Consider the matrices A = 2 1 2 2 2 −2 3 1 1 and x = x1 x2 x3
(a) Show that the equation Ax = x can be rewritten as (A − I)x = 0 and use this result to solve Ax = x for x.
(b) Solve Ax = 4

Answer:

The number of ways Tony and Maria can both be selected for the committee is 8 ways.

Step-by-step explanation:

i) Tony and Maria have to be on the committee together or not at all.

ii) Let us consider Tony and Maria as combined as one person.

 Therefore now we can say that the number of eligible people for the committee = 9 - 1 = 8.

iii) therefore the number of ways that both Tony and Maria can be selected for the committee are

 = 8C1   =  [tex]\hspace{0.2cm}\binom{8}{1} = \frac{8!}{1! (8-1)!} = \frac{8!}{1!\times 7!} = \frac{8}{1} = \hspace{0.1cm}8 \hspace{0.1cm}ways[/tex]

The number of ways to select exactly one of Tony and Maria is 70.

It is given that,

Explanation:

Excluding Tony and Maria from the 9 people. The number of remaining people is 7.

Exactly one of Tony and Maria be selected for the committee. So, one person is selected from 2 and 3 people are selected from the remaining 7 people.

[tex]\text{Number of ways}=^2C_1\times ^7C_3[/tex]

[tex]\text{Number of ways}=\dfrac{2!}{1!(2-1)!}\times \dfrac{7!}{3!(7-3)!}[/tex]

[tex]\text{Number of ways}=2\times 35[/tex]

[tex]\text{Number of ways}=70[/tex]

Thus, the number of ways to select exactly one of Tony and Maria is 70.

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Answer:

The value of the house after adding the garage is $135,700.

Step-by-step explanation:

Given,

value of house before adding garage = $118,200.00

we need to find the value of house after adding two car garage.

Solution,

Since after adding two car garage the value of the house increased by 15%.

So firstly we will find out the 15% of the value of the house after adding garage.

So we can say that;

15% of the value of the house after adding garage is equal to 15 divided by 100 the multiplied with the value of the house before adding garage.

15% of the value of the house after adding garage = [tex]\frac{15}{100}\times118,000=\$17,700[/tex]

Now, The value of the house after adding garage is equal to the sum of value of house before adding garage and 15% of value of house before adding garage.

We can frame it in equation form as;

The value of the house after adding garage = [tex]\$118,000+\$17,000=\$135,700[/tex]

Hence The value of the house after adding the garage is $135,700.

Answer:

12

Step-by-step explanation:

Given: Diagonal of square= [tex]3\sqrt{2}[/tex]

To find the perimeter of square, we need to find the length of sides of square.

∴ Using the formula of diagonal to find side of square.

Formula; [tex]Diagonal= s\sqrt{2}[/tex]

Where, s is side of square.

⇒ [tex]3\sqrt{2} = s\sqrt{2}[/tex]

Dividing both side by √2

⇒[tex]s= \frac{3\sqrt{2} }{\sqrt{2} }[/tex]

∴[tex]s= 3[/tex]

Hence, Length of side of square is 3.

Now, finding the perimeter of square.

Formula; [tex]Perimeter= 4s[/tex]

⇒[tex]Perimeter= 4\times 3[/tex]

∴ [tex]Perimeter= 12[/tex]

Hence, Perimeter of square is 12.

The perimeter of a square with a diagonal of 3 square root 2 units is 12 units.

To find the perimeter of a square with a diagonal of 3√2, we can use the fact that the diagonal of a square divides it into two congruent right triangles. The length of each leg of the triangle is equal to one side of the square. We can use the Pythagorean theorem to find the length of the side of the square. Let's assume that the length of the side of the square is s.

Using the Pythagorean theorem, we have s2 + s2 = (3√2)2. Simplifying, we get 2s2 = 18. Dividing by 2, we have s2 = 9. Taking the square root of both sides, we get s = 3.

The perimeter of the square is equal to 4 times the length of one side, so the perimeter is 4 * s = 4 * 3 = 12 units.

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Answer: Charlotte finished the word search in 16 minutes.

Step-by-step explanation:

Frank finished the word search in 32 minutes.

Terrence finished a word search in 3/4 the time it took Frank. This means that the time it took Terrence to finish the word search would be

3/4 × 32 = 24 minutes.

Charlotte finished the word search in 2/3 the time it took Terrence. This means that the time it took Charlotte to finish the word search would be

2/3 × 24 = 16 minutes

Answer:

Vanilla milkshake = 1/3 and Chocolate milkshake = 2/3

Step-by-step explanation:

Given data:

Total ounce of milkshake = 12 ounce

Vanilla milkshake = 4

Chocolate is the rest which can be interpreted as 12 - 4= 8 ounce

Representing as fractions

Vanilla milkshake = 4/12 (Reducing to lowest terms that is diving numerator and denominator by common factor in this case 4)

Vanilla milkshake = 1/3

Chocolate milkshake = 8/12 (Reducing to lowest terms that is diving numerator and denominator by common factor in this case 4)

Chocolate milkshake = 2/3

Answer:

Step-by-step explanation:

On pet day 18 children brought a pet to school.

2/3 of the pets were dogs. This means that the number of pets that were dogs would be

2/3 × 18 = 12 dogs

1/9 of the pets were cats. This means that the number of pets that were cats would be

1/9 × 18 = 2 dogs

Total number of dogs and cats is

12 + 2 = 14

The number of animals that were neither dogs nor cats is

18 - 14 = 4

Final answer:

There were 12 dogs, 2 cats, and 4 animals that were neither dogs nor cats among the 18 pets that children brought to school on pet day.

Explanation:

On pet day, 18 children brought a pet to school. To find out how many dogs were there, we calculate two-thirds of 18, and for the cats, we calculate one-ninth of 18.

Dogs: ⅓ × 18 = 12 dogs

Cats: ⅙ × 18 = 2 cats

To determine how many animals were neither dogs nor cats, we subtract the number of dogs and cats from the total number of pets: 18 - (12 + 2) = 4 animals neither dogs nor cats.

So, there were 12 dogs, 2 cats, and 4 animals that were neither dogs nor cats.

Answer:

The answer to your question is  [tex]\frac{3b}{a}[/tex] , check your options, maybe you forgot one.

Step-by-step explanation:

Original operation

                                [tex]\frac{6a^{2}}{5b^{2}} \frac{45b^{3}}{18a^{3}}[/tex]

Process

1.- Simplify 6 and 18 and 45 and 5

   [tex]\frac{6}{18} = \frac{3}{9} = \frac{1}{3}[/tex]

  [tex]\frac{45}{5} = \frac{9}{1} = 9[/tex]

 Result

  [tex]\frac{9}{3} = \frac{3}{1} = 3[/tex]

2.- Simplify a² and a³

    [tex]\frac{a^{2}}{a^{3}} = \frac{1}{a}[/tex]

3.- Simplify b³ and b²

     [tex]\frac{b^{3}}{b^{2}} = b[/tex]

4.- Join the results [tex]\frac{3b}{a}[/tex]

Answer:

[tex]m\angle CED= 64\°[/tex]  

[tex]m\angle ACD=124\°[/tex]  

Step-by-step explanation:

In the figure given:

∠ABC = 93°

∠BAC = 31°

∠CDE = 60°

To find ∠CED and ∠ACD.

Solution:

In triangle ABC, we are given two vertex angles. We can find the third angle as angle sum of triangle = 180°.

∠ABC = 93° , ∠BAC = 31°

∠BCA=  [tex]180\°-(93\°+31\°)[/tex]

∠BCA = 56°

[tex]m\angle BCA+m\angle ACD=180\°[/tex]    [Supplementary angles forming a linear pair]

[tex]m\angle ACD=180\°-56\°[/tex]

[tex]m\angle ACD=124\°[/tex]   (Answer)

In triangle CDE:

[tex]m\angle CDE+m\angle CED = m\angle ACD[/tex]   [Exterior angle theorem :Exterior angle of a triangle is equal to sum of opposite interior angles ]

[tex]60\°+m\angle CED = 124\°[/tex]

[tex]m\angle CED= 124\°-60\°[/tex]

[tex]m\angle CED= 64\°[/tex]      (Answer)

Answer:

m\angle CED= 64\°  

m\angle ACD=124\°  

Step-by-step explanation:

get an A!

The area of the purple band is [tex]A = 4 - x^2[/tex]

Since both the shape contains the square so here the area square should be applied,

A  = Total purple area - Total green area

[tex]A = (2)^2 - (x)^2\\\\A = 4 - x^2[/tex]

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Final answer:

The area of the purple band around the green square is calculated by finding the difference between the area of the larger purple square and the green square. It is determined by subtracting the green square's area (x²) from the total area of the larger square ((x + 4)²), resulting in an area of 8x + 16 square inches for the purple band.

Explanation:

To calculate the area of the purple band on the rug, we must first determine the dimensions of both the inner green square and the larger purple square that includes the band. The side length of the green square is given as x inches, and we know the width of the purple band surrounding it is 2 inches. To find the side length of the larger purple square, we add twice the width of the purple band (2 inches on each side) to the side length of the green square, giving us x + 2 + 2 or x + 4 inches.

The area of the larger purple square is therefore (x + 4)² square inches. The area of the inner green square is x² square inches. To find the area of just the purple band, we subtract the area of the green square from the area of the larger purple square.

So, the area of the purple band is (x + 4)² - x² square inches. Expanding this expression, we get x² + 8x + 16 - x² which simplifies to 8x + 16 square inches. Therefore, the area of the purple band around the green square is 8x + 16 in²

To Determine:

So, here is the complete proof of the Pythagorean theorem.

Given: Δ ABC is a right triangle, with

a right angle at ∠C

Prove: A²+B² =C²

Answer:

Note: All the answers for the questions marks are filled with in bold text.

Statement

1. ΔABC is a right triangle, with a right angle at ∠C

2. Draw an altitude from point C to line AB

3. ∠CDB and ∠CDA are right angles.

4. ∠BCA ≅ ∠BDC

5. ∠B ≅ ∠B

6. AA Similarity Postulate          

7.   [tex]\frac{a}{x}\:=\:\frac{c}{a}[/tex]

8. a² = cx

9. ∠CDA ≅ ∠BCA

10. ∠A ≅ ∠A

11. ΔCBA ~ ΔDBA      

12.  [tex]\frac{b}{y}\:=\:\frac{c}{b}[/tex]

13. b² = cy

14. a² + b² = cx + cy

15. [tex]\left(CB\right)^2+\left(CA\right)^2=\left(AB\right)\left(DB+BA\right)[/tex]

16. x + y = c

17. a² + b² = c²

Reason

1. Given

2. From a point not on a line, exactly one perpendicular can be drawn through the point to the line.

3. Definition of altitude

4. All right angles are congruent.

5. Reflexive Property

6. AA Similarity Postulate

7. Polygon Similarity Postulate

8. Cross Multiply and Simplify

9. All right angles are Congruent

10. Reflexive Property

11. AA similarity Postulate

12. Polygon Similarity Postulate

13. Cross Multiply and Simplify

14. Addition Property of Equality

15. Distributive Property

16. Segment Addition Postulate

17. Substitution Property

Keywords: statement, reason

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Answer:

Interval data

Step-by-step explanation:

Brian's score is an interval data because it appears within the GMAT range of score, which is 200-800

Answer:

72 = 2h

Step-by-step explanation:

6 dozens = 6×12 = 72

72 = 2h

Each hen lays eggs, so h hens will lay 2×h eggs

The equation you've made implies every hen lays 36 eggs

72 = 2h is the equation,

Which simplifies to

h = 36

Answer:

1. 1/2-2/5

2. 3/4-2/5

3. 1/2-3/4

Step-by-step explanation:

For you to be able to create such fractions you must make sure that the numerator is divisible by 20.

Clearly the number 2,4 and 5 can divide 20.so therefore any fraction formed with them will give the denominator 20

The similarity statement relating the three triangles in the diagram is 'YHB ~ YDH ~ HDB'. This is because they are all similar triangles, sharing the same shape but differing in sizes due to scale.

In this case, the answer would be 'YHB ~ YDH ~ HDB'. We are looking for a similarity statement, which states that all three triangles are similar. Similar triangles are triangles that have the same shape, but can be different sizes, i.e. they are scaled versions of each other. Since HD is a perpendicular drawn from right angle H in triangle YHB, this creates two triangles (YDH, HDB) that are respectively similar to the original triangle YHB as each of the two triangles include one of the acute angles of triangle YHB and their own right angles. Therefore, YHB ~ YDH ~ HDB.

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Answer:

the gcf is 21r^2t^3

Step-by-step explanation:

21r^2t^3(3+2rt^2)

The greatest common factor of the expression 63r²t³+42r³t⁵ is 21r²t³, determined by finding the highest common power of each factor.

To find the greatest common factor (GCF) of the expression 63r²t³+42r³t⁵, we need to identify the highest powers of each factor that divide both terms.

Thus, the GCF of the expression 63r²t³+42r³t⁵ is 21r²t³.

Answer:

[tex]V=24,501.42\ m^3[/tex]

Step-by-step explanation:

we know that

The volume of the cylinder is equal to

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

where

r is the radius of the circular base of cylinder

h is the height of the cylinder

we have

[tex]r=34/2=17\ m[/tex] ----> the radius is half the diameter

[tex]h=27\ m\\\pi=3.14[/tex]

substitute the given values in the formula

[tex]V=(3.14)(17)^{2} (27)=24,501.42\ m^3[/tex]

Answer:

Step-by-step explanation

Givens

r is the radius = 17

h is the height of the cylinder = 27

pi = 3.14

Unknowns = Volume

Anwser = 24,501.42 m^3

Answer: the ratio that represents Cos C = a/b

Step-by-step explanation:

Triangle ABC is a right angle triangle.

From the given right angle triangle,

AC represents the hypotenuse of the right angle triangle.

With m∠C as the reference angle,

BC represents the adjacent side of the right angle triangle.

AB represents the opposite side of the right angle triangle.

To determine m∠C, we would apply

the cosine trigonometric ratio.

Cos θ = adjacent side/hypotenuse. Therefore,

Cos C = a/b

Answer:

a) Positively skewed

b) Median

Step-by-step explanation:

We are given the following in the question:

For a particular data, mean levels of attraction are much higher than the median and the mode for these data.

[tex]\text{Mean} > \text{Median}\\\text{Mean} > \text{Mode}[/tex]

a) Shape of data

Since the mean of data is greater than the median and mode of the data, thus, is a skewed data.

For a positively skewed data:

[tex]\text{Mean} > \text{Median} > \text{Mode}[/tex]

Thus, the given data is positively skewed data.

b) Measure of central tendency

Since it is a positively skewed data, median is a better measurement of central of tendency.

Advantage of median:

Final answer:

The data likely has a right-skewed distribution, making the median the most suitable measure of central tendency due to the impact of outliers.

Explanation:

a. What is the shape of the distribution for the data in this study?
The data distribution is likely skewed to the right, as the mean is much higher than the median and mode, indicating a long tail to the right.

b. What measure of central tendency is most appropriate for describing these data? Why?
In this case, the median is the most appropriate measure of central tendency because it is less affected by extreme values compared to the mean. Since the mean is much higher than the median and mode, it suggests outliers are pulling the mean upwards.

Answer:

Step-by-step explanation:

We would apply the formula for determining simple interest which is expressed as

I = PRT/100

Where

I = interest at the end of t years

r represents the interest rate.

P represents the principal or initial amount deposited.

t represents the number of years of investment.

From the information given,

P = 1000

R = 2%

T = 5 years

Therefore,

I = (1000 × 2 × 5)/100

I = $100

The total amount in the account after 5 years would be

1000 + 100 = $1100

The sum of -4 and -8 is -12, so the numerical expression for the given phrase is 36 / (-12) and the simplified expression is -3.

To write a numerical expression for the phrase 'The quotient of 36 and the sum of -4 and -8,' we need to divide 36 by the sum of -4 and -8. The sum of -4 and -8 is -12, so the numerical expression is 36 / (-12).

Simplifying this expression, we get -3. Therefore, the simplified expression is -3.

Answer:

[tex]a.\ 12[/tex]

Step-by-step explanation:

[tex]\frac{5}{6}y-8=2\\\\Add\ 8\ both\ sides\\\\\frac{5}{6}y-8+=2+8\\\\\frac{5}{6}y=10\\\\Multiply\ by\ 6\ both\ the\ sides\\\\\frac{5}{6}y\times 6=10\times 6\\\\5y=60\\\\divide\ by\ 5\ both\ the\ sides\\\\\frac{5y}{5}=\frac{60}{5}\\\\y=12[/tex]

Answer:

Step-by-step explanation:

a) The two populations were i) the pregnant mothers who took the drug ii) the pregnant mothers who did not take the drugs

b) The data must have been obtained in a survey because experiment was not done.

c) 63 out of 3980 developed abnormalities in I case.

Hence out of 1000 abnormalities estimated = [tex]\frac{63}{3980} *1000\\=15.829\\[/tex]

i.e. approximately 16

d) Mothers who did not take drug

(information incomplete)

e) Medical hypothesis testing requires accurate results and hence sample sizes should be very large.

The mentioned study compares two populations: women exposed to DES during their mother's pregnancy and women who weren't. The data seems to be from a survey, is calculated with available information to be around 15.8 per 1000 women for the first population and half that for the second. Medical studies use large samples for higher statistical reliability.

a. The two populations in this study are women whose mothers took the drug DES during pregnancy and women whose mothers did not take the drug DES during pregnancy.

b. The data is most likely obtained through a survey, since medical research often relies on observations, health histories, and existing data rather than conducting an experiment. This would also protect subjects' safety and uphold ethical considerations.

c. To provide a descriptive statistic, we would use the rate of occurrence in the sample to estimate the rate in the overall population. In the sample of 3980 women, 63 developed tissue abnormalities. This is a rate of (63/3980) * 1000 ≈ 15.8 per 1000 women.

d. This question does not provide specific data for the second population, but based on the statement that women from the first group are twice as likely to develop abnormalities, we can estimate that the occurrence of abnormalities in the second population would be half as frequent. This would be approximately 7.9 out of 1000 women.

e. Large sample sizes are often used in medical studies to ensure the results are statistically significant and more reliable. This helps to avoid anomalies and provides a more accurate representation of the population.

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Answer:

5x - 4

Step-by-step explanation:

2( x + 4 ) + 3( x - 4)

Distribute the 2 into ( x + 4 ) so it comes out to be 2x + 8

now distribute 3 into ( x - 4 ) = 3x - 12

now combine like terms 2x + 8 + 3x - 12

2x and 3x can combine into 5x

-12 and 8 can combine into -4

so it comes out to be 5x - 4

Answer:

pictured and shown and solved

Answer:

Therefore the measure of∠ A is 60.07.

Step-by-step explanation:

Given:

In Right Angle Triangle ABC

∠ B = 90°

BC = 13   ....Side opposite to angle A

AC = 15  .... Hypotenuse

To Find:

m∠A = ?

Solution:

In Right Angle Triangle ABC ,Sine Identity,

[tex]\sin A = \dfrac{\textrm{side opposite to angle A}}{Hypotenuse}\\[/tex]

Substituting the values we get

[tex]\sin A = \dfrac{BC}{AC}=\dfrac{13}{15}=0.8666\\\\A=\sin^{-1}(0.8666)=60.065\\\\m\angle A=60.07\°[/tex]

Therefore the measure of∠ A is 60.07

Question was Incomplete;Complete question is given below;

Trey drove to the mountains last weekend. There was heavy traffic on the way there, and the trip took 12 hours. When Trey drove home, there was no traffic and the trip only took 8 hours. If his average rate was 20miles per hour faster on the trip home, how far away does Trey live from the mountains?

Do not do any rounding.

Answer:

Trey lives 480 miles from the mountain.

Step-by-step explanation:

Given:

Time taken to drove the mountain =12 hours

Time taken to return back from mountain = 8 hours.

Let the speed at which he drove to mountain be denoted by 's'.

Speed on the trip to home = [tex]s+20[/tex]

We need to find the distance Trey live from the mountains.

Solution:

Let the distance be denoted by 'd'.

Now we know that;

Distance is equal to speed times Time.

framing in equation form we get;

distance from home to mountain [tex]d=12s[/tex]

Also distance from mountain to home [tex]d = (s+20)8=8s+160[/tex]

Now distance is same for both the trips;

so we can say that;

[tex]12s=8s+160[/tex]

Combining the like terms we get;

[tex]12s-8s=160\\\\4s=160[/tex]

Dividing both side by 4 we get;

[tex]\frac{4s}{4}=\frac{160}{4}\\\\s=40\ mph[/tex]

Speed while trip to mountain = 40 mph

Speed while trip to home = [tex]s+20=420+20=60\ mph[/tex]

So Distance [tex]d=12s=12\times40 = 480\ miles[/tex]

Hence Trey lives 480 miles from the mountain.

Model A's optimal price is $5, with a $3 increase, and Model B's optimal price is $6, with a $4 increase.

Let's walk through the step-by-step calculations for both Model A and Model B.

Model A:

Given Function:

f(x) = -12.5x^2 + 75x + 200

Completing the Square to Find Vertex:

f(x) = -12.5(x^2 - 6x - 16)

f(x)/(-12.5) = (x^2 - 6x - 16)

f(x)/(-12.5) + 16 + 9 = (x-3)^2

f(x)/(-12.5) + 25 = (x-3)^2 - 25

f(x) = (-12.5)[(x-3)^2 + 312.5]

Vertex Form:

f(x) = (-12.5)(x-3)^2 + 312.5

The vertex is at (3, 312.5).

Optimal Price Calculation:

The x-coordinate of the vertex indicates the optimal increase, which is $3. So, the optimal price is 3 + 2 = $5.

Y-intercept:

f(x) = -12.5(9) + 312.5 = 200

The y-intercept is $200.

X-intercepts:

Factorizing the original equation, we get x = 8 and x = -2.

Model B:

Symmetry Axis Calculation:

Midpoint between x-intercepts x = (-2 + 10)/2 = 4.

Optimal Price Calculation:

The optimal increase is $4, resulting in a price of 4 + 2 = $6.

Y-intercept:

The y-intercept is graphically determined to be between $180 and $210.

In summary, Model A's optimal price is $5, achieved with a $3 increase, while Model B's optimal price is $6, attained with a $4 increase.

Answer:

C = 1.75M + 3.25

Step-by-step explanation:

Let E represent Economy

Let L represent Luxury

Let M be the number of miles driven

Let C be how much it cost to rent a luxury car than economy

E = -0.70M + 14.95

L = 1.05M + 18.20

C = L - M

C =(1.05M + 18.20) - (-0.70M + 14.95)

C = 1.05M + 18.20 + 0.70M - 14.95

collect like terms

C = 1.05M + 0.70M + 18.20 - 14.95

C = 1.75M + 3.25

The equation representing how much more it costs to rent a luxury car than an economy car, given the number of miles driven, is C = 1.75M + 3.25. This is derived by subtracting the equation for economy car costs from the equation for luxury car costs and simplifying.

The value of C, which is the cost difference between renting a luxury and an economy car, can be found by finding the difference between the charge equations for the two types of cars. To find C in terms of M, subtract the equation for the economy car (E) from the equation for the luxury car (L).

So, C = L - E

Substitute the given equations for L and E into the equation for C:

C = (1.05M + 18.20) - (-0.70M + 14.95)

Simplify the equation by distributing the negative sign on the right side of the equation:  

C = 1.05M + 18.20 + 0.70M - 14.95

Combine like terms:

C = 1.75M + 3.25

This equation represents how much more it costs to rent a luxury car than an economy car based on the number of miles driven.

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Answer:

Third option: Definition of Congruent angles.

Step-by-step explanation:

For this exercise it is important to know the definition Congruent angles.

Congruent angles are defined as those angles that have equal measure.

The symbol used for Congruent angles is the following:

≅

Keep the explanation above on mind.

In this case, you know that [tex]\angle ABC[/tex] measures 90 degrees (This is also known as "Rigth angle"):

[tex]\angle ABC=90\°[/tex]

And you can observe in the table attached that the measure of [tex]\angle ABD[/tex] is also 90 degrres:

[tex]\angle ABD=90\°[/tex]

Therefore, since they have exactly the same measure, these angles are congruent. Then:

[tex]\angle ABC[/tex] ≅ [tex]\angle ABD[/tex]

Based on this, you can identify that the missing reason number 8 is: Definition of Congruent angles.

Answer:

In this case the Central Tendency measure that would be appropriate to report is Mode.

Step-by-step explanation:

Central Tendency measures are listed as follows:

i.) Mean

ii) Median

iii) Mode.

In the case that the data collected of a population is qualitative and not quantitative then the best Central Tendency measure to qualify the data is Mode of the data.

In the given example the data collected is of the students' racial classification which is not quantitative and purely qualitative. Therefore in case it is proper to take the Central Tendency measure to be reported as the Mode.