Solve for x. Round to the nearest tenth of a
degree, if necessary.
S
to
3.9
8.5
R
Q

By completing squares we will get the expression:

(x + 7)² = -4

Here we have the quadratic equation:

x² + 22x = 8x - 53

First, move all the terms to the left side, then we will get:

x² + 22x - 8x + 53 = 0

x² + 14x + 53  =0

Now, remember the perfect square trinomial:

(a + b)² = a² + 2ab + b²

Then we can rewrite our expression as:

x² + 2*7*x + 53  =0

Now we can add and subtract 7², then we will get.

x² + 2*7*x + 7² - 7² + 53  =0

(x + 7)² - 7² + 53 = 0

(x + 7)² - 49 + 53 = 0

(x + 7)² + 4 = 0

(x + 7)² = -4

That is the expression we wanted.

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The case are appropriate with the given:

(x-3)² = 8(y-4)

y² = 4²x

(y-2)² = 4(x-3)

Analyze each given equation to determine the appropriate case and find the vertex, focus, opening of the graph, directrix, and latus rectum.

1. (x-3)² = 8(y-4)

This equation represents a parabola with its vertex form given by (h, k) = (3, 4).

Case 1: The coefficient of (y - k) is positive, indicating an upward-opening parabola.

Vertex: The vertex is (3, 4).

Focus: The focus can be found using the formula (h, k + 1/4a), where a = coefficient of (y - k). In this case, the focus is (3, 4 + 1/4 × 8) = (3, 6).

Directrix: The directrix is a horizontal line located at y = k - 1/4a. In this case, the directrix is y = 4 - 1/4 × 8 = 2.

Latus Rectum: The latus rectum is the length of the chord passing through the focus and perpendicular to the axis of symmetry. In this case, it is equal to 4a = 4 × 8 = 32 units.

2. y² = 4²x

This equation represents a parabola with its vertex form given by (h, k) = (0, 0).

Case 2: The coefficient of x is positive, indicating a right-opening parabola.

Vertex: The vertex is (0, 0).

Focus: The focus can be found using the formula (h + 1/4a, k), where a = coefficient of x. In this case, the focus is (0 + 1/4 × 4, 0) = (1, 0).

Directrix: The directrix is a vertical line located at x = h - 1/4a. In this case, the directrix is x = 0 - 1/4 × 4 = -1.

Latus Rectum: The latus rectum is the length of the chord passing through the focus and perpendicular to the axis of symmetry. In this case, it is equal to 4a = 4 × 4 = 16 units.

3. (y-2)² = 4(x-3)

This equation represents a parabola with its vertex form given by (h, k) = (3, 2).

Case 3: The coefficient of (x - h) is positive, indicating a right-opening parabola.

Vertex: The vertex is (3, 2).

Focus: The focus can be found using the formula (h + 1/4a, k), where a = coefficient of (x - h). In this case, the focus is (3 + 1/4 × 4, 2) = (4, 2).

Directrix: The directrix is a vertical line located at x = h - 1/4a. In this case, the directrix is x = 3 - 1/4 × 4 = 2.

Latus Rectum: The latus rectum is the length of the chord passing through the focus and perpendicular to the axis of symmetry. In this case, it is equal to 4a = 4 × 4 = 16 units.

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Answer:

  -14

Step-by-step explanation:

You want to know the value of -a +b -(-c) when a=3, b=-6, c=-5.

We can simplify the given expression by recognizing that a double negative is a positive:

  = -a +b +c

The value is found by substituting the given numbers and doing the arithmetic:

  = -(3) +(-6) +(-5)

  = -3 -6 -5 = -(3 +6 +5) = -14

The value of the expression is -14.

__

Additional comment

Your calculator can help you find the value of a numeric expression.

<95141404393>

The value of the mathematical expression −a +b − −c with given values a =3, b = −6, and c= −5 is -4.

The given mathematical expression is −a +b − −c. In this equation, we are given the values a =3, b = −6, and c= −5. We substitute these values into the equation to solve it. Thus, it becomes: -3 + (-6) - −(-5). When solved, -3 + (-6) +5 equals -4

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The decimal equivalent to the fraction of a cup of flour added to each case is approximately 0.923.

To find the decimal equivalent of the fraction of a cup of flour added to each case, we can divide the total amount of flour (36 cups) by the number of cases (39).

Decimal equivalent = Total amount of flour / Number of cases

Decimal equivalent = 36 cups / 39 cases

Calculating this division:

Decimal equivalent ≈ 0.923

Therefore, the decimal equivalent to the fraction of a cup of flour added to each case is approximately 0.923.

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The equation of the circle with a center at (-4, 6) and a point on the circumference at (-2, 9) is (x + 4)² + (y - 6)² = 13.

Given:

Center of the circle: (-4, 6)

Point on the circumference: (-2, 9)

The center of the circle (h, k) is given as (-4, 6), which means h = -4 and k = 6.

The formula for the distance between the center and the point on the circumference can be used to determine the radius (r).

The formula for distance is provided by:

d = √((x₂ - x₁)² + (y₂ - y₁)²)

where (x₁, y₁) is the center (-4, 6), and (x₂, y₂) is the point on the circumference (-2, 9).

Putting in the values:

d = √((-2 - (-4))² + (9 - 6)²)

= √((2)² + (3)²)

= √(4 + 9)

= √13

So, the radius (r) is √13.

Now, we can substitute the values of h, k, and r into the general equation of the circle:

(x - h)² + (y - k)² = r²

(x - (-4))² + (y - 6)² = (√13)²

(x + 4)² + (y - 6)² = 13

Therefore, the equation of the circle is (x + 4)² + (y - 6)² = 13.

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The length of segment x in the two intersecting chords is determined as 4.

The value of segment x is calculated by applying intersecting chord theorem, which states that the angle at tangent is half of the arc angle of the two intersecting chords.

Also this theory states that the product of two segments of a chord is equal to the product of the two segments of second intersecting chord in a circle.

From the diagram, we can set up the following equations and solve for length x;

5 (x) = 10(2)

5x = 20

x = 20/5

x = 4

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Answer:

6.98

Step-by-step explanation:

5.9454+1.03 =6.9754 which is rounded to 6.98

The perimeter of triangle A"B"C" is found as 43.78.

We first determine length of each side:

The distance formula has that :

AB = √[(4 - (-4))² + (8 - 0)²]

= √128

= 8√2

BC = √[(8 - 4)² + (-12 - 8)²]

= √320

= 8√5

CA = √[(-4 - 8)² + (0 - (-12))^²]

= √320

= 8√5

cos(ABC) = (AB² + BC² - CA²) / (2 * AB * BC)

cos(ABC) = (128 + 320 - 320) / (2 * 8√2 * 8√5)

cos(ABC) = 1 / (4√2)

ABC = [tex]cos^{-1}[/tex](1 / (4√2))

ABC = 78.46°

A"B" = AB = 8√2

B"C" = BC = 8√5

C"A" = CA = 8√5

We then find the perimeter of triangle A"B"C" is:

A"B" + B"C" + C"A"

= 8√2 + 8√5 + 8√5

= 8√2 + 16√5

= 43.78

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Answer: y = 5x-2

Step-by-step explanation:

The correct system of expressions is (3) x ≥ 0, y ≥ 0, x+y ≥ 42, x ≤ 28, y ≤ 40, 2.25x + 17.00y = z

The interest accrued on a certificate of deposit (CD) can be compounded quarterly by using the following formula:

P(1 + r/n)nt = A

Where:

The total of the accrued interest and principle is denoted by the letter A.

P denotes the principle of the initial investment.

The yearly interest rate is expressed in decimals as r.

n is the number of interest compoundings every year.

t is the age in years.

Celine Hocking invests $3,500 at a 4.5% annual interest rate with quarterly compounding in this case. Let's calculate the interest after one year:

Given in decimal form, 0.045 is equal to P = $3,500 and r = 4.5%.

(Quatrio-annual compounding) t = 1 (1 year), and n = 4.

A = 3500(1 + 0.045/4)^(4*1)

A = 3500(1 + 0.01125)^4 A = 3500(1.01125)^4 A ≈ 3500(1.045564)

A ≈ $3,668.47

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1)

Surface area of rectangular prism:

SA = 2( hw + lw + lh )

h= height of prism .

w = width of prism .

l = length of prism .

Substitute the values,

h= 4m

w=1.2 m

l= 13.4 m

SA = 2(4*1.2 + 13.4*1.2 + 13.4*4)

SA = 148.96 m²

2)

Surface area of triangular prism:

SA = (S1 +S2 + S3)L + bh

h= height of prism .

b = base of prism .

S1 , S2 , S3 = sides of triangle .

Substitute the values,

SA = (3+ 4 + 0.5)4 + 0.5*3

SA = 31.5 in²

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Slope = 8/3

Given,

The table of linear function.

x :  -3  , 0 , 3,  6

y :  -5 , -3 , -1,  1

Slope = change in y over change in x

Slope = -3-(-5)/0 - (-3)

Slope = 8/3

Passing point = 0,-3

y-y1 = m (x - x1)

y-(-3) = 8/3(x - 0)

y + 3 = 8/3  x

y = 8/3  x -3

Hence, slope = 8/3 and linear function = y = 8/3  x -3

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The perimeter of the triangle is √34 + √65 + √101, and the area is 2.5.

We have,

To calculate the perimeter and area of a triangle, we need to know the coordinates of its vertices.

Let's calculate the perimeter and area for the given triangles.

a) Triangle with vertices A(2,2), B(7,-1), C(3,-8)

To calculate the perimeter, we need to find the lengths of the three sides of the triangle and sum them up.

Side AB:

Length AB = √((x2 - x1)² + (y2 - y1)²)

= √((7 - 2)² + (-1 - 2)²)

= √(5² + (-3)²)

= √(25 + 9)

= √34

Side BC:

Length BC = √((x2 - x1)² + (y2 - y1)²)

= √((3 - 7)² + (-8 - (-1))²)

= √((-4)² + (-7)²)

= √(16 + 49)

= √65

Side AC:

Length AC = √((x2 - x1)² + (y2 - y1)²)

= √((3 - 2)² + (-8 - 2)²)

= √(1² + (-10)²)

= √(1 + 100)

= √101

Perimeter = AB + BC + AC

= √34 + √65 + √101

To calculate the area, we can use the formula for the area of a triangle:

Area = 1/2 x base x height

We can consider any side of the triangle as the base and find the height corresponding to that base.

Let's consider side AB as the base.

Height h = distance between point C and the line AB

Using the formula for the distance between a point (x0, y0) and a line Ax + By + C = 0:

h = |(Ax0 + By0 + C)| / √(A² + B²)

Equation of line AB:

(x2 - x1) (y - y1) - (y2 - y1)(x - x1) = 0

(7 - 2)(y - 2) - (-1 - 2)(x - 2) = 0

5(y - 2) + 3(x - 2) = 0

5y - 10 + 3x - 6 = 0

5y + 3x - 16 = 0

Plugging in the coordinates of point C (3, -8):

h = |(53 + 3(-8) - 16)| / √(5² + 3²)

= |-5| / √(25 + 9)

= 5 / √34

Area = 1/2 x AB x h

= 1/2 x √34 x (5 / √34)

= 1/2 x 5

= 2.5

Therefore,

The perimeter of the triangle is √34 + √65 + √101, and the area is 2.5.

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The complete question.

Calculate the perimeter and area of ​​the triangles whose vertices are a) A(2,2) B(7,-1) ((3,-8) ​

The real roots of the function using false-position method are 1/2 and 5/8

The real root of the function f(x) = x³ - 2x - 5 = 0 using the False-Position method can be calculated as;

1. Choose two initial guesses, a and b, such that f(a) and f(b) have opposite signs. In this case, we can choose a = 0 and b = 1.

[tex]c = \frac{(a * f(b) - b * f(a))}{(f(b) - f(a)}[/tex]

If f(c) = 0, then c is the root of the equation. Otherwise, replace a with b and b with c.

[tex]c = \frac{(0 * f(1) - 1 * f(0))}{(f(1) - f(0))} = \frac{1}{2}[/tex]

Since f(1/2) = -0.25, we can replace a with 1 and b with 1/2.

[tex]c = \frac{\frac{1}{2} * f(1) * f(\frac{1}{2}) }{(f(1) - f(\frac{1}{2}) } = \frac{5}{8}[/tex]

Since f(5/8) = 0.0625, we can stop here and say that the root of the equation is 5/8, which is approximately 0.625.

Where the roots of the equation are 1/2 and 5/8

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The rocket's maximum height is 800 feet. It reached its maximum height at 5 seconds after launch.

To solve this problem, we need to find the vertex of the parabola represented by the equation Y= -16t^2+160t. The vertex of a parabola is the point where the parabola changes direction, from increasing to decreasing or vice versa.

The vertex of the parabola is given by the following formula:

(-b/2a, c - b^2/4a)

In this case, the value of b is 160 and the value of a is -16. Plugging these values into the formula, we get the following:

(-160/2(-16), 800 - 160^2/4(-16))

(5, 800)

Therefore, the rocket reached its maximum height at 5 seconds after launch. The maximum height is 800 feet.

The average rate of change of [tex]\(g(x) = x\)[/tex] over the interval [tex]\([-9, -2]\) is 1.[/tex]

To find the average rate of change of a function over an interval, we calculate the difference in the function values at the endpoints of the interval and divide it by the difference in the corresponding x-values.

In this case, the function [tex]\(g(x) = x\)[/tex] and the interval is [tex]\([-9, -2]\)[/tex]. We can determine the average rate of change as follows:

[tex]\[ \text{Average rate of change} = \frac{g(-2) - g(-9)}{-2 - (-9)} \][/tex]

By substituting the function values, we get:

[tex]\[ \text{Average rate of change} = \frac{-2 - (-9)}{-2 - (-9)} \][/tex]

Simplifying the numerator and denominator, we have:

[tex]\[ \text{Average rate of change} = \frac{7}{7} = 1 \][/tex]

Therefore, the average rate of change of [tex]\(g(x) = x\)[/tex] over the interval [tex]\([-9, -2]\)[/tex] is [tex]1[/tex].

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Answer:

  870 miles

Step-by-step explanation:

You want to know the number of miles that a car can travel on 29 gallons of gas if it travels 570 miles on 19 gallons, and the miles are proportional to the gallons.

Increasing the number of gallons by a factor of 29/19 will increase the number of miles by that same factor. The car will be able to travel ...

  (29/19)(570) = 870 . . . . miles

The car can travel 870 miles if it has 29 gallons of gas.

__

Additional comment

We could do this solving the equation ...

  (570 mi)/(19 gal) = (x mi)/(29 gal)

Multiplying by 29 gal gives ...

  x = (29/19)(570) = 870 . . . . . as above

<95141404393>

Based on the ratio of the digit in its hundred places compared to the same digit in its hundredth place, the number of times the value of the digit in the hundred places is greater than the value of the digit in the hundredth place is 10,000.

The ratio is the relative size of one value, number, or quantity compared to another.

The ratio is computed as the quotient of one quantity divided by another.

We can express a ratio as a fraction, decimal, percentage, or in its standard form (:).

For instance, using the digit 4 in its hundred places will give 400.  The same digit in its hundredth place gives 0.04.

The ratio of 400 to 0.04 = 10,000 (400 ÷ 0.04)

Thus, in this instance, we can conclude that 400 is 10,000 times greater than 0.04.

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The value of 0.045 × 2.05 / 0.0025 in standard form is 3.69 × 10.

To perform the calculation and express the answer in standard form, follow these steps:

Multiply 0.045 by 2.05:

0.045 × 2.05 = 0.09225

Divide the result by 0.0025:

0.09225 / 0.0025

= 36.9

Convert the answer to standard form by writing it as a decimal multiplied by a power of 10:

36.9 = 3.69 × 10

Therefore, the value of 0.045 × 2.05 / 0.0025 in standard form is 3.69 × 10.

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The volume of the  triangular prism is determined as 2,520 yd².

The volume of the  triangular prism is calculated by applying the following formula as follows;

V = ¹/₂bhl

where;

The volume of the  triangular prism is calculated as;

V =  ¹/₂  x 12 yd x 14 yd x 30 yd

V = 2,520 yd²

Thus, the volume of the  triangular prism is calculated by applying the following formula for volume of  triangular prism.

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Mrs. Tilmos should use 8 1/2 cups of fertilizer for her triangle garden.

To answer the question, we must first calculate the area of Mrs. Tilmos' triangle garden. We can do this using the equation for the area of a triangle (A=1/2 × b × h). In this equation, b is the base and h is the height.

For this garden, the base (b) is 13 feet and the height (h) is twice the base, so h=2 × b or h=26 feet. Using the equation, we can calculate that the area of the garden is:

A = 1/2 × b × h

A = 1/2 × 13 × 26

A = 169 square feet

Now that we have the area of the garden, we can calculate how many cups of fertilizer to use. We know that the fertilizer's directions say to use 1 cup per every 20 square feet, so we have to divide 169 by 20.

169 / 20 = 8.45

We can round this result up to 8 1/2 cups of fertilizer to use.

Therefore, Mrs. Tilmos should use 8 1/2 cups of fertilizer for her triangle garden.

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Answer:

Step-by-step explanation:

Ok so, as given in the table, the cost of the 3rd tank is 800$, we can thus determine the cost of 1 cubic inch for a tank and use that to determine the cost of the other tanks.

[tex]volume =L\times W\times H\\=24\times 24\times 24\\=13824 in^3[/tex]

thus, to compute the cost of one cubic inch, we can divide the cost by the volume of the 3rd tank (in cubic inches)

[tex]\frac{800}{13824}=0.0579[/tex] [tex]usd/in^3[/tex]

now, we can determine the dimensions of the second tank that will make it cost 150$

[tex]0.0579\times 18\times W\times 24=150[/tex]

[tex]W=\frac{150}{0.0579\times 18\times 24} =6 in.[/tex]

Now to determine the cost of the first tank, we can multiply its volume by the cost per cubic inch:

[tex]cost=0.0579\times36\times 36\times 36=2700[/tex]$

As of question B, the cost will increase as the width increases.

For Question C, we can find the volume of the sphere by the formula:

[tex]volume=\frac{4}{3}\pi r^3 =\frac{4}{3}\pi \times 24^3=57905 in^3[/tex]

To find the cost of that tank, we just multiply its volume by the cost per unit volume:

[tex]cost=57905\times 0.0579=3352[/tex]$ (roughly)

Answer:

Since there is only one 9 on the spinner, the probability of spinning a 9 is 1/3.

P(9) = 1/3

Answer:

x = 2nπ ± π/3 or x = 2nπ ± cos⁻¹(3/4)

The transformation used to create the pattern in the top figure is reflection.

A reflection is a transformation that flips an object over a line, called the line of reflection. It produces a mirror image of the original object. In the case of the pattern in the top figure, we can observe that the pattern is symmetric about a specific line of reflection.

To determine if a reflection was used, we examine the pattern for mirror symmetry. Mirror symmetry means that one half of the pattern is a reflection of the other half. In the top figure, if we were to fold the pattern along a vertical line, the two halves would align perfectly, indicating mirror symmetry.

This mirror symmetry suggests that a reflection has been applied to create the pattern. Each shape in the pattern appears to have been reflected across the line of reflection to create its corresponding mirrored shape.

Other transformations such as translation, rotation, and glide reflection do not exhibit the same mirror symmetry as a reflection. A translation involves shifting an object without changing its orientation, a rotation involves rotating an object around a fixed point, and a glide reflection is a combination of a translation and a reflection.

In conclusion, based on the mirror symmetry observed in the top figure's pattern, the transformation used to create the pattern is a reflection.

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The equation that models the exponential decline is y = 46348587109809610(0.9861)ˣ if the population over the past 20 years is provided.

Let's suppose the equation that models the exponential decline is: y=a(bˣ).

From the data given, we can calculate the value of a and b:

a = 46348587109809610

b = 0.9861

y = 46348587109809610(0.9861)ˣ

Therefore, the equation that models the exponential decline is y = 46348587109809610(0.9861)ˣ if the population over the past 20 years is provided.

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Answer:

A

Step-by-step explanation:

5 - [tex]\frac{3}{2}[/tex] x ≥ [tex]\frac{1}{3}[/tex]

multiply through by 6 ( the LCM of 2 and 3 ) to clear the fractions

30 - 9x ≥ 2 ( subtract 30 from both sides )

- 9x ≥ - 28

divide both sides by - 9 , reversing the symbol as a result of dividing by a negative quantity.

x ≤ [tex]\frac{28}{9}[/tex]

Answer:

A) 4.24 miles/hour

Step-by-step explanation:

To find Abigail’s jogging speed in miles per hour, we need to divide the distance she jogged by the time it took her to jog that distance.

Therefore, Abigail’s jogging speed is 10.6 miles / 2.5 hours = 4.24 miles/hour.

Answer: A) 4.24 miles/hour

Step-by-step explanation:

The jogging speed can be calculated by dividing the total distance by the total time. So, to find Abigail's speed in miles per hour, we would divide the total miles (10.6) by the total hours (2.5).

10.6 miles ÷ 2.5 hours = 4.24 miles/hour

Therefore, the answer is:

A) 4.24 miles/hour

Answer:

13 pints

Step-by-step explanation:

quarts times 2.

6.5•2=13

Answer:

13 pints.

Step-by-step explanation:

The conversion rate from pints to quarts is 1:2 meaning every 1 pints will result in 2 quarts. Therefore with this formula, (6 1/2)*2 brings the result to 13.

You're welcome!

Applying the Pythagorean theorem, we can say that the length of the diagonal walkway that Max is designing is 20 feet.

The first step is to understand that the design of the diagonal walkway forms a right triangle, so we can use the Pythagorean theorem, whose square of the length of the diagonal, also called the hypotenuse, will be equal to the sum of the squares on both sides.

Therefore, substituting the values in the formula C²=A²+B² where A corresponds to the width, B to the length and C as the diagonal, we have:

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