Practice
Example 1
PROOF Write each proof.
1. Given: AB || CD, ∠CBD ≅ ∠ADB
Prove: ΔABD ≅ ΔCDB

The number of sheep originally on the farm will be equal to 200.

The Latin phrase "per centum," which means "by the hundred," is where the English word "percentage" comes from.

Percentage segments are those with a numerator of 100. In other words, it is a connection where the whole is always deemed to be valued 100.

As per the given information in the question,

Let the number of animals originally at the farm is x.

60% of the animals are cows.

So, 0.6x are cows.

If there are 60% cows, then there are 40% of sheep.

So, 0.4x are sheep.

The quantity of sheep multiplies when many cows and sheep were introduced to the property.

As a result, the farm now has the same amount of sheep that it did before.

Number of sheep added = 0.4x

There are total 260 animals added to the farm.

So,

Number of sheep added = 260-0.4x

Total number of animals = x + 260

The percentage of cows increased to 20%.

Initial number of cows at farm + cows added = Initial number of cows at farm + 20% of the total cows initially at the farm.

0.6x + (260-0.4x) = 0.6x + 0.2(0.6x)

0.6x+260-0.4x = 0.6x+0.12x

0.2x+260 = 0.6x+0.12x

0.2x+260 = 0.72x

260 = 0.52x

x = 260/0.52

x = 500

Thus, the farm's initial animal population was 500.

0.6x = the amount of cows that were first on the farm= 0.6(500) = 300 cows

0.4x = the amount of sheep that were initially on the farm = 0.4(500) = 200 sheep

0.4x = Sheep added = 200

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Answer:

y = x - 15

Step-by-step explanation:

y-2 = 1(x-17)

y-2 = x - 17

y = x -17 + 2

y = x - 15

The ordered pair that is NOT a point on the graph of f(x) = (1/2)*x - 7 is  (2,8) , the correct option is (b)  .

In the question ,

it is given that ,

the function is given as ;

f(x) = (1/2)*x - 7 ,

Option(a) ,

the ordered pair is (1 , [tex]-6\frac{1}{2}[/tex] )

To satisfy the function f(x)  , on putting x = 1 , we should get y =  [tex]-6\frac{1}{2}[/tex] .

f(1) =  (1/2) - 7 = (1 - 14)/2 = -13/2 .

So , this point lies on graph of f(x) .

Option(b) ,

the ordered pair is (2,8) ,

f(2) = (1/2)*2 - 7 = 1 - 7 = -6 ≠ 8 .

So , this point does not lie on graph of f(x) .

Option(c) ,

the ordered pair is (-2,-8) ,

f(-2) = (1/2)*(-2) - 7 = -1 - 7 = -8 = -8 .

So , this point lies on graph of f(x) .

Option(d) ,

the ordered pair is (0,-7) ,

f(2) = (1/2)*0 - 7 = - 7 = -7 .

So , this point lies on graph of f(x) .

Therefore , the ordered pair (2,8) is not on the graph .

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The rule that describes the translation will be as  T<-3, 2>

Given,

Translation;-

Translation is the act of moving a shape or a figure from one location to another. A figure can move in translation up, down, right, left, or anywhere else in the coordinate system. Only the object's position changes during translation; its size stays the same.

Here,

In order to move from A = (-3,3) to A' = (-6, 5), we must first move 3 units to the left and 2 units up.

Step 1's change causes x to become x-3.

Step 2's change causes y to become y+2.

When you combine these, (x, y) becomes (x-3, y+2).

That is equivalent to writing T<-3, 2>.

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E(max(X, Y )) = PxPy max(x, y)p(x, y)

= max(0, 0)p(0, 0)+max(0, 5)p(0, 5)+· · ·+max(10, 10)p(10, 10)+max(10, 15)p(10, 15)

= 0 ∗ 0.02 + 5 ∗ 0.06 + · · · + 10 ∗ 0.14 + 15 ∗ 0.01 = 9.6.

fX(x) = R 10(2x + y − 2xy)dy = x +12, 0 ≤ x ≤ 1.

fY (y) = R 10

(2x + y − 2xy)dx = 1, 0 ≤ y ≤ 1.

Since f(x, y) 6= fX(x)fY (y), X and Y are not independent.

Column=A column is a recurring piece or article in a newspaper, magazine or other publication, where a writer expresses their own opinion in few columns allotted to them by the newspaper organisation. Columns are written by columnists.

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The statement given is true. When approximating the area below a curve, using midpoint rectangles always produces the best approximation.

A good way to estimate areas with rectangles is to make each rectangle cross the curve at the midpoint of that rectangle's top side. Midpoints always the best approximate for the area under the curve using a finite number of rectangles.

The midpoint rule, also known as the rectangle method or mid-ordinate rule, is used to estimate the area under a simple curve. There are other methods to estimate the area, such as the left rectangle or right rectangle sum, but the midpoint rule gives the better approximate compared to the two methods.

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The teeth on the key are busted and now they must be readjusted and the code is 12y×2x^2.

Coding in Math is a series of the independent, standalone modules to that is  use coding to reinforcement and extended by to the  students' understandings of to the  math.! As students learning major programming is  concepts, they will development math-related to the  projects that is  demonstrate their proficiency by math and to the computer science.

18xy^2z^2=2xy^2 z×9xz

20xy^2z= 2xy^2z×10x^2

11xy^z=xyz^2×11

180y=9y×2x

24xy =12y×2x^2

(Take the greatest common factor)

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Answer

P(A∩B) = 0.021

Step-by-step explanation:

For independent events P(A∩B) is the same as P(A)*P(B) so for this 0.42*0.05 = 0.021

Answer:

24. y= -1/3x+1

25. y= -2/-1x+3

40% percentage of your portfolio in the zero-coupon bond with a maturity of 5 years and 60% of your portfolio in the perpetuity. This would result in an overall duration of 10 years.

To achieve your target duration of 10 years, you would need to allocate your investments in a way that has an overall duration of 10 years. You could do this by investing 40% of your portfolio in the zero-coupon bond with a maturity of 5 years and 60% percentage of your portfolio in the perpetuity. This would result in an overall duration of 10 years.

Duration of zero-coupon bond = 5 years

Duration of perpetuity = infinity

Weighted duration of portfolio = (0.4 x 5) + (0.6 x infinity) = 10 years.

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Sell Order 5: Price: $20, Quantity: 400

Since this order has the same price as Sell Order 2, it will be executed fifth.

1: Sell Order 3: Price: $30, Quantity: 300

2: Sell Order 1: Price: $25, Quantity: 500

3: Sell Order 4: Price: $25, Quantity: 100

4: Sell Order 2: Price: $20, Quantity: 200

5: Sell Order 5: Price: $20, Quantity: 400

1: Sell Order 3: Price: $30, Quantity: 300

Since this order has the highest price, it will be executed first.

2: Sell Order 1: Price: $25, Quantity: 500

Since this order has the second highest price, it will be executed second.

3: Sell Order 4: Price: $25, Quantity: 100

Since this order has the same price as Sell Order 1, it will be executed third.

4: Sell Order 2: Price: $20, Quantity: 200

Since this order has the third highest price, it will be executed fourth.

5: Sell Order 5: Price: $20, Quantity: 400

Since this order has the same price as Sell Order 2, it will be executed fifth.

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the orthogonal projection of [-2 2] onto the line through [-1 5] and the origin is [-1.25 3.75] using unit vector.

First, we need to calculate the unit vector of the line. This can be done by subtracting the two given points and dividing the result by the magnitude of the vector:

[-1 5] - [0 0]

= [-1 5]

|[-1 5]| = sqrt(1^2 + 5^2) = sqrt(26) = 5.1

Unit vector = [-1/5.1 5/5.1] = [-0.196 0.980]

Next, we need to calculate the dot product of the vector [-2 2] and the unit vector [-0.196 0.980]. This will give us the scalar projection.

(-2)(-0.196) + (2)(0.980) = -0.392 + 1.960 = 1.568

Finally, we can calculate the orthogonal projection by multiplying the scalar projection by the unit vector.

[-0.196 0.980] (1.568) = [-0.301 1.527]

Rounding to two decimal places, the orthogonal projection of [-2 2] onto the line through [-1 5] and the origin is [-1.25 3.75].

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The two column proof below has shown us that the LP ≅ LQ by definition of segment congruence.

We are told that Point M is the midpoint of PQ, and LM is the perpendicular bisector of PQ. Thus, the two-column proof to show that LP = LQ is as follows;

Statement 1; PM ≅ QM, LM ⊥ PQ

Reason 1; Given

Statement 2; LM ≅ LM

Reason 2; Reflexive Property of Congruence

Statement 3; ∠PML ≅ ∠QML

Reason 3; Right angle congruence theorem

Statement 4; ΔPML ≅ ΔQML

Reason 4; SAS congruence theorem

Statement 5; ∠PML ≅ ∠QML

Reason 5; Corresponding parts of congruent triangles are congruent

Statement 6; LP ≅ LQ

Reason 6; Definition of segment congruence

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Answer:

C)  Domain:  all real numbers x except x = ±2

E)  f(x) → ∞ as x → -2⁻ and as x → 2⁺, f(x) → -∞ as x → -2⁺ and as x → 2⁻

Step-by-step explanation:

Given function:

[tex]f(x)=\dfrac{3x^2}{x^2-4}[/tex]

The domain of a function is the set of all possible input values (x-values).

A rational function is undefined when the denominator is equal to zero.

The denominator of the given function is zero when:

[tex]\implies x^2-4=0[/tex]

[tex]\implies x^2=4[/tex]

[tex]\implies \sqrt{x^2}=\sqrt{4}[/tex]

[tex]\implies x= \pm 2[/tex]

Therefore the domain of the function is:

The excluded x-values are x = -2 and x = 2.

To find the behaviour of the function near the excluded x-values, input values of x that are very near either side of excluded values:

[tex]x \rightarrow -2^-: \quad f(-2.001)=\dfrac{3(-2.001)^2}{(-2.001)^2-4}=3002.250...[/tex]

[tex]x \rightarrow -2^+: \quad f(-1.999)=\dfrac{3(-1.999)^2}{(-1.999)^2-4}=-2997,750...[/tex]

[tex]x \rightarrow 2^-: \quad f(1.999)=\dfrac{3(1.999)^2}{(1.999)^2-4}=-2997.750...[/tex]

[tex]x \rightarrow -2^+: \quad f(2.001)=\dfrac{3(2.001)^2}{(2.001)^2-4}=3002.250...[/tex]

Therefore, the behaviour of the function near the excluded x-values:

Answer:

Below

Step-by-step explanation:

I'll do the first one....you can then do the second one

-4x-2y = -12

4x+8y = -24       add these two equations together ( this will eliminate 'x')

    6y = -36

       y = - 6      use this value in either equation to compute x = 6

HINT: On the next one....multiply either equation by -1    then add together

The sum of the first 15 terms of the geometric sequences will be 76,456.33.

The aggregate to boundless GP implies, the number of terms in a limitless GP.

Let a₁ be the first term, n be the total number term, and r be a common ratio.

Then the sum of the geometric sequence will be

Sₙ = [a₁ (1 - rⁿ)] / (1 - r)

The geometric sequence is given below.

7, -14, 28, -56, 112,...

The first term of the geometric sequence is 7 and the common ratio of the geometric sequence is given as,

r = - 14 / 7

r = - 2

Then the sum of the first 15 terms of the sequences will be given as,

S₁₅ = [7 (1 - (-2)¹⁵)] / (1 - (-2))

S₁₅ = [7(1 + 32,768)] / 3

S₁₅ = -229,369 / 3

S₁₅ = 76,456.33

The sum of the first 15 terms of the geometric sequences will be 76,456.33.

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A completely randomized design is an appropriate design to use in this situation because There is no blocking variable, and incentive plans will be randomly assigned to the workers.

A completely randomized design (CRD) is the simplest design for comparative experiments because it employs only two basic experimental design principles: randomization and replication.

By randomization, we mean that the experimental units' run sequence is determined at random.

Treatments are assigned to experimental units or plots in a completely random manner in CRDs. CRD can be used for either single-factor or multifactor experiments.

According to the question,

The quality control manager of a large factory wants to find out the number of defective items . For that 10 workers are selected Randomly and these worker will work on item according 3 incentive plans Means Replication is 3.

Hence , A completely randomized design is an appropriate design to use in this situation because There is no blocking variable, and incentive plans will be randomly assigned to the workers.

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The efficiency for a steel specimen immersed in a phosphating tank is the weight of the phosphate.

a) The equation of the estimated regression line is ŷ = - 16.592 + 0.1017x.

b) A point estimate is 1.9174 for true average efficiency ratio when tank temperature is 182.

c) The values of the residuals from the least squares line for the four observations for which temperature is 182 are -1.0774, -0.1874, 0.1226, 0.7826..

We have given the accompanying data on tank temperature (x) and efficiency ratio (y).

a) We have to find out the equation of the estimated regression line. Consider the calculations shown in the above table:

ΣΧ= 4308 ; ΣΥ= 39.99 ; ΣΧΥ = 7229.47 ;

ΣX² = 773790

Determine the slope and intercepts using the following formulas,

Slope b₁ = (n ΣΧΥ - ΣX ΣΥ)/(nΣΧ² - (ΣX)²)

=> b₁ = [24 (7229.47) - (4308) (39.99)]/(24(773790)-(4308)²)

=> b₁ = 1230.36/12096 = 0.10171626984

=> b₁ ≈ 0.1017

Intercept, b₀ = ΣY /n - b₁ΣΧ /n

=> b₀ = 39.99/24 - 0.1017×4308/24

= -16.592

Thus, the estimated regression equation of the form, ŷ = b₀ + b₁ x

ŷ = - 16.592 + 0.1017x

b) Calculate the point estimate for true average efficiency ratio when tank temperature is 182.

y = -16.592 + 0.1017x

=> y= -16.592 +0.1017 (182)

=> y= 1.9174

Therefore, the point estimate for true average efficiency ratio when tank temperature is 182 is 1.9174..

c)We have to find out the values of the residuals from the least squares line for the four observations for which temperature is 182.

From part (b), the value of ŷ is 1.9174.

The table below shows the residuals.

Vi ŷ e = y₁ - y

0.84 1.9174 -1.0774

1.73 1.9174 -0.1874

2.04 1.9174 0.1226

2.70 1.9174. 0.7826

Hence, we got all the required values.

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Answer:

Step-by-step explanation:

9z + 27 = 45

9z= 18

z= 18/9

z = 2

Answer:

[tex]z=2[/tex]

Step-by-step explanation:

[tex]9(z+3)=45\\9z+27=45\\9z=45-27\\9z=18\\z=\frac{18}{9}\\ z=2[/tex]

Hope it helps you

By SSS rule of congruence

∠QPR ≅ ∠SPR

"Information available from the question"

In the question:

Use the given information to prove that ∠QPR ≅ ∠SPR.

Given: QR ≅ SR

PQ ≅ PS

Prove: ∠QPR ≅ ∠SPR

Now, According to the question:

In triangle PQR and triangle PRS

We can see that

QR ≅ SR (given)

PQ ≅ PS (given)

PR = PR (Common line)

By SSS rule of congruence

When all the sides of two triangles are congruent, the angles of those triangles must also be congruent. This method is called side-side-side, or SSS for short.

Hence, ∠QPR ≅ ∠SPR

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Answer:

The graph of f(x) = x^2 is shown.

Use the parabola tool to graph g(x).

9(x)= (1 + x)^2 – 2

graph the parabola by first plotting its vertex and then plotting a second point on the parabola

Answer:

Step-by-step explanation:

Given the points

   (4, 2)

   (3, 5)

We know that the slope-intercept form of the line equation is

where m is the slope and b is the y-intercept

substituting m=-3 and the point (4, 2) to get the y-intercept i.e. b

2=(-3)4 + b

2 = -12 + b

b = 2+12

b = 14

Now, substituting b=14 and m=-3 in the slope-intercept form to determine the equation of a line in the slope-intercept.

y=(-3)x+(14)

y=-3x+14

Thus, the equation of the line containing (4,2) and (3,5)  in the slope-intercept form will be:

   y=-3x+14

Answer:

based on the data i can draw a conclusion that child two and child one both grow taller as their time (months) increases. child 2 grows at a faster rate than child 1.

Step-by-step explanation:

MGFs means moment generating function. Every distribution function has a unique mgf

Make X a random number. We say that X has a moment generating function, and the function

Mₓ(t) = E(e^tx)   is referred to as the moment generating function of X

However, not all random variables have a moment generating function. However, every random variable has a unique function—an additional transform with properties comparable to those of the mgf.

The second creating capability (mgf) is a capability frequently used to describe the circulation of an irregular variable.

The moment-generating function is extremely useful for the following reasons:

1. Moments can be easily calculated with it; At zero, its derivatives are equivalent to the random variable's moments;

2. The mgf of a probability distribution is what makes it unique.

The question is incomplete so, I've answered in general

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Answer:

Step-by-step explanation:

The set {x | x < 0 or x > 3 } can be written in interval notation as (-∞, 0) ∪ (3, ∞).

The set {x | x < 0 or x > 3 } consists of all real numbers that are either less than 0 or greater than 3. This set can be represented as the union of the set of all numbers less than 0, which can be written as (-∞, 0), and the set of all numbers greater than 3, which can be written as (3, ∞).

Therefore, the set {x | x < 0 or x > 3 } can be written in interval notation as (-∞, 0) ∪ (3, ∞).

The age of the spear head to the nearest year is 8 years.

A percentage is a value or ratio that may be stated as a fraction of 100. If we need to calculate a percentage of a number, we should divide it's entirety and then multiply it by 100.

Since the archaelogist in Turkey discovers a spear head that contains 55% of its original amount of C-14. The amount will be:

= 55% × 14

= 0.55 × 14

= 7.7

The age of the spear will be 8 years.

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Complete question

An archaelogist in Turkey discovers a spear head that contains 55% of its original amount of C-14. Find the age of the spear head to the nearest year.

The conclusion is that there is no sufficient evidence at the 0.02 level of significance that the new technique lengthens the training time.

Given,

The mean time taken to receive advanced driving license, through traditional methods [tex]\mu = 90[/tex] hours

The researcher believes the new technique may lengthen the training time

Researcher conducts hypothesis test of  n = 190 students

The researcher fails to reject the null hypothesis at significance level, [tex]\alpha = 0.02[/tex]

Test hypothesis :-

[tex]H_0[/tex] :The mean time it takes for the new technique to receive an advanced driving license is 90 hours, i.e. [tex]\mu = 90[/tex].

[tex]H_a[/tex]: The mean time it takes for the new technique to receive an advanced driving license is greater than 90 hours, i.e. [tex]\mu > 90[/tex].

Failure to reject the null hypothesis means that our sample did not provide enough evidence to conclude that the effect exists. However, the lack of evidence does not prove that the effect does not exist.

As a result, the average time for the new technique to obtain an advanced driving licence is less than 90 hours.

Thus,, the researcher's belief that the new technique would lengthen training time was incorrect.

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