Drag the tiles to the correct boxes to complete the pairs. Not all tiles will be used.

Match the properties to the regular polygons they describe.

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 height of the rectangular pyramid is 7 ft.

First, we know that the volume of a rectangular pyramid is given by:

[tex]V = B*H/3[/tex]

Where B is the base, H is the height.

First, we know that the volume is 56 ft³, and the base is 24ft², then we can replace these two in the volume equation to get:

[tex]56 ft^3 = 24ft^2*H/3[/tex]

Solving that for H, we get:

[tex]H = 3*(56ft^3/24ft^2) = 7ft[/tex]

The height of the rectangular pyramid is 7 ft.

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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]

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The required length of the hypotenuse of the triangle is √19.

To find the required length of the hypotenuse of the triangle:

Legs in the right triangle refer to the perpendicular and base.

Hypotenuse:

[tex]= \sqrt{perpendicular^{2}+base^{2} } \\= \sqrt{3^{2}+\sqrt{10^{2} } } \\=\sqrt{19}[/tex]

Therefore, the required length of the hypotenuse of the triangle is √19.

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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:

  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:

m<DCF = 155°

Step-by-step explanation:

m<DCF = m<D + m<E

m<DCF = 72° + 83°

m<DCF = 155°

The statement is false as the sample is a few ATMs of the total ATMs in US.

The sample is only 860 ATMs and not total ATMs in US thus, a sample of 860 ATMs have taken and conducted the survey.

Thus, the statement is false as the sample is a few ATMs of the total ATMs in US.

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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)

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6) The solution of the system of linear equations is (x, y, z) = (2, 5, 3).

11) The average depth of the water is 0.35 kilometers.

12) The dollar value of the deposit after 10 years is $ 1205.85.

13) Three horses need a consumption of 930 pounds of hay for the month of July.

14) The circle with equation x² + y² + 8 · y + 8 · y + 28 = 0 has a radius of 2.

15) Anna has a mean time of 7.39 minutes.

In this question we must analyze and resolve on algebraic equations such as linear equations or conic sections. 6) We must use algebra properties to solve on the system of linear equations. First, we clear x in the first equation:

x = y - z

Then, we apply it in the two remaining equations:

- 5 · (y - z) + 3 · y - 2 · z = - 1

2 · (y - z) - y + 4 · z = 11

- 2 · y + 3 · z = - 1

y + 2 · z = 11

Second, we clear y in the second expression and we substitute into the first expression:

y = 11 - 2 · z

- 2 · (11 - 2 · z) + 3 · z = -1

- 22 + 7 · z = - 1

z = 3

y = 5

x = 2

The solution of the system of linear equations is (x, y, z) = (2, 5, 3).

11) There is a function of the speed of a tsunami in terms of the average depth of the water. The former variable is known and we need to clear d in the formula to find the missing value:

145 = 356 · √d

d = (145 / 356)²

d = 0.345 km

The average depth of the water is 0.35 kilometers.

12) The compound interest model is shown below:

C' = C · (1 + r / 100)ˣ      (1)

Where:

If we know that C = 500, r = 4.5 and x = 20, then the resulting capital after 10 years is:

C' = 500 · (1 + 4.5 / 100)²⁰

C' = 1205.85

The dollar value of the deposit after 10 years is $ 1205.85.

13) The situations indicates a direct variation as the amount of hay is directly proportional to the number of days. Hence, we have the following linear model for the hay consumption of one horse:

y = m · x        (2)

y = 10 · x

Where:

The total consumption of three horses during July is equal to the product of the number of horses and consumed hay by one horse during one month:

y' = 3 · [10 · (31)]

y' = 930

Three horses need a consumption of 930 pounds of hay for the month of July.

14) There is the general equation of the circumference and we must transform it into its vertex form to find information about the radius. This can done by algebraic procedures:

x² + y² + 8 · y + 8 · y + 28 = 0

(x² + 8 · y) + (y² + 8 · y) = - 28

(x² + 8 · y + 16) + (y² + 8 · y + 16) = 4

(x + 4)² + (y + 4)² = 2²

The circle with equation x² + y² + 8 · y + 8 · y + 28 = 0 has a radius of 2.

15) We need to sum all the times described in the table and divide it by the number of data to find the mean time for a 1-kilometer race:

x = (7.25 + 7.40 + 7.20 + 7.10 + 8.00 + 8.10 + 6.75 + 7.35 + 7.25 + 7.45) / 10

x = 7.39

Anna has a mean time of 7.39 minutes.

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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:

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

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.

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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]

[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: 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

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.

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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]

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Using a calculator, the critical value for the t-distribution with a confidence level of 99% and 49 df is of Tc = 2.4049.

It is found using a calculator, with two inputs, which are given by:

In this problem, the inputs are given as follows:

Hence, using a calculator, the critical value for the t-distribution with a confidence level of 99% and 49 df is of Tc = 2.4049.

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The factors illustrate they the sides of the rectangles are:

The first equation given is x² - 3x + 2.

= x² - 3x + 2.

= x² - x - 2x + 2

= x(x - 1) - 2(x - 1)

= (x - 1)(x - 2)

Therefore, the side lengths are (x - 1) and (x - 2).

In the rectangular figure, the length and width should be the expressions above.

The second equation given is 2x² + x - 6.

= 2x² + x - 6.

= 2x² + 4x - 3x - 6

= 2x(x + 2) - 3(x + 2)

= (2x - 3)(x + 2)

Therefore, the side lengths are (2x - 3) and (x + 2).

In the rectangular figure, the length and width should be the expressions above.

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

Answer:

23

Step-by-step explanation:

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.

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.

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Answer:q = 6+43r/9

Step-by-step explanation:

Assuming R=9q-43r-6 is what you meant
R+6 = 9q - 43r
R + 6 + 43r = 9q
q = 6+43r/9

The area under the curve y = x² + 4 on the interval [-3, 2] exists [tex]$\frac{95}{3}[/tex].

The area under a curve between two points exists seen by accomplishing a definite integral between the two points. To estimate the area under the curve y = f(x) between x = a and x = b, integrate y = f(x) between the limits of a and b. This area can be estimated by utilizing integration with given limits.

The equation that represents the curve exists

y = x² + 4

To estimate the area under the curve in the interval of [-3, 2] will be

[tex]$&\text { area }=\int_{-3}^{2} x^{2}+4 d x \\[/tex]

[tex]$&=\left[\frac{x^{3}}{3}+4 x\right]_{-3}^{2} \\[/tex]

[tex]$&=\frac{1}{3}\left[x^{3}\right]_{-3}^{2}+4[x]_{-}^{2} _{3} \\[/tex]

simplifying the above equation, we get

[tex]$&=\frac{1}{3}\left[(2)^{3}(-3)^{3}\right]+4[(2)-(-3)] \\[/tex]

[tex]$&=\frac{1}{3}[8+27]+4[2+3] \\[/tex]

[tex]$&=\frac{1}{3}(35)+20 \\[/tex]

[tex]$&=\left(\frac{95}{3}\right)[/tex]

Therefore, the area under the curve y = x² + 4 on the interval [-3, 2] exists [tex]$\frac{95}{3}[/tex].

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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

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Answer:

C = 10π ft

Step-by-step explanation:

the circumference (C) of a circle is calculated as

C = 2πr ( r is the radius )

to find r use the area formula, that is

A = πr² = 25π ( divide both sides by π )

r² = 25 ( take square root of both sides )

r = [tex]\sqrt{25}[/tex] = 5

then

C = 2π × 5 = 10π ft

120=!3÷!6

!2-!5=!3

to 120 modes