A circle is circumscribed around a rectangle with sides lengths 6 and 8 what is the area of the circle?

A. 16[tex]\pi[/tex]
B. 20[tex]\pi[/tex]
C. 24[tex]\pi[/tex]
D. 25[tex]\pi[/tex]
E. 30[tex]\pi[/tex]

Answer:

see explanation

Step-by-step explanation:

(a)

let the two consecutive odd integers ne n and n + 2 , then

n + 2(n + 2) = 85

n + 2n + 4 = 85

3n + 4 = 85 ( subtract 4 from both sides )

3n = 81 ( divide both sides by 3 )

n = 27

n + 2 = 27 + 2 = 29

the 2 consecutive odd integers are 27 and 29

--------------------------------------------------------------------

(b)

let the first integer be n , then 2nd integer is 3n and 3rd is 3n + 5 , so

n + 3n + 3n + 5 = 40

7n + 5 = 40 ( subtract 5 from both sides )

7n = 35 ( divide both sides by 7 )

n = 5

3n = 3 × 5 = 15

3n + 5 = 15 + 5 = 20

the 3 integers are 5, 15, 20

For a circle k(O) with diameter AB and CD ⊥ AB, it is proved that AD·CB=AC·CD, using similarity of triangles.

Given:

circle k(O) with diameter AB

CD ⊥ AB

To Prove: AD·CB=AC·CD

Proof:

In ΔADC and ΔCDB,

∠ADC = ∠CDB = 90°  

[∵Both are right angle triangles]

CB = CB [Common side]

⇒ AC / CB = CD / DB

Thus, ΔACD is similar to ΔCDB by RHS similarity.

Therefore, we can write,

AD/CD = AC/CB [Since corresponding sides of similar triangles are proportional]

⇒ AD·CB = AC·CD

Hence proved.

Learn more about similarity here:

https://brainly.com/question/25882965

#SPJ1

The parallel sides AB, PQ, and CD, gives similar triangles, ∆ABD ~ ∆PQD and ∆CDB ~ ∆PQB, from which we have;

[tex] \frac{1}{x} + \frac{1}{y}= \frac{1}{z}[/tex]

From the given information, we have;

According to the ratio of corresponding sides of similar triangles, we have;

[tex] \frac{x}{z} = \mathbf{\frac{BD}{QD} }[/tex]

[tex] \frac{y}{z} = \frac{BD}{ BQ} [/tex]

Which gives;

[tex] \mathbf{\frac{y}{z}} = \frac{BD }{ BD - Q D} [/tex]

[tex] \frac{z}{y} = \frac{BD - QD }{ BD } = 1 - \frac{Q D }{ BD}[/tex]

QD × x = BD × z

BD × z = (1 - QD/BD) × y = y - (QD × y/BD)

Therefore;

BD × z = y - (QD × y/BD)

BQ × y = y - (QD × y/BD)

BQ × y = y - (z × y/x) = y × (1 - z/x)

(1 - z/x) = BQ

BD × z = y × (1 - z/x)

BD = (y × (1 - z/x))/z

Therefore;

QD × x = y × (1 - z/x)

(BD-BQ) × x = y × (1 - z/x)

(BD-(1 - z/x)) × x = y × (1 - z/x)

BD = (y × (1 - z/x))/x + (1 - z/x)

BQ + QD = (1 - z/x) + (y × (1 - z/x))/x

BD = BQ + QD

(y × (1 - z/x))/x + (1 - z/x) = (y × (1 - z/x))/z

(1 - z/x)×(y/x + 1) =(1 - z/x) × y/z

Dividing both sides by (1 - z/x) gives;

y/x + 1 = y/z

Dividing all through by y gives;

(y/x + 1)/y = (y/z)/y

Therefore;

[tex] \frac{1}{x} + \frac{1}{y}= \frac{1}{z}[/tex]

Learn more about characteristics similar triangles here:

https://brainly.com/question/1799826

#SPJ1

Answer:

18

Step-by-step explanation:

The arrows on lines AB and CD indicate that these 2 shapes are similar and these 2 lines are corresponding so :

Linear scale factor :

12 ÷ 10 = 1.2 or 6/5

Let's use the fraction form

Using the linear scale factor we can make an equation to solve for x :

2x+4 = 6/5(x+8)

Expand the brackets :

2x+4 = 6/5x + 9.6

Subtract 4 from both sides :

2x = 6/5x + 5.6

Subtract 6/5x from both sides :

4/5x = 5.6

Divide both sides by 4/5 :

x = 7

Now substitute this value into the expression for the length of AE :

AE = 2(7) + 4

AE = 14 + 4

AE = 18

Hope this helped and have a good day

Answer:

116 units

Step-by-step explanation:

AE + ED = 180 because they make a straight angle and a straight angle is 180 degrees or half of a circle 360/2.

AE = 2x + 4 and ED = x +8 added together they equal 180

2x + 4 + x + 8 = 180  Combine the like terms

3x + 12 = 180  Subtract 12 from both sides of the equation

3x = 168  Divide both sides by 3

x = 56

Now that we know that x is 56 we can plug that in for AE to find its length

2x + 4

2(56) + 4

112 + 4

116

Answer:

Step-by-step explanation:

The rational root theorem tells you any rational roots of the expression will be found from the constant and the leading coefficient:

  rational roots = ±{divisor of 25} / {divisor of 3}

The list of divisors in each case is pretty short, so this is ...

  rational roots = ±{1, 5, 25) / {1, 3} = ±{1/3, 1, 5/3, 5, 25/3, 25}

We find the only actual rational root is x = 5/3 when we graph the function.

(Factoring out that root, we find the remaining roots are ±i√5, irrational imaginary values.)

Answer:

m<DCF = 155°

Step-by-step explanation:

m<DCF = m<D + m<E

m<DCF = 72° + 83°

m<DCF = 155°

The solution to the given differential equation [tex]\frac{dy}{dx} = e^{4x+5y}[/tex] is , [tex]\frac{e^{-5y}}{-5}=\frac{e^{4x}}{4} +c[/tex], where c is constant of integration.

For given question,

We have been given a differential equation [tex]\frac{dy}{dx} = e^{4x+5y}[/tex]

We know that for any real number a, m, n,

[tex]a^{m + n} = a^m \times a^n[/tex]

⇒ dy/dx = [tex]e^{4x}[/tex] × [tex]e^{5y}[/tex]

Separating the variables (x and its differential in one side and y and its differential in another side )

⇒ [tex]\frac{1}{e^{5y}}[/tex] dy = [tex]e^{4x}[/tex] dx

⇒ [tex]e^{-5y}[/tex] dy = [tex]e^{4x}[/tex] dx

Integrating on both the sides,

⇒ [tex]\int e^{-5y}[/tex] dy = [tex]\int e^{4x}[/tex] dx

We know that, [tex]\int e^{ax}\, dx=\frac{e^{ax}}{a} +C[/tex]

⇒ [tex]\int e^{4x}\, dx=\frac{e^{4x}}{4} +C[/tex]

and [tex]\int e^{-5y}\, dy=\frac{e^{-5y}}{-5} +C[/tex]

So the solution is, [tex]\frac{e^{-5y}}{-5}=\frac{e^{4x}}{4} +c[/tex], where c is constant of integration.

Therefore, the solution to the given differential equation [tex]\frac{dy}{dx} = e^{4x+5y}[/tex] is , [tex]\frac{e^{-5y}}{-5}=\frac{e^{4x}}{4} +c[/tex], where c is constant of integration.

Learn more about the differential equation here:

https://brainly.com/question/14620493

#SPJ4

There are 27720 strings.

Strings of 12 ternary digits, 3 possibilities for each digit in the string, gives a total of 3^12 = 531,441 ternary strings of length 12.

There are

C(12,3) = 12! / (9! 3!) = 12 * 11 * 10/ 3 * 2 = 220 possible combinations of positions for the three 0s.

Of the 9 remaining positions, there are

C(9,4) = 9! / (5! 4!)

= 9 * 8 * 7 * 6/ (4 * 3 * 2 * 1)

= 126 possible combinations of positions for the four 2s.

The remaining 5 places are, of course, occupied by the five 1s.

So there that contain exactly three 0s, five 1s, and 4 2s are 220 * 126= 27720 such strings.

Learn more about strings here: https://brainly.com/question/27128699

#SPJ4

Answer:

$613.04

Step-by-step explanation:

Compound Interest Formula:

   [tex]A = P(1+\frac{r}{n})^{nt}[/tex]

   n = number of compounds

   t = time

   r = interest rate

   P = principle amount (original amount)

   A = final amount

Since it's compounded quarterly, that means there will be 4 compounds per year, so n=4. The interest rate has to be converted to the decimal value, and this is done by simply dividing it by 100 to get r=0.06.

Plug Values into equation:

[tex]A = 300(1+\frac{0.06}{4})^{12*4}[/tex]

Simplify inside parenthesis

[tex]A = 300(1.015)^{48}[/tex]

Calculate exponent

[tex]A \approx 300(2.043478)[/tex]

Multiply values

[tex]A \approx 613.04348[/tex]

Round

[tex]A = 613.04[/tex]

The image on the left represents a function. The image on the right does not as a function cannot have multiple variables for a single X quantity.

Answer: A

Step-by-step explanation:

[tex](-5,3)\\(-3, 1)\\(-1,-1)\\(1, -1)\\(3, 1)\\(5, 3)[/tex]

Answer:g(x)

Step-by-step explanation:1/5 g (x)

The term "altitude" is the same as "height of a triangle". It is perpendicular to the base. Since we can rotate the triangle to have any side be horizontal, there are effectively 3 possible bases. Hence, there are 3 heights. It all depends how you look at it.

Let h1, h2, and h3 be the three altitudes or heights.

Without loss of generality, we'll focus on the first two heights h1 and h2. Their respective bases are b1 and b2.

If we use b1 as the base, then the area is...

area = 0.5*base*height = 0.5*b1*h1

Similarly, the other base gives the area of:

area = 0.5*b2*h2

------------------------

Since both formulas refer to the same area (because we're working with the same triangle), we can set the expressions equal to one another

0.5*b1*h1 = 0.5*b2*h2

b1*h1 = b2*h2

Let's see what happens when b1 = b2, so,

b1*h1 = b2*h2

b1*h1 = b1*h2

b1h1 - b1h2 = 0

b1(h1 - h2) = 0

b1 = 0 or h1 - h2 = 0

b1 = 0 or h1 = h2

If the bases b1 and b2 were equal, then either those bases must be 0 which isn't possible, or the altitudes must be equal. However, the initial premise is that the heights must be different from one another.

Therefore, the bases b1 and b2 can't be the same length.

We could follow the same steps and logic to conclude that if the altitudes h1 and h3 were different, then the bases b1 and b3 can't be the same. Similarly, we would conclude that b2 and b3 can't be the same. This is where the "without loss of generality" kicks in.

In other words, we only need to focus on one subcase to extend the logic to the other cases, without having to actually do every single step. That would be a bit tedious busywork.

In conclusion, we've shown that if the heights are different, then their respective bases must be different. This leads to wrapping up the proof that we have a scalene triangle.

Side note: I used an indirect proof or proof by contradiction. I assumed that a non-scalene triangle was possible and it led to a contradiction of h1 = h2.

The function f(x) = ∛x - 3 is positive at the interval (3, oo)

The function is given as:

f(x) = ∛x - 3

The function is positive when the function value is greater than 0

This is represented as:

f(x) > 0

So, we have the following inequality expression

∛x - 3 > 0

Take the cube of both sides

x - 3 > 0

Add 3 to both sides

x > 3

Express as an interval notation

(3, oo)

Hence, the function f(x) = ∛x - 3 is positive at the interval (3, oo)

Read more about function interval at:

https://brainly.com/question/1600302

#SPJ1

[tex] \qquad \qquad \bf \huge\star \: \: \large{ \underline{Answer} } \huge \: \: \star[/tex]

A function should be one - to - one and onto in order to have inverse.

and to find the point on its inverse function we swap the value of x - coordinate and y - coordinate.

like (x , y) becomes (y , x)

The only way we get (y , x) is by taking image of point (x , y) about line : y = x

[tex] \qquad \large \sf {Conclusion} : [/tex]

we can conclude that the graph of a function and it's inverse is symmetric about equation (line) : y = x

Answer:

  no solution

Step-by-step explanation:

The Law of Sines tells you the relationship between sides and angles of a triangle.

The Law of Sines tells you ...

  sin(A)/a = sin(B)/b = sin(C)/c

Using the given information, this would tell us ...

  sin(55°)/12 = sin(B)/b = sin(C)/27

Then angle C would be ...

  sin(C) = (27/12)sin(55°) ≈ 1.843

There is no angle whose sine is greater than 1. The triangle we seek does not exist. (Side BC is too short relative to side AB.)

Answer:

20 pounds

Step-by-step explanation:

i took the test trust me i got 2 points

Answer:cost(weight)

Step-by-step explanation:

The probability for at least 1 page is 0.667 .

Probability is the department of arithmetic concerning numerical descriptions of the way likely an event is to occur, or how probable it is that a proposition is actual. The probability of an event is a number between 0 and 1, where, roughly speaking, zero shows the impossibility of the event, and 1 suggests truth.

Possibility = the wide variety of methods of achieving success. The whole wide variety of possible outcomes. As an example, the opportunity of flipping a coin and its being heads is ½ because there's 1 way of having a head and the total quantity of viable outcomes is 2 (a head or tail). We write P(heads) = ½.

The form opportunity is from old French probability (14 c.) and at once from Latin probabilities (nominative probabilitas) "credibility, possibility," from probabilistic (see likely). The mathematical sense of the time period is from 1718.

As there's an average of 1 misprint in line with the web page, the possibility of a minimum of 5 misprints is 1 - P(0, 1, 2, 3, or 4 misprints)

P(0) = e-1/0!

P(1) = e-1/1!

P(2) = e -1/2!

P(3) = e-1/3!

P(4) = e-1/4!  

P(0, 1, 2, 3, or 4 misprints) = e-1(1 + 1 + half of + 1/6 + 1/24) = 2 17/24 e-1 = .99634

Then, P(5 or more misprints) is 1 - .99634 = .00366

Then, the P(at the least 1 page in a 300-page ebook has at least 5 misprints) = 1 - P(0 pages have at least 5 misprints).

P(0 pages have at the least 5 misprints) = P(300 of 300 pages have 0, 1, 2, 3, or 4 misprints) =.99634 300

(you could view this as C(300, 300) p300 in case you desire) = 0.332881690573629

Then, P(1 or greater pages have at least 5 misprints) = 1 - 0.332881690573629 = 0.667118309426371

Learn more  about probability here https://brainly.com/question/24756209

#SPJ4

The amount of stone in each sidewalk is 3/16

The given parameters are

Stones = 3/4 tons

Side walks = 4

The amount of stone in each sidewalk is calculated using

Amount = Stones/Side walks

Substitute the known values in the above equation

Amount = 3/4 / 4

Evaluate the quotient

Amount = 3/16

Hence, the amount of stone in each sidewalk is 3/16

Read more about proportions at:

https://brainly.com/question/1781657

#SPJ1

Answer: 15

Step-by-step explanation:

First step is to find the slope of the line given

y = -x - 2

y = mx + c where m is the slope of the line so the slope of the line is -1

If 2 lines are perpendicular to one another, the product of the slopes is -1 so the slope of the perpendicular line is -1/-1 = 1

y = 1x + c or y = x + c

As the line passes through the coordinate (-5,10), we can substitute the x and y value of the coordinates into the equation

10 = -5 + c

c = 15

So the equation of the line is y = x + 15

The correct simplification of 8x2(5x 2x2 − 3) is [tex]16x^{4} +40x^{3} -24x^{2}[/tex]

The correct simplification of 8x2(5x 2x2 − 3).

[tex]8x^{2} (5x+2x^{2+1} -3)=8*2x^{2+2} -8*3x^{2}[/tex]

                                [tex]=40x^{3} +16x^{4} -24x^{2}[/tex]

Rearranging the above expression in descending order of power, we get:[tex]16x^{4} +40x^{3} -24x^{2}[/tex]

The correct simplification of 8x2(5x 2x2 − 3) is [tex]16x^{4} +40x^{3} -24x^{2}[/tex]

To learn more about simplification, refer to:

https://brainly.com/question/723406

#SPJ4

Answer:

y = - [tex]\frac{1}{3}[/tex] x + 5

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

calculate m using the slope formula

m = [tex]\frac{y_{2-y_{1} } }{x_{2-x_{1} } }[/tex]

with (x₁, y₁ ) = (0, 5) and (x₂, y₂ ) = (3, 4) ← 2 points on the line

m = [tex]\frac{4-5}{3-0}[/tex] = [tex]\frac{-1}{3}[/tex] = - [tex]\frac{1}{3}[/tex]

the line crosses the y- axis at (0, 5 ) ⇒ c = 5

y = - [tex]\frac{1}{3}[/tex] x + 5 ← equation of line

[tex]\huge\text{Hey there!}[/tex]

[tex]\large\text{Keep in mind that \boxed{of} means multiplication in mathematics and}\\\large\text{percentages run out of 100.}[/tex]

[tex]\mathsf{3\%\ of\ x = 10}[/tex]

[tex]\mathsf{\dfrac{3}{100}\ of\ x = 10}[/tex]

[tex]\mathsf{\dfrac{3}{100}\times x = 10}[/tex]

[tex]\mathsf{\dfrac{3}{100}x = 10}[/tex]

[tex]\textsf{Find the reciprocal of }\mathsf{\dfrac{3}{100}}\textsf{ and multiply that particular number on both sides.}[/tex]

[tex]\mathsf{\dfrac{3}{100}= \dfrac{100}{3}}[/tex]

[tex]\textsf{So, that means we have multiply by }\mathsf{\dfrac{100}{3}}\textsf{ to both of your sides.}[/tex]

[tex]\mathsf{\dfrac{100}{3}\times\dfrac{3}{100}x = 10\times \dfrac{100}{3}}[/tex]

[tex]\textsf{Cancel out: }\mathsf{\dfrac{100}{3}\times\dfrac{3}{100}}\textsf{ because it gives you 1}[/tex]

[tex]\textsf{Keep: }\mathsf{10\times\dfrac{100}{3}}\textsf{ because it gives you the value of x or simply understanding of}\\\textsf{the \boxed{\mathsf{x-value}}\ .}[/tex]

[tex]\mathsf{x = \dfrac{100}{3}\times10}[/tex]

[tex]\mathsf{x = \dfrac{100}{3}\times\dfrac{10}{1}}[/tex]

[tex]\mathsf{x = \dfrac{100\times10}{3\times1}}[/tex]

[tex]\mathsf{x = \dfrac{1,000}{3}}[/tex]

[tex]\mathsf{x\approx 333 \dfrac{1}{3}}[/tex]

[tex]\huge\textbf{Therefore, your answer should be:}[/tex]

[tex]\huge\boxed{\mathsf{x = }\frak{\dfrac{1,000}{3}}}\huge\checkmark[/tex]

[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]

[tex]r = \frac{15}{7} [/tex]

Answer:  [tex]h(\text{x}) = -4\text{x}^2 - 6\text{x} + 6[/tex]

Work Shown:

[tex]h(\text{x}) = f(\text{x}) - g(\text{x})\\\\h(\text{x}) = ( 8\text{x}^2 - 2\text{x} + 3) - ( 12\text{x}^2 + 4\text{x} - 3 )\\\\h(\text{x}) = 8\text{x}^2 - 2\text{x} + 3 - 12\text{x}^2 - 4\text{x} + 3 \\\\h(\text{x}) = (8\text{x}^2-12\text{x}^2) + (- 2\text{x} - 4\text{x}) + (3 + 3) \\\\h(\text{x}) = -4\text{x}^2 - 6\text{x} + 6 \\\\[/tex]

Answer:

8/9

Step-by-step explanation:

I'm pretty sure it's that

The answer is 8/9.

First, find the HCF of 24 and 27.

Hence, the HCF of the numbers is 3.

Then, divide both sides of the fraction by 3.

Answer:

81

Step-by-step explanation:

If a number is divisible by 11, then you can make those numbers by multiplying 11 by any number. Like 33 is divisible by 11 because 3×11 is 33.

Of course, 33 is too small, we are looking for 3-digit numbers...

see image.

The values of [tex]x[/tex] are [tex]1+\sqrt{89}[/tex] and [tex]1-\sqrt{89}[/tex].

To find the values of x:

Given equation: [tex](x+7)(x-9)=25[/tex]

Then: [tex]x(x-9)+7(x-9)=25[/tex]

Using the distributive property: [tex]a.(b+c)=a.b+a.c[/tex]

[tex]x^{2} -9x+7x-63=25[/tex]

Combine like terms:

[tex]x^{2} -2x-63=25[/tex]

Subtract 25 from both sides and obtain:

[tex]x^{2} -2x-88=0[/tex]

Using completing square form:

Add and subtract [tex](\frac{2}{2} )^{2} =1[/tex] we have:

[tex]x^{2} -2x-88+1-1=0\\(x-1)^{2} -89=0[/tex]

Add 89 to both sides we have:

[tex](x-1)^{2} =89[/tex]

Taking square roots on both sides, obtain:

[tex]x-1=[/tex] ± [tex]\sqrt{89}[/tex]

Add 1 to both sides we have:

[tex]x=1[/tex]±[tex]\sqrt{89}[/tex]

Therefore, the values of [tex]x[/tex] are [tex]1+\sqrt{89}[/tex] and [tex]1-\sqrt{89}[/tex].

Know more about square roots here:

https://brainly.com/question/428672

#SPJ4

The complete question is given below:

Use completing the square to solve (x + 7)(x – 9) = 25 for x.

The (f + g)(x) = 15x - 2 is represent the total round trip time.

According to the statement

we have given that the f(x) represents the time it takes a boat to travel upstream and g(x) represents the time it takes a boat to travel downstream.

And we have to find the total round trip time.

So, For this purpose, the given information is:

f(x) = 3x − 10 and g(x) = 12x + 8

And

The total round trip time is represented by the term (f + g)(x)

We know that the

(f + g)(x) = f(x) + g(x)

Substitute the given values in it

So, (f + g)(x) = f(x) + g(x)

(f + g)(x) = (3x - 10) + (12x + 8)

(f + g)(x) = 15x - 2

Hence, The (f + g)(x) = 15x - 2 is represent the total round trip time.

Learn more about Upstream and downstream here

https://brainly.com/question/2108925

#SPJ1

Considering the definition of an inequality, the number of people who can go to the amusement park is 3.

An inequality is the existing inequality between two algebraic expressions, connected through the signs:

An inequality contains one or more unknown values ​​called unknowns, in addition to certain known data.

Solving an inequality consists of finding all the values ​​of the unknown for which the inequality relation holds.

In this case, you know that:

Being "p" the number of people who can go to the amusement park, the inequality that expresses the previous relationship is

$9.75 + $16.25×p ≤$65

Solving:

$16.25×p ≤$65 - $9.75

$16.25×p ≤$55.25

p ≤$55.25÷ $16.25

p≤ 3.4

Finally, the number of people who can go to the amusement park is 3.

Learn more about inequality:

brainly.com/question/17578702

brainly.com/question/25275758

brainly.com/question/14361489

brainly.com/question/1462764

#SPJ1