Which expression is equivalent to
(x-1)^2 x^2+-6
———— * ————
x^2-x-12 x^2+6x-5
If no denominator equals 0?

Answer:

Step-by-step explanation:

The circumference is the perimeter of a circle, which is equal to 131.88ft.

Circumference = [tex]2\pi r^{2}[/tex]

[tex]131.88=2\pi r^{2} \\r^{2} =\frac{131.88}{2\pi } \\r=\sqrt{20.989} \\r=4.581[/tex]

This means that the diameter is:

[tex]d=2r\\d=9.163[/tex]

So our area is:

[tex]A=\pi r^{2} \\A=\pi (4.581)^{2} \\A=65.940[/tex]

Hope this helped!

Answer:

[tex] \frac{y - y1}{m} = x - x1 \\ x1 = x - \frac{y - y1}{m} [/tex]

so,A is the correct answer...

A 95% confidence interval for the average is 3898.

In statistics, the confidence interval formula is used to express the degree of uncertainty surrounding a sample estimate of a population parameter.

Formula of confidence interval:

If n ≥ 30,

Confidence Interval = m ± z(SD/√n)

where,

n = Number of terms

m = Sample average

SD = Standard Deviation

Value for the confidence interval in the z table is z.

Table of Confidence interval  and Z

CI      z

0.70 1.04

0.75 1.15

0.80 1.28

0.85 1.44

0.90  1.645

0.92 1.75

0.95 1.96

0.96 2.05

0.98 2.33

0.99 2.58

According to the question

random sample of student (n) =  99

average savings = 3800 dollars

Standard deviation =  500 dollars

95% confidence interval for the average needed

i.e , z = 1.96

Now, By using Formula of confidence interval:

Confidence Interval = m ±  z(SD/√n)

substituting the values

Confidence Interval = 3800 ± 1.96(500/√99)

                                 = 3800 ± 98.49

                                 = 3898.49

values to the nearest whole number

Confidence Interval = 3898

Hence,  A 95% confidence interval for the average is 3898 .

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The principal from difference between Simple interest and Compound interest of a certain sum of money is 50.4 at 6% p.a. for 2 years is $14,000.

Let the principal be P.

Simple interest = P x 0.06 x 2 = 0.12P

Compound interest = P(1 + 0.06)^2 - P = 1.1236P - P = 0.1236P

Given: Compound interest - Simple interest = 50.4

0.1236P - 0.12P = 50.4

0.0036P = 50.4

P = 50.4 / 0.0036

P = 14000

Therefore, the principal is $14,000.

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Answer:

5 + 10[tex]\sqrt{3}[/tex]

= 22.3205080757

Step-by-step explanation:

Answer:

≈ 13,7 inches

Step-by-step explanation:

Just use trigonometry (sin or cos) from the right triangles ∆ABC and ∆ADC

Answer:

The formula for calculating the size of an interior angle is: interior angle of a polygon = sum of interior angles ÷ number of sides. The sum of exterior angles of a polygon is 360°. The formula for calculating the size of an exterior angle is: exterior angle of a polygon = 360 ÷ number of sides.

Step-by-step explanation:

The given triangles are  similar by AA~, SSS~, SAS~, or not similar.

What is similar triangles in maths?

Triangles that are similar to one another in terms of shape but not necessarily size are called congruent triangles. Examples of related objects are all equilateral triangles and squares with any side length. To put it another way, if two triangles are comparable, then their corresponding angles and sides are congruent and equal in size.

7.  In triangles AEB & BDC

 AE/BD = EB/DC

We can conclude that, the corresponding sides are  in common ratio, so triangles are  similar.

Hence, The given triangles are  similar.

8. Given a figure of two triangles, ΔLKM and ΔLJN

Here,

              LK/LJ = LM/LN

             21 - 6/21 = 10/14

                 15/21 = 10/14

                 5/7 = 5/7

We can conclude that, the corresponding sides are in common ratio, so triangles are  similar.

Hence, The given triangles are  similar.

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Answer:

Common difference = 2

Step-by-step explanation:

We can find the common difference by subtracting any two consecutive terms with the smaller number being subtracted from the larger:

Thus, to find the common difference, d, we can subtract -13 from -15:

d = -15 - (-13) = -15 + 13 = 2

Answer:

Let the length of the rectangle be x and the width of the rectangle be y.

Then, we can use the Pythagorean Theorem to find the value of x and y:

x2 + y2 = 62

x2 + y2 = 36

We can use trigonometry to find the value of x and y:

x = 6 sin 30°

y = 6 cos 30°

Therefore, the length of the rectangle is 6 sin 30° cm and the width of the rectangle is 6 cos 30° cm.

The volume of the sphere is approximately 33.51 cubic centimeters.

The volume of a sphere can be calculated using the formula:

V = (4/3)π[tex]r^{3}[/tex]

where r is the radius of the sphere

V is the volume of the sphere

Since the diameter of the sphere is 4 centimeters, the radius is half of that, or 2 centimeters.

Plugging this value into the formula, we get:

V = (4/3)π([tex]2^{3}[/tex])

V = (4/3)π(8)

V = (32/3)π

So the answer is (a)  32/3π[tex]cm^{3}[/tex]

the volume of the sphere is approximately 33.51 cubic centimeters.

The volume of the sphere is a measure of how much space the sphere occupies. The formula for the volume of the sphere tells us that the volume is proportional to the cube of the radius. This means that as the radius increases, the volume of the sphere increases much more rapidly. This is why a small increase in the radius can result in a much larger increase in the volume of the sphere.

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The dimensions of the box are 30 cm by 50 cm by 25 cm.

Area is a measure of the amount of space enclosed by a two-dimensional object, such as a shape or a surface. It is usually expressed in square units, such as square meters (m²), square inches (in²), or square centimeters (cm²), depending on the unit system used.

Let x be the side length of the squares that are cut out of each corner of the cardboard.

When the squares are cut out and the sides are folded up, the resulting box will have dimensions (80-2x) cm by (100-2x) cm by x cm.

The area of the base of the box is:

(80-2x) cm × (100-2x) cm = 8000 - 360x + 4x² cm²

We know that the area of the base is 4000 cm², so we can set this equal to the expression we just found:

8000 - 360x + 4x² cm² = 4000 cm²

Simplifying and rearranging, we get a quadratic equation:

4x² - 360x + 4000 = 0

Dividing both sides by 4, we get:

x² - 90x + 1000 = 0

This quadratic can be factored as:

(x - 40)(x - 25) = 0

Therefore, the possible values of x are x = 40 cm and x = 25 cm.

If we use x = 40 cm, then the dimensions of the box are:

Length = 80 - 2x = 0 cm (not possible)

Width = 100 - 2x = 20 cm

Height = x = 40 cm

If we use x = 25 cm, then the dimensions of the box are:

Length = 80 - 2x = 30 cm

Width = 100 - 2x = 50 cm

Height = x = 25 cm

Therefore, the dimensions of the box are 30 cm by 50 cm by 25 cm.

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Therefore, the probability of Mrs. Vo winning is 1 - 91/300 - 8/21 = 11/30.

Probability refers to the likelihood or chance that a particular event or outcome will occur. It is a mathematical concept that quantifies the degree of uncertainty associated with an event or situation. Probability is typically expressed as a number between 0 and 1, where 0 indicates that an event is impossible, and 1 indicates that it is certain to occur.

by the question.

let's consider the first case: Mrs. Vo spins a number greater than 65 on her first spin. There are a total of 8 numbers greater than 65 on the wheel: 70, 75, 80, 85, 90, 95, 100. Since there are 21 sections on the wheel, the probability of spinning a number greater than 65 on the first spin is 8/21.

The number of possible outcomes for Mrs. Vo's spin is 21, since there are 21 equal portions on the wheel. However, we need to exclude the outcomes that would cause Mrs. Vo's sum to exceed 100, since that would result in an automatic loss. The largest possible outcome on the wheel is 100, so Mrs. Vo cannot spin a number greater than or equal to 35 on her first spin.

The number of outcomes that would give Mrs. Vo a winning sum of at least 66 is the number of outcomes greater than 65, which are 70, 75, 80, 85, 90, 95, and 100. There are 7 such outcomes.

If Mrs. Vo spins a number less than or equal to 65 on her first spin, the probability of this happening is 13/21. The probability of her second spin being between 1 and 35 is 35/100 (since there are 35 numbers between 1 and 35 on the wheel). So, the probability of her total sum being less than or equal to 65 is (13/21) * (35/100) = 91/300.

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Answer:

sin(theta) = -3/5.

Step-by-step explanation:

We know that tan(theta) = 3/4, which means that the ratio of the opposite side to the adjacent side of a right triangle with angle theta is 3/4. We can use the Pythagorean theorem to find the hypotenuse of the triangle:

hypotenuse^2 = opposite^2 + adjacent^2

hypotenuse^2 = (3^2) + (4^2)

hypotenuse^2 = 9 + 16

hypotenuse^2 = 25

hypotenuse = 5

Since 180 degrees < theta < 270 degrees, we know that theta is in the third quadrant, where the sine function is negative. Therefore, sin(theta) = -opposite/hypotenuse = -3/5.

So, sin(theta) = -3/5.

Answer:

2. Where is the point of discontinuity of y =

=

O x = 5

O x = 49

O x=-10

O x = -5

-10x+49

x-5

?

After answering the provided question, we can conclude that Therefore, inequality the answer is (D) x > -3 and y > 1.

In mathematics, an inequality is a non-equal relationship between two expressions or values. As a result, imbalance leads to inequality. In mathematics, an inequality connects two values that are not equal. Inequality is not the same as equality. When two values are not equal, the not equal sign is commonly used (). Different inequalities, no matter how small or large, are used to contrast values. Many simple inequalities can be solved by modifying the two sides until only the variables remain. But a number of things contribute to inequality: Negative values are divided or added on both sides. Trade off the left and right.

domain and range of the inequality

[tex]y - 1 < \sqrt{x + 3}\\(y - 1)^2 < x + 3\\(y - 1)^2 - 3 < x\\[/tex]

Therefore, the answer is (D) x > -3 and y > 1.

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A sphere's volume may be calculated using the formula [tex]137.25[/tex] cm cube.

V = 4/3 r3, where V is the volume and r is the radius, is the formula for a sphere's volume. 1/2 of a sphere's radius makes up its radius. Therefore, you may determine the radius first, followed by the volume, to get the surface of a circle given its diameter.

The radius of a sphere is the distance between its center and its outside surface. The amount of space a satellite's surface covers is known as its surface area, and it is equal to 4r2.

The diameter of the sphere, [tex]d = 6.4[/tex] cm

Radius,[tex]r = 3.2[/tex] cm

We need to find the volume of the sphere. The formula for the volume of a sphere is given by :

[tex]V=4/3pir^{3}[/tex]

[tex]4/3pi*(3.2)^{3}[/tex]

[tex]=137.25 cm^{3}[/tex]

The radius of the bowling ball is [tex]6.4 cm/2 = 3.2 cm[/tex], and its volume is [tex]V[/tex] bowling [tex]= 4/3 pi(3.2)^{3} * 8 = 1,100.8[/tex] cubic cm.

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Answer:

A

Step-by-step explanation:

Answer:

An exponential function can be used to model this situation.

In this case, we know that each hour following the initial dose, 80% of the concentration remains in the body from the preceding hour. This means that the amount of penicillin remaining in the body is decreasing exponentially.

We can use the formula for exponential decay, which is:

A = A0 * e^(-kt)

where A is the amount of penicillin remaining in the body at time t, A0 is the initial amount of penicillin (in this case, 500 milligrams), k is the decay constant, and e is the mathematical constant approximately equal to 2.718.

The decay constant, k, can be calculated using the following formula:

k = ln(0.8)

where ln is the natural logarithm.

So the exponential function that can be used to model this situation is:

A = 500 * e^(-0.2231t)

where t is the time in hours since the initial dose was administered.

This function will give us the amount of penicillin remaining in the body at any time t, based on the initial dose and the rate of decay.

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Answer:

30.5 or 61/2 cm²

Step-by-step explanation:

So the area of the figure is shown by this equation.

Area = Area of triangle - Area of rectangle cut out

        = (0.5)(9)(9) - (2)(5)

        = 40.5 - 10

        = 30.5 or 61/2 cm²

The area of the blue paper obtained by subtracting the 6 circle areas from the rectangle is 20.64 sq. cm.

A circle is made up of all points in a single plane that are equidistant from one another. Only the bordering points make up the circle. The radius is the distance from the centre to the outside of the circle.

The diameter is the length of a line that connects the circle's ends to its middle point. The radius is half as large as the diameter.

Given data:

length l = 12 cm

width w = 8 cm

radius r = 4/2 = 2 cm.

Area of blue page = Total area of rectangle - 6*area of circle

Area of blue page = length * width - 6*π*r²

Area of blue page = 12*8 - 6*3.14*2²

Area of blue page = 20.64 sq. cm.

Thus, the area of the blue paper obtained by subtracting the 6 circle areas from the rectangle is 20.64 sq. cm.

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Answer:

1/4

Step-by-step explanation:

As given the figure, 3 out of 4 rectangles are shaded,

So,

4/4-3/4=1/4

The total space that the cargo will take up is 7 x 10⁷ cubic feet.

In geometry, a prism is a three-dimensional solid shape that consists of two congruent bases (which can be any polygon) and a set of parallel line segments that connect the corresponding vertices of the two bases. The bases are usually parallel and of the same shape and size, and the line segments connecting the vertices are known as the lateral faces of the prism.

Prisms are commonly found in architecture and engineering, where they are used to create structures such as buildings, bridges, and towers. They are also used in optics, where they are used to refract and reflect light, and in mathematics, where they are used to study geometric properties and relationships.

To find the total space that the energy prisms will take up, we need to multiply the number of prisms by the amount of space each prism takes up. We can use scientific notation to simplify the calculation:

3.5 x 10⁵ energy prisms * 2 x 10² cubic feet per prism

= (3.5 * 2) x 10⁵⁺² cubic feet

= 7 x 10⁷ cubic feet

Therefore, the total space that the cargo will take up is 7 x 10⁷ cubic feet.

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We can conclude after answering the provided question that Therefore, diameter m GFH = 21° and m GZH = 116°.

In geometry, the diameter of a circle is any straight line segment that has an endpoint on the circle and travels through its centre. Another way to put it is the longest chord of a circle. Either concept can be used to define the diameter of a sphere. The diameter is the length of the line that runs tangent to the two points at either end of the circle. If you consider length to be the distance between two points, diameter equals length. The diameter of a circle is the distance between its two furthest points.

We know that angle EGF is a right angle (90°) because EG is a diameter.

Given that m FG = 116°, we can calculate m FGE by subtracting it from 180° (the sum of triangle angles):

m FGE = 180 degrees - m FG = 180 degrees - 116 degrees = 64 degrees

m GFH = 180° - m ZGF - m ZGFH = 180° - 90° - 69° = 21°

Finally, we can calculate m GZH by subtracting m FGE from 180° (the sum of triangle angles):

180° m GZH - 180° m FGE - 64° = 116°

Therefore, m GFH = 21° and m GZH = 116°.

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If b is not equal to 0, an ordered pair of numbers a and b, denoted as a / b, is a ratio and the ratio in the given situation is 3a:2b = 6:1.

An ordered pair of numbers a and b, represented as a / b, is a ratio if b is not equal to 0.

A proportion is an equation that sets two ratios at the same value.

For instance, if there is 1 boy and 3 girls, you may express the ratio as 1: 3 (there are 3 girls for every boy), meaning that there are 1 in 4 boys and 3 in 4 girls.

Ratios contrast two figures by ordinarily dividing them. A/B would be your formula if you were comparing one data point (A) to another data point (B).

So, we know that:

a+b/a-b = 5/3

Now, solve as follows:

3 ( a + b) = 5 ( a - b)

3 a + 3 b = 5 a - 5 b

5 b + 3 b = 5 a - 3 a

8 b = 2 a

2 a = 8 b

a = 4 b

3 a = 3 × 4 b

3 a = 12 b

3 a: 2 b = 12 b : 2 b

Then: 3a:2b = 6:1

Therefore, if b is not equal to 0, an ordered pair of numbers a and b, denoted as a / b, is a ratio and the ratio in the given situation is 3a:2b = 6:1.

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Correct question:

A+b/a-b=5/3 what are the following ratios? 3a:2b

The area of the triangle with vertices (-1, 1), (-1, 5), and (4, 1) is 10 square units.

The area of a triangle is a measure of the amount of space inside the triangle. It is defined as half the product of the base of the triangle and its corresponding height.

If a triangle has base length b and corresponding height h, then its area A is given by the formula:

A = (1/2) × b ×h.

In the given question,

To find the area of the triangle with vertices (-1, 1), (-1, 5), and (4, 1), we can use the formula: A = (1/2) × b ×h.

where the base is the distance between two vertices of the triangle and the height is the perpendicular distance from the third vertex to the base.

We can choose any two points as the endpoints of the base, but it may be easiest to use the points (-1, 1) and (4, 1), since the height will be the vertical distance from the third vertex (which is ( -1, 5)) to the base.

The distance between (-1, 1) and (4, 1) is: base = 4 - (-1) = 5

To find the height, we need to determine the perpendicular distance from the point (-1, 5) to the line containing the base. This can be done by finding the equation of the line containing the base and then finding the distance from the point (-1, 5) to that line.

The equation of the line containing the base is: y = 1

since the line is horizontal and passes through the point (4, 1) and (-1, 1).

The distance from the point (-1, 5) to the line y = 1 is:

height = |5 - 1| = 4

where the absolute value is taken to ensure that the height is positive.

Now, we can substitute the values of the base and height into the formula for the area:

Area = (1/2) × base × height

= (1/2) * 5 * 4

= 10

Therefore, the area of the triangle with vertices (-1, 1), (-1, 5), and (4, 1) is 10 square units.

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The evaluation of expression 7√3 / (√10 + √3) - 2√5 / (√6 + √5) - 3√2 / (√15 + 3√2) and simplification to get the final result of 2√5 - 2√6 + 6.

To evaluate the expression is

7√3 / (√10 + √3) - 2√5 / (√6 + √5) - 3√2 / (√15 + 3√2)

we need to rationalize the denominators of each fraction. To do this, we can multiply both the numerator and denominator of each fraction by the conjugate of the denominator.

For the first fraction, we can multiply the numerator and denominator by (√10 - √3):

7√3(√10 - √3) / [(√10 + √3)(√10 - √3)]

Simplifying the denominator gives:

7√3(√10 - √3) / 10 - 3

For the second fraction, we can multiply the numerator and denominator by (√6 - √5):

-2√5(√6 - √5) / [(√6 + √5)(√6 - √5)]

Simplifying the denominator gives:

-2√5(√6 - √5) / 6 - 5

For the third fraction, we can multiply the numerator and denominator by (√15 - 3√2):

-3√2(√15 - 3√2) / [(√15 + 3√2)(√15 - 3√2)]

Simplifying the denominator gives:

-3√2(√15 - 3√2) / 15 - 9

Now, we can simplify each expression separately and then combine them:

7√3(√10 - √3) / 7 = √30 - 3

-2√5(√6 - √5) / 1 = -2√6 + 2√5

-3√2(√15 - 3√2) / 6 = -√30 + 9

Combining the expressions gives:

√30 - 3 - 2√6 + 2√5 - √30 + 9

Simplifying further gives:

2√5 - 2√6 + 6

Therefore, the value of the expression 7√3 / (√10 + √3) - 2√5 / (√6 + √5) - 3√2 / (√15 + 3√2) is 2√5 - 2√6 + 6.

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Answer:

16

Step-by-step explanation:

When you square a square root (√1^2) it just cancels out. :)