Determine algebraically, the solution interval for the quadratic inequality 2x²-7x≤-3

Interval
Test Point
Substitution
True/False?
Solution:

The function rule that describes the given translation is T-9, 1(x, y).

The first value in the function rule represents the horizontal translation, while the second value represents the vertical translation. In this case, the square is translated 1 unit to the right, indicating a positive horizontal translation.

Additionally, the square is translated 9 units down, indicating a negative vertical translation. Therefore, the correct function rule is T-9, 1(x, y).

In the coordinate plane, the x-coordinate represents the horizontal position, and the y-coordinate represents the vertical position. When we apply the function rule T-9, 1 to the coordinates of the square, we subtract 9 from the y-coordinate and add 1 to the x-coordinate.

This results in the square being moved 9 units down and 1 unit to the right from its original position.

The negative sign in front of the 9 indicates a downward movement, and the positive sign in front of the 1 indicates a rightward movement. Hence, the translation is accurately described by the function rule T-9, 1(x, y).

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Answer:

C

Step-by-step explanation:

Answer:

a. 5π cm

b. 144π in²

c. 6 ft

Step-by-step explanation:

a.

The circumference of a circle is given by:

Circumference pf circle= πd

where d is the diameter.

In this case, d = 5 cm,

Therefore, Circumference of circle = π*5=5π cm

b.

The area of a circle is given by:
Area of circle=πr²,

where r is the radius. In this case, the diameter is d = 24 in,

so, the radius is r = d/2 = 24/2=12in

Therefore, Area of circle=π*12²=144π in²

c.

The area of a circle is given by:

Area of circle=πr²,

where r is the radius. In this case, Area is 36π ft²

Now substituting value

36π=πr²

dividing both side by π, we get

36=r²

[tex]r=\sqrt{36}=6[/tex]

r=6 ft.

Therefore, Radius is 6 ft.

Answer:

21 - 12p

Step-by-step explanation:

I hope this helps and I'm super sorry if I'm wrong!

Answer:

Fractions are like pancakes: you can flatten them horizontally or vertically. Horizontal flattening means you shrink the x-value by multiplying it by a huge number before doing anything else. Vertical flattening means you squish the y-value by multiplying the whole function by a tiny number. For example, if f (x) = x^2, then f (2x) is horizontally flattened by 2 and f (0.5x^2) is vertically flattened by 0.5. Don't worry, it's not rocket science, it's just math.

The row operation on the matrix [tex]\left[\begin{array}{ccc|c}2&0&0&16\\0&8&0&3\\0&0&5&6\end{array}\right][/tex] is [tex]\left[\begin{array}{ccc|c}1&0&0&8\\0&8&0&3\\0&0&5&6\end{array}\right][/tex]

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

[tex]\left[\begin{array}{ccc|c}2&0&0&16\\0&8&0&3\\0&0&5&6\end{array}\right][/tex]

The row operation is given as

1/2R₁

This means that we divide the entries on the first row by 2

Using the above as a guide, we have the following:

[tex]\left[\begin{array}{ccc|c}2&0&0&16\\0&8&0&3\\0&0&5&6\end{array}\right] = \left[\begin{array}{ccc|c}1&0&0&8\\0&8&0&3\\0&0&5&6\end{array}\right][/tex]

Hence, the row operation on the matrix is [tex]\left[\begin{array}{ccc|c}2&0&0&16\\0&8&0&3\\0&0&5&6\end{array}\right] = \left[\begin{array}{ccc|c}1&0&0&8\\0&8&0&3\\0&0&5&6\end{array}\right][/tex]

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1. The indefinite integral of (v + 1) / (2v - 20)^5 dv is -1 / (8(2v - 20)^4) + C.

2. The indefinite integral of x^2 / (x - 5) dx is (1/2) x^2 + 5x + 25 ln|x - 5| + C.

3. The indefinite integral of x cos(8x^2) dx is (1/16) sin(8x^2) + C.

4. The indefinite integral of 176 / e^(-x) + 1 dx is 176 ln|1 + e^x| + C.

1. To find the indefinite integral of (v + 1) / (2v - 20)^5 dv:

Let u = 2v - 20. Then du = 2 dv.

The integral becomes:

(1/2) ∫ (1/u^5) du

Now we can integrate using the power rule:

(1/2) ∫ u^(-5) du

Applying the power rule, we get:

(1/2) * (u^(-4) / -4) + C

= -1 / (8u^4) + C

Substituting back u = 2v - 20:

= -1 / (8(2v - 20)^4) + C

Therefore, the indefinite integral of (v + 1) / (2v - 20)^5 dv is -1 / (8(2v - 20)^4) + C.

2. To find the indefinite integral of x^2 / (x - 5) dx:

We can use polynomial long division to simplify the integrand:

x^2 / (x - 5) = x + 5 + 25 / (x - 5)

Now we can integrate each term separately:

∫ x dx + ∫ (5 dx) + ∫ (25 / (x - 5) dx)

Using the power rule, we get:

(1/2) x^2 + 5x + 25 ln|x - 5| + C

Therefore, the indefinite integral of x^2 / (x - 5) dx is (1/2) x^2 + 5x + 25 ln|x - 5| + C.

3. To find the indefinite integral of x cos(8x^2) dx:

We can use the substitution method. Let u = 8x^2, then du = 16x dx.

The integral becomes:

(1/16) ∫ cos(u) du

Integrating cos(u), we get:

(1/16) sin(u) + C

Substituting back u = 8x^2:

(1/16) sin(8x^2) + C

Therefore, the indefinite integral of x cos(8x^2) dx is (1/16) sin(8x^2) + C.

4. To find the indefinite integral of 176 / e^(-x) + 1 dx:

We can simplify the integrand by multiplying the numerator and denominator by e^x:

176 / e^(-x) + 1 = 176e^x / 1 + e^x

Now we can integrate:

∫ (176e^x / 1 + e^x) dx

Using u-substitution, let u = 1 + e^x, then du = e^x dx:

∫ (176 du / u)

Integrating 176/u, we get:

176 ln|u| + C

Substituting back u = 1 + e^x:

176 ln|1 + e^x| + C

Therefore, the indefinite integral of 176 / e^(-x) + 1 dx is 176 ln|1 + e^x| + C.

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Hello!

[tex]32 = 25x^2 - 4\\\\32 + 4 = 25x^2\\\\36 = 25x^2\\\\25x^2 - 36 = 0\\\\x = \dfrac{-b \±\sqrt{b^2 - 4ac} }{2a} \\\\\\x = \dfrac{-0 \±\sqrt{0^2 - 4 \times 25 \times (-36) } }{2 \times 25} \\\\\\x = \dfrac{\±60}{50} \\\\\boxed{x = \±\frac{6}{5} }[/tex]

The solutions of the system of equations are (2, -1) and (-4, 17)

The given system of equations is:

y = x² - x - 3

y = -3x + 5

To find the solutions, we need to solve these equations simultaneously.

Set the equations equal to each other:

x² - x - 3 = -3x + 5

Simplify and rewrite the equation in standard form:

x² - x + 3x - 3 - 5 = 0

x² + 2x - 8 = 0

Factor the quadratic equation:

(x - 2)(x + 4) = 0

Now we can solve for x by setting each factor equal to zero:

x - 2 = 0 or x + 4 = 0

Solving for x, we get:

x = 2 or x = -4

To find the corresponding y-values, we substitute these x-values into either of the original equations. Let's use equation 1):

For x = 2:

y = (2)² - 2 - 3 = 4 - 2 - 3 = -1

For x = -4:

y = (-4)² - (-4) - 3 = 16 + 4 - 3 = 17

As a result, the system of equations has two solutions: (2, -1) and (-4, 17).

The right responses are therefore (2, -1) and (-4, 17).

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The correct answer is By finding the quotient of the bases to be one third and canceling common factors. Option D.

The Quotient of Powers Property states that when dividing two powers with the same base, you can subtract the exponents. In the given expression, we have three to the fourth power divided by nine.

To simplify this expression using the Quotient of Powers Property, we first need to recognize that nine can be written as three squared, since 3 multiplied by itself gives 9.

So, we have (3^4) / (3^2). According to the Quotient of Powers Property, we subtract the exponents: 4 - 2.

This gives us 3^(4-2), which simplifies to 3^2. Therefore, the expression three to the fourth power all over nine equals three squared.

It states that we find the quotient of the bases to be one third and cancel common factors. In this case, the bases are 3 and 3, and their quotient is indeed one third. Additionally, there are no common factors that can be canceled, as the expression does not contain any variables or additional terms.

Therefore, By finding the quotient of the bases to be one third and canceling common factors. accurately describes the steps involved in simplifying the expression using the Quotient of Powers Property.

We find the quotient of the bases (one third) and cancel common factors (which is not applicable in this case). Option D is correct.

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Note the complete question is

Explain how the Quotient of Powers Property was used to simplify this expression. (1 point)

Three to the fourth power all over nine equals three squared

A.) By simplifying 9 to 32 to make both powers base three and adding the exponents

B.) By simplifying 9 to 32 to make both powers base three and subtracting the exponents

C.) By finding the quotient of the bases to be one third and simplifying the expression

D.) By finding the quotient of the bases to be one third and cancelling common factors

Answer:

D. Right skewed

Step-by-step explanation:

To determine the shape of the distribution, we can examine the given data:

12, 9, 4, 8, 25, 6, 8, 5, 18, 13

One way to determine the shape of the distribution is by visualizing it using a histogram or a box plot. However, without the exact frequency of each value, we cannot create an accurate visual representation.

Alternatively, we can examine the skewness of the distribution. Skewness is a measure of the asymmetry of a distribution. If the data is skewed to the left, it is left-skewed or negatively skewed. If it is skewed to the right, it is right-skewed or positively skewed. If the data is symmetric and evenly distributed, it is considered a symmetrical distribution.

Let's calculate the skewness of the given data to determine the shape:

Skewness = (3 * (mean - median)) / standard deviation

First, let's calculate the mean, median, and standard deviation of the data:

Mean = (12 + 9 + 4 + 8 + 25 + 6 + 8 + 5 + 18 + 13) / 10 = 10.8

Median = the middle value when the data is arranged in ascending order:

4, 5, 6, 8, 8, 9, 12, 13, 18, 25

Median = (8 + 9) / 2 = 8.5

Next, let's calculate the standard deviation:

Step 1: Calculate the squared differences from the mean for each value:

(12 - 10.8)^2, (9 - 10.8)^2, (4 - 10.8)^2, (8 - 10.8)^2, (25 - 10.8)^2, (6 - 10.8)^2, (8 - 10.8)^2, (5 - 10.8)^2, (18 - 10.8)^2, (13 - 10.8)^2

Step 2: Calculate the sum of squared differences:

(1.44 + 2.88 + 45.76 + 8.64 + 228.01 + 22.09 + 8.64 + 32.49 + 47.04 + 4.84) = 411.73

Step 3: Calculate the variance:

Variance = sum of squared differences / (n - 1) = 411.73 / (10 - 1) = 45.75

Step 4: Calculate the standard deviation:

Standard deviation = square root of variance = √45.75 = 6.76 (approximately)

Now we can calculate the skewness:

Skewness = (3 * (mean - median)) / standard deviation

Skewness = (3 * (10.8 - 8.5)) / 6.76

Skewness = 6.4 / 6.76

Skewness ≈ 0.95

Since the skewness is positive (0.95), the data is right-skewed or positively skewed. Therefore, the appropriate shape of the distribution is:

D. Right skewed

The percentile of 6.2 in the given data set is 40%.

To determine the percentile of 6.2 in the given data set, we can use the following steps:

Arrange the data set in ascending order:

4.2, 4.6, 5.1, 6.2, 6.3, 6.6, 6.7, 6.8, 7.1, 7.2

Count the number of data points that are less than or equal to 6.2. In this case, there are 4 data points that satisfy this condition: 4.2, 4.6, 5.1, and 6.2.

Calculate the percentile using the formula:

Percentile = (Number of data points less than or equal to the given value / Total number of data points) × 100

In this case, the percentile of 6.2 can be calculated as:

Percentile = (4 / 10) × 100 = 40%

The percentile of 6.2 in the sample data set is therefore 40%.

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The equation of the line of fit is y = 15x + 30.

To determine the equation of the line of fit, we can use the given data points (0,30) and (2,60). We can use the slope-intercept form of a linear equation, which is y = mx + b, where m represents the slope and b represents the y-intercept.

Using the two data points, we can calculate the slope (m) as the change in y divided by the change in x:

m = (60 - 30) / (2 - 0) = 30 / 2 = 15

Now that we have the slope, we can substitute one of the data points into the equation to solve for the y-intercept (b). Let's use the point (0,30):

30 = 15(0) + b

30 = 0 + b

b = 30

Therefore, the equation of the line of fit is y = 15x + 30. This means that for every additional hour of study time (x), the grade percent (y) increases by 15, and the line intersects the y-axis at 30.

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Answer:

A) No solutions

Step-by-step explanation:

First of all, we know that option B will always be incorrect. You cannot have two solutions. To illustrate this, try drawing two lines. You will find that they will either intersect once (one solution), or they will not intersect, (no solutions, parallel lines), or they are the same line and thus they will always intersect (infinitely many solutions).

With that in mind, let's solve the equation.

4x-4-5x=7-x+5

First, combine all like terms.

-x-4=12-x

Now add 4 to both sides to leave x by itself.

-x=16-x

This statement cannot be true. Therefore, this equation has no solutions (parallel lines. One line starts from 0, or the origin. That line is -x. The other line starts from 16. That line is -x+16.)

Hope this helps!

a) There are 2400 animals on the farm.

b) The number of sheep on the farm is 700.

c) The percentage of rabbits in relation to the total number of animals is 37.5%.

d) The angle that represents the number of pigs is 75 degrees.

a) To determine the total number of animals on the farm, we add up the number of rabbits, sheep, cattle, and pigs:

Total number of animals = 900 (rabbits) + 700 (sheep) + 300 (cattle) + 500 (pigs) = 2400 animals.

b) The number of sheep on the farm is given as 700.

c) To calculate the percentage of rabbits in relation to the total number of animals, we divide the number of rabbits by the total number of animals and multiply by 100:

Percentage of rabbits = (900 / 2400) * 100 = 37.5%.

d) To calculate the angle that represents the number of pigs, we need to find the proportion of the total number of animals that pigs make up, and then convert it to an angle on the pie chart.

Proportion of pigs = 500 / 2400 = 0.2083.

To find the angle in degrees, we multiply the proportion by 360 degrees:

Angle representing pigs = 0.2083 * 360 = 75 degrees.

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Answer: 337

Step-by-step explanation: it is 337 because if you subtract it all you get that

1. The limit of (sin3x)/(2x - sinx) as x approaches 0 is -27.

2. The limit of x^(-1)lnx as x approaches infinity is -1.

3. The limit of x/e^x as x approaches infinity is 0.

1. To find the limit of (sin3x)/(2x - sinx) as x approaches 0 using L'Hôpital's rule, we can differentiate the numerator and denominator separately and take the limit again:

Let's differentiate the numerator and denominator:

Numerator: d/dx (sin3x) = 3cos3x

Denominator: d/dx (2x - sinx) = 2 - cosx

Now, we can find the limit of the differentiated function as x approaches 0:

lim x->0 (3cos3x)/(2 - cosx)

Again, differentiating the numerator and denominator:

Numerator: d/dx (3cos3x) = -9sin3x

Denominator: d/dx (2 - cosx) = sinx

Taking the limit as x approaches 0:

lim x->0 (-9sin3x)/(sinx)

Now, substituting x = 0 into the function gives:

(-9sin0)/(sin0) = 0/0

Since we obtained an indeterminate form of 0/0, we can apply L'Hôpital's rule again.

Differentiating the numerator and denominator:

Numerator: d/dx (-9sin3x) = -27cos3x

Denominator: d/dx (sinx) = cosx

Taking the limit as x approaches 0:

lim x->0 (-27cos3x)/(cosx)

Now, substituting x = 0 into the function gives:

(-27cos0)/(cos0) = -27/1 = -27

Therefore, the limit of (sin3x)/(2x - sinx) as x approaches 0 is -27.

2. To find the limit of x^(-1)lnx as x approaches infinity using L'Hôpital's rule, we can differentiate the numerator and denominator separately and take the limit again:

Let's differentiate the numerator and denominator:

Numerator: d/dx (lnx) = 1/x

Denominator: d/dx (x^(-1)) = -x^(-2) = -1/x^2

Now, we can find the limit of the differentiated function as x approaches infinity:

lim x->∞ (1/x)/(-1/x^2)

Simplifying the expression:

lim x->∞ -x/x = -1

Therefore, the limit of x^(-1)lnx as x approaches infinity is -1.

3. To find the limit of x/e^x as x approaches infinity using L'Hôpital's rule, we can differentiate the numerator and denominator separately and take the limit again:

Let's differentiate the numerator and denominator:

Numerator: d/dx (x) = 1

Denominator: d/dx (e^x) = e^x

Now, we can find the limit of the differentiated function as x approaches infinity:

lim x->∞ (1)/(e^x)

Since the exponential function e^x grows much faster than any polynomial function, the denominator goes to infinity much faster than the numerator. Therefore, the limit of (1)/(e^x) as x approaches infinity is 0.

Thus, the limit of x/e^x as x approaches infinity is 0.

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Answer:

be more clear of what u mean edit the question so we can tell what u mean and answer correctly

Step-by-step explanation:

no explanation

The fraction that converts to a terminating decimal number is C. 3/8.

To determine which fraction converts to a terminating decimal number, we need to analyze the denominator of each fraction. A fraction will result in a terminating decimal if its denominator has only prime factors of 2 and/or 5.

Let's examine each option:

A. 1/6: The denominator is 6, which can be factored into 2 * 3. Since 3 is not a factor of 2 or 5, this fraction does not convert to a terminating decimal.

B. 2/9: The denominator is 9, which can be factored into 3 * 3. Since 3 is not a factor of 2 or 5, this fraction does not convert to a terminating decimal.

C. 3/8: The denominator is 8, which can be factored into 2 * 2 * 2. Since all the factors are 2, this fraction does convert to a terminating decimal.

D. 4/7: The denominator is 7, which cannot be factored into 2 or 5. Therefore, this fraction does not convert to a terminating decimal.

Based on our analysis, the fraction that converts to a terminating decimal number is C. 3/8.

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Answer:

The recursive formula for an=1/8(2)n-1 is:

a1=1/8 an+1=1/8(2)(n)

This formula defines a sequence where each term is equal to 1/8 of the previous term multiplied by 2.

Step-by-step explanation:

Answer:

[tex]\textsf{a)} \quad \textsf{Radius = $\overline{HG}$}[/tex]

[tex]\textsf{b)} \quad \textsf{Chord = $\overline{GF}$}[/tex]

[tex]\textsf{c)} \quad \textsf{Diameter = $\overline{JF}$}[/tex]

[tex]\textsf{d)} \quad \textsf{Secant = $\overleftrightarrow{GF}$}[/tex]

[tex]\textsf{e)} \quad \textsf{Tangent = $\overleftrightarrow{GK}$}[/tex]

[tex]\textsf{f)} \quad \textsf{Point of tangency = $\overset{\bullet}{G}$}[/tex]

[tex]\textsf{g)} \quad \textsf{Circle $H$}[/tex]

Step-by-step explanation:

The radius is the distance from the center of a circle to any point on its circumference. The center of the circle is point H. Therefore, the radius of the given circle is line segment HG.

A chord is a straight line joining two points on the circumference of the circle. There are two chords in the given circle:  line segments GF and JF. Therefore, a chord of the given circle is line segment GF.

The diameter of a circle is a straight line segment passing through the center of a circle, connecting two points on its circumference.

Therefore, the diameter of the given circle is line segment JF.

A secant is a straight line that intersects a circle at two points.

Therefore, the secant of the given circle is line GF.

A tangent is a straight line that touches a circle at only one point.

Therefore, the tangent line of the given circle is line GK.

The point of tangency is the point where the line touches the circle.

Therefore, the point of tangency of the given circle is point G.

A circle is named by its center point. Therefore, as the center point of the circle is point H, the name of the circle is "Circle H".

The x-intercepts, y-intercept, vertex, and axis of symmetry for the given function g(x) = x² + 4x + 3 have been plotted on the graph below.

In Mathematics and Geometry, the graph of a quadratic function would always form a parabolic curve because it is a u-shaped. Based on the given quadratic function, we can logically deduce that the graph would be a upward parabola because the coefficient of x² is positive and the value of "a" is greater than zero (0).

Since the leading coefficient (value of a) in the given quadratic function g(x) = x² + 4x + 3 is positive 1, we can logically deduce that the parabola would open upward and the x-intercept (roots) is given by the ordered pair (-3, 0) and (-1, 0).

In conclusion, the vertex is given by the ordered pair (-2, -1) and the minimum value is -1.

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Answer:

x-intercepts = (-3, 0) and (-1, 0)

y-intercept = (0, 3)

Vertex = (-2, -1)

Axis of symmetry:  x = -2

Step-by-step explanation:

Given quadratic function:

[tex]g(x) = x^2 + 4x + 3[/tex]

[tex]\hrulefill[/tex]

To find the x-intercepts of the given function g(x), set g(x) equal to zero and solve for x.

Set g(x) = 0:

[tex]x^2 + 4x + 3 = 0[/tex]

Factor the quadratic equation:

[tex]x^2+x+3x+3=0[/tex]

[tex]x(x+1)+3(x+1)=0[/tex]

[tex](x + 3)(x + 1) = 0[/tex]

Set each factor equal to zero and solve for x:

[tex]x + 3 = 0 \implies x = -3[/tex]

[tex]x + 1 = 0 \implies x = -1[/tex]

Therefore, the x-intercepts are at (-3, 0) and (-1, 0).

[tex]\hrulefill[/tex]

To find the y-intercept of g(x), substitute x = 0 and solve for y.

[tex]\begin{aligned}x=0 \implies y&= (0)^2 + 4(0) + 3\\y& = 0 + 0 + 3 \\y&= 3\end{aligned}[/tex]

Therefore, the y-intercept is at (0, 3).

[tex]\hrulefill[/tex]

The x-value of the vertex of a parabola in the form y = ax² + bx + c is x = -b/2a.

For the function g(x), a = 1, and b = 4.

Therefore, the x-coordinate of the vertex is:

[tex]\textsf{$x$-coordinate of vertex} = \dfrac{-b}{2a} = \dfrac{-4}{2(1)} = -2[/tex]

To find the y-coordinate of the vertex, substitute x = -2 into function g(x):

[tex]\begin{aligned}x=-2 \implies y &= (-2)^2 + 4(-2) + 3\\y& = 4-8 + 3 \\y&= -1\end{aligned}[/tex]

Therefore, the vertex is at (-2, -1).

[tex]\hrulefill[/tex]

The axis of symmetry of a vertical parabola is the x-coordinate of its vertex.

As the x-coordinate of the vertex is -2, the axis of symmetry is x = -2.

[tex]\hrulefill[/tex]

Answer:

Para resolver este problema, podemos plantear un sistema de ecuaciones. Si definimos "p" como el número de velas pequeñas y "g" como el número de velas grandes, podemos expresar la información del problema de la siguiente manera:

p + g = 20 (la tienda vendió un total de 20 velas) 10.98p + 27.98g = 355.60 (el ingreso total por la venta de velas fue de $355.60)

Podemos resolver este sistema de ecuaciones utilizando el método de sustitución. Despejando "p" de la primera ecuación, obtenemos:

p = 20 - g

Luego, sustituimos esta expresión de "p" en la segunda ecuación:

10.98(20 - g) + 27.98g = 355.60

220.20 - 10.98g + 27.98g = 355.60

17.00g = 135.40

g = 8

Por lo tanto, la tienda vendió 8 velas grandes.

Step-by-step explanation:

Using the concept of perimeter of polygon, the perimeter of figure C is 27cm shorter than total perimeter of A and B

To solve this problem, we have to know the perimeter of the polygon C.

The perimeter of a polygon is the sum of all the lengths of the outer edges of the figure, that is, we must find the length of all the edges of the polygon, and then add these lengths to obtain the perimeter.

The perimeter of the figures are;

Using the concept of perimeter of a rectangle;

a. figure A = 2(4 + 11) = 30cm

b. figure B = 2(8 + 4) = 24cm

c figure C = 11 + 4 + 8 + 4 = 27cm

Now, we can add A and B and then subtract c from it.

30 + 24 - 27 = 27cm

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Answer:

C) -1/3

Step-by-step explanation:

Slope=rise/run

Slope=-1/3

The best answer that matches the calculated values is C. sin θ = -1/3, tan θ = -3/(2√2)

Let's break down the given values and find the values of sine and tangent.

We are given:

cos θ = √(8/9)

csc θ < 0

Using the Pythagorean identity, sin^2θ + cos^2θ = 1, we can find the value of sin θ.

sin^2θ + (√(8/9))^2 = 1

sin^2θ + 8/9 = 1

sin^2θ = 1 - 8/9

sin^2θ = 1/9

Taking the square root of both sides, we get:

sin θ = ±1/3

Since csc θ is negative (csc θ < 0), we can conclude that sin θ is negative. Therefore, sin θ = -1/3.

Next, let's find the value of tan θ.

tan θ = sin θ / cos θ

tan θ = (-1/3) / (√(8/9))

tan θ = -√9/√8

tan θ = -√9/√(4*2)

tan θ = -√9/(2√2)

tan θ = -3/(2√2)

So, the values are:

sin θ = -1/3

tan θ = -3/(2√2)

The best selection from the available options that matches the calculated values is:

C. sin θ = -1/3, tan θ = -3/(2√2)

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A

no, because they are both right triangles and

the one on the left is 88° at the right anglr

We conclude that there is no valid domain for the given function g(x, y) = sin^-1(x² + y² - 3). Thus, the concept of circles with radii does not apply in this case.

To determine the domain of the function g(x, y) = sin^-1(x² + y² - 3), we need to examine the range of the arcsine function. The arcsine function, [tex]sin^{(-1)[/tex](z), is defined for values of z between -1 and 1, inclusive. Therefore, for the given function, we have:

-1 ≤ x² + y² - 3 ≤ 1

Rearranging the inequality, we get:

-4 ≤ x² + y² ≤ -2

Now, let's analyze the inequalities separately:

x² + y² ≤ -2:

This inequality is not possible since the sum of squares of two non-negative numbers (x² and y²) cannot be negative. Therefore, there are no points that satisfy this inequality.

x² + y² ≤ -4:

Similarly, this inequality is also not possible since the sum of squares of two non-negative numbers cannot be less than or equal to -4. Therefore, there are no points that satisfy this inequality either.

Based on the analysis, we conclude that there is no valid domain for the given function g(x, y) = sin^-1(x² + y² - 3). Thus, the concept of circles with radii does not apply in this case.

It's important to note that the arcsine function has a restricted range of -π/2 to π/2, and for a valid domain, the input of the arcsine function must be within the range of -1 to 1. In this particular case, the given expression x² + y² - 3 exceeds the range of the arcsine function, resulting in no valid domain.

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