2}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left(\cfrac{8+5}{2}~~,~~\cfrac{2+6}{2} \right)\implies \left( \cfrac{13}{2}~,~\cfrac{8}{2} \right)\implies \left(6\frac{1}{2}~,~4 \right)[/tex]

Answer:5

Step-by-step explanation:

To solve the expression 2 + 3x16 - 2x21 - 3, we apply the order of operations rule (PEMDAS) without any parentheses or exponents to handle. The expression simplifies to 5.

The student is asking to solve a mathematical expression using the correct order of operations. The proper order to solve math expressions is Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This rule is often remembered by the acronym PEMDAS.

Let's solve the expression step by step:

First calculate any operations inside parentheses. In the given problem, there are none.

Next, perform all multiplication and division operations from left to right. 3x16 equals 48, and 2x21 equals 42.

Subtract and add from left to right. So, 2 + 48 - 42 - 3 = 5.

Therefore, the expression

2 + 3x16 - 2x21 - 3= 5

.