Samantha is the best middle manager the company has. She constantly puts countless hours into her job and is one of the best and brightest around. Samantha has not received a raise or promotion for many years and believes it is because she is a woman experiencing unseen discrimination. It is clear that Samantha is feeling the effects of a_____.

(i) The length of the longer side is 3 cm

(ii) The length of the longer side is 6 cm

(iii) The length of the longer side is 15 cm

Step-by-step explanation:

A rectangle has sides in the ratio 1 : 3, we need to find the length of the longer side if:

Let us use the ratio method to solve the problem

(i)

∵ The ratio of the two sides of the rectangle is 1 : 3

∵ The length of the shorter side is 1 cm

→  Shorter    :    Longer

→  1               :    3

→  1               :    x

By using cross multiplication

∴ 1 × x = 1 × 3

∴ x = 3

∵ x represents the length of the longer side

∴ The length of the longer side = 3 cm

The length of the longer side is 3 cm

(ii)

∵ The ratio of the two sides of the rectangle is 1 : 3

∵ The length of the shorter side is 2 cm

→  Shorter    :    Longer

→  1               :    3

→  2               :    x

By using cross multiplication

∴ 1 × x = 2 × 3

∴ x = 6

∵ x represents the length of the longer side

∴ The length of the longer side = 6 cm

The length of the longer side is 6 cm

(iii)

∵ The ratio of the two sides of the rectangle is 1 : 3

∵ The length of the shorter side is 5 cm

→  Shorter    :    Longer

→  1               :    3

→  5               :    x

By using cross multiplication

∴ 1 × x = 5 × 3

∴ x = 15

∵ x represents the length of the longer side

∴ The length of the longer side = 15 cm

The length of the longer side is 15 cm

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Answer: The rate in still water is 8m/s

The rate in current period is is 6 m/s

Step-by-step explanation:

In a canoe race, A team paddles downstream 480 m in 60 seconds. The same team makes the trip upstream and 80 seconds.

We observe that it took the team more time paddling upstream than paddling downstream even though it was the same distance.

Let us assume that on paddling downstream, they paddled in the same direction with the current. This means that they paddled on still water. On paddling upstream, they paddled in the opposite direction of the current.

Let the speed of the boat or teammates be

x m/s

Let the speed of the current be

y m/s

Distance = speed × time

Distance travelled on still water or downstream

= (x+y) × 60 = 60(x+y)

Distance travelled on upstream

= (x-y) × 80 = 80(x-y)

Since the distance is 480 miles for both upstream and downstream,

60(x+y) = 480

x + y = 480/60 = 8 - - - - - -1

80(x-y) = 480

x - y = 480/80 = 6 - - - - - -2

Adding equation 1 and 2,it becomes

2x = 14

x = 14/2 = 7 m/s

y = 8 - x = 8-7

y = 1 m/s

Rate in still water = x +y = 7+1 = 8m/s

Rate in current period = x - y = 7 - 1 = 6m/s

Answer:

Step-by-step explanation:

For cost and revenue functions C(x) = 36400-83.2x and R(x) = 1248-8.32x², you want to know the selling price limits that will let the company make a profit.

Profit is the difference between revenue and cost.

  P(x) = R(x) -C(x)

  P(x) = 1248-8.32x² -(36400-83.2x) . . . . . . use the given functions

  P(x) = -8.32(x² -160x +4375) . . . . . . . . remove common factor

  P(x) = -8.32(x -35)(x -125) . . . . . . factor

The profit will be zero when the factors are zero, for x = 35 and x = 125.

We have to assume the demand function is found by dividing the revenue by the number of chairs sold.

  R(x) = x(price) = x(1248 -8.32x)

Then the price is ...

  price = 1248 -8.32x . . . . . . . . . . . where x is the number of chairs sold

When selling 125 chairs, the price is ...

  1248 -8.32(125) = 208 . . . . . dollars

When selling 35 chairs, the price is ...

  1248 -8.32(35) = 956.80 . . . . dollars

For the company to make a profit, the selling price can go no lower than $208 and no higher than $956.80.

__

Additional comment

These prices will result in 0 profit, as the number of chairs sold makes the revenue equal to the cost. If we require sales of 36 to 124 chairs, so profit is positive, then the price limits are $216.32 and $948.48. Profit will be maximized when 80 chairs are sold for $582.40 each.

Answer: the number of adult tickets is 210

The number if student tickets is 270

Step-by-step explanation:

Let x represent the number of adult tickets that were purchased.

Let y represent the number of student tickets that were purchased.

At the ritz, concert tickets for adults cost $6 and tickets for students cost $4. If the cost of total tickets purchased is $2340, then,

6x + 4y = 2340 - - - - - - - -1

Total number of tickets purchased is 480. This means that

x + y = 480

x = 480 - y

Substituting x = 480 - y into equation 1, it becomes

6(480 - y) + 4y = 2340

2880 - 6y + 4y = 2340

- 6y + 4y = 2340 - 2880

-2y = - 540

y = - 540/-2 = 270

x = 480 - 270

x = 210

Answer:

$7,661

Step-by-step explanation:

Closing balance = Opening balance + purchases - Issued items

Given

Office supplies account on January 1 = $6,791 - Opening balance

Purchases = $3,205

Supplies on hand on January 31 = $2,155 - Closing balance

Substituting into the formula above

2155 = 6791 + 3025 - Issued items

Issued items = 6791 + 3025 - 2155

                     = $7,661

The amount to be used for the appropriate adjusting entry is $7,661

Final answer:

The adjusting entry for the used office supplies for the month of January is $7,841, which is calculated by subtracting the supplies on hand at the month's end from the sum of the starting balance and purchases made during the month.

Explanation:

To calculate the adjusting entry for office supplies, you need to calculate the cost of supplies that were used during the month. Start with the balance of supplies on hand at the beginning of the month, add the purchases made during the month, and then subtract the balance of supplies on hand at the end of the month.

The calculation is as follows:

Starting balance on January 1: $6,791

Add purchases during January: $3,205

Subtract ending balance on January 31: $2,155

The adjusting entry for supplies used = (Starting balance + Purchases) - Ending balance
= ($6,791 + $3,205) - $2,155
= $9,996 - $2,155
= $7,841

Therefore, the adjusting entry to record the office supplies used would be for $7,841.

The total weight of the smaller and the bigger cat Alberto has is 26 1/12 pounds.

A fraction is written in the form of p/q, where q ≠ 0.

Fractions are of two types they are proper fractions in which the numerator is smaller than the denominator and improper fractions where the numerator is greater than the denominator.

Given, Alberto has 2 cats.

The smaller cat weighs 10 3/4 pounds and the larger cat weighs 15 1/3 pounds.

Therefore, The weights of the cats together is the sum of their individual

weights which is,

= (10 3/4 + 15 1/3) pounds.

= (43/4 + 46/3) pounds.

= [(3×43 + 4×46)/12] pounds.

= (129 + 184)/12 pounds.

= 313/12 pounds.

= 26 1/12 pounds.

So, Together the cats weigh 26 1/12 pounds.

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Answer:

(x-4)² - 11

Step-by-step explanation:

You find half of 8 which is 4 and half of x² which is x. this forms (x - 4).

However this would expand as

x²-8x+16 which isn't the expression. So to make it 5, you have to take away 11 leaving you with

(x-4)²-11

Answer:

(x - 4)^2 - 11.

Step-by-step explanation:

x^2 - 8x + 5

Note that x^2 - 8x = (x - 4)^2 - 16 so we have:

(x - 4)^2 - 16 + 5

= (x - 4)^2 - 11.

To get (x - 4)^2 - 16 I used the identity:

x^2 + ax = ( x + a/2)^2 - a^2/4    with a = -8.

Answer:

The equation [tex]7\,\sqrt{x} =14[/tex] is a radical equation.

Step-by-step explanation:

If the equations given are (as I can read them from your typing):

a) [tex]x+\sqrt{5} =12[/tex]

b) [tex]x^2=16[/tex]

c) [tex]3+x\,\sqrt{7} =13[/tex]

d) [tex]7\,\sqrt{x} =14[/tex]

The only radical equation is the last one : [tex]7\,\sqrt{x} =14[/tex], because it is the only one where the unknown appears inside the root. The name "radical equations" is associated with the fact that the unknown is contained inside the root and therefore the process involved in solving for the unknown will need to include the elimination of the root via algebraic methods to free the unknown.

Notice that the options a) and c) have roots, but what appears inside them are numbers (5 and 7 respectively), and not an unknown like "x". Equation b) doesn't contain a root, and wouldn't classify as a radical equation.

A radical equation is one which contains roots in it, specially those which has root over variables or things whose values changes.

Thus, by above definition, we will have the fourth option: [tex]7\sqrt{x} = 14[/tex] as a radical equation.

A radical equation is one which contains roots in it, specially those which has root over variables or things whose values changes.

Since only in the fourth option we see there's root over x which is a variable here, thus the  fourth option: [tex]7\sqrt{x} = 14[/tex] is a radical equation.

Rest of the options, although containing roots, aren't having variables inside the root, thus they aren't classified as radical equations.

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Answer:

Range of the average number of tours is between 150 and 200 including 150 and 200.

Step-by-step explanation:

Given:

The profit function is modeled as:

[tex]p(x)=-2x^2+700x-10000[/tex]

The profit is at least $50,000.

So, as per question:

[tex]p(x)\geq50000\\-2x^2 + 700x-10000\geq 50000\\-2x^2+700x-10000-50000\geq 0\\-2x^2+700x-60000\geq 0\\\\\textrm{Dividing by 2 on both sides, we get}\\\\-x^2+350x-30000\geq 0[/tex]

Now, rewriting the above inequality in terms of its factors, we get:

[tex]-1(x-150)(x-200)\geq 0\\(x-150)(x-200)\leq 0[/tex]

Now,

[tex]x<150,(x-150)(x-200)>0\\x>200,(x-150)(x-200)>0\\For\ 150\leq x\leq200,(x-150)(x-200)\leq 0\\\therefore x=[150,200][/tex]

Therefore, the range of the average number of tours he must arrange per day to earn a monthly profit of at least $50,000 is between 150 and 200 including 150 and 200.

Answer: The correct answer is D). Between 150 and 200; exclusive

Step-by-step explanation:

Given profit function p(x) of a tour operator is modeled by

p(x)=[tex](-2)x^{2} +700x-10000[/tex]

Where, x is the average number of tours he arranges per day.

To find number of tours to arrange per day to get monthly profit of at least 50,000$:

Now, he should make at-least 50000$ profit.

we can write as p(x)>50000$

[tex](-2)x^{2} +700x-10000\geq50000[/tex]

[tex](-2)x^{2} +700x-60000\geq0[/tex]

Roots are x is 150 and 200

(x-150)(x-200)>0

Case 1 : x>150 and x>200

x>150 also satisfy the x>200.

Case2: x<100 and x<200

x<200 also satisfy the x<100

Thus, the common range is 150<x<200

The correct answer is D). Between 150 and 200; exclusive

Answer:  between 150 and 200; inclusive

Step-by-step explanation:

The answer is 'inclusive' NOT 'exclusive.'

Answer:

x  =  140  ft

w = 70 ft

A(max)  =  9800 ft²

Step-by-step explanation:

We have:

280 feet of fence to enclose three sides of a rectangular area

perimeter of the rectangle ( 3 sides ) is

p  =  L  =  x  +2w       w   = (L - x ) / 2       w   =  ( 280  -  x ) / 2

where:

x is the longer side

w is the width

A(x,w)  = x*w         ⇒   A(x)  =  x* ( 280 - x ) / 2  ⇒ A(x)  = (280x -x²)/2

Taking derivatives on bth sides of the equation

A´(x)  = ( 280 -2x)*2 /4          A´(x)  = 0      ( 280 -2x)  =  0

280 -2x  = 0     x = 280/2

x  =  140  ft

And   w  = ( 280 - x ) / 2  ⇒  w  =(  280  -140  )/ 2

w = 70 ft

A(max)  =  9800 ft²

95% confidence interval would be (58.96, 69.04).

Step-by-step explanation:

Since we have given that

Number of dogs' weight = 20

Mean = 64 ounces

Standard deviation = 11.5 ounces

We need to find the 95% confidence interval.

So, z = 1.96

so, interval would be

[tex]\bar{x}\pm z\dfrac{\sigma}{\sqrt{n}}\\\\=64\pm 1.96\times \dfrac{11.5}{\sqrt{20}}\\\\=64\pm 5.04\\\\=(64-5.04,64+5.04)\\\\=(58.96,69.04)[/tex]

Hence, 95% confidence interval would be (58.96, 69.04).

Answer:

28.108 K.

Step-by-step explanation:

Given: Pressure (P)= 1.2atm

           Number of moles (n)=  2.6 moles

           Volume (V)= 5.00-L

Now finding the temperature (T).

Formula; T= [tex]\frac{P\times V}{n\times R}[/tex]

R is a constant factor which makes other factors work together.

There is a numerical value for R which we use is [tex]0.0821 \times \frac{L.atm}{mole.K}[/tex]

∴ Temperature (T)= [tex]\frac{1.2\times 5}{2.6\times 0.0821 \frac{L.atm}{mol.K} }[/tex]

⇒ Temperature (T)= [tex]\frac{6}{0.21346} = 28.1083\ K[/tex]

∴ Temperature is 28.108 K

Answer:

  m∠A = m∠D = 40°

Step-by-step explanation:

Angles A and D are corresponding angles in the congruent triangles, so have the same measure. You set one measure equal to the other:

  x + 20 = 2x

To solve this, subtract x from both sides:

  20 = x

Then both angle measures are 2x = 40°.

Answer: You would spend 160 hours in total of ten weeks.

Step-by-step explanation: Just add 12.5 + 3.5 which = 16. Then multiply 16 times 10 which is 160, and that is your answer.

Answer:

The answer is b.)  -5.2 degrees

Step-by-step explanation:

to find the mean of this problem you have to add all numbers and then divide it by how many numbers there is.

so you have to add  -42+ -17+14+-4+23 and that'll equal -26

so you take -26 and divide it by 5 because thats how many numbers their are to divide

-26 divided by 5 is (-5.2)

Answer:

(Solution: X=16 and Y=-10)

Explanation:

》Total money that will be spent on flour and corn meal altogether(T)

=$25

》Since it is not mentioned that whether corn and flour are bought in same quantity or not, we will assume them of different quantity.

i.e.,Suppose

》X pound of flour is bought

&

》Y pound of corn is bought.

So,

》Cost of flour(F)=$1.50X

》Cost of corn(C)=$2.50Y

So total cost will be sum of cost of flour and corn altogether,

Writing it in equation(linear),

F+C=T

Also,

Total pounds=6

ie,

The system of linear equations is x + y = 6 and 1.5x + 2.5y = 25.

A connection between a number of variables results in a linear model when a graph is displayed. The variable will have a degree of one.

The linear equation is given as,

y = mx + c

Where m is the slope of the line and c is the y-intercept of the line.

Assume you purchase flour and corn dinner in mass to make flour tortillas and corn tortillas flour cost $1.50 per pound and corn feast cost $2.50 per pound would you like to spend veiling $25 on flour and corn feast yet you want somewhere around 6 pounds by and large

Let x be the number of pounds of flour and y be the number of pounds of corn meal. Then the system of linear equalities is given as,

x + y = 6

1.5x + 2.5y = 25

The system of linear equations is x + y = 6 and 1.5x + 2.5y = 25.

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Answer:

6 minutes

Step-by-step explanation:

Machine A produces x boxes in 10 minutes

In one minute, the machine produces x/10 boxes

Machine B produces 2x boxes in 5 minutes

In one minute, the machine produces 2x/5 boxes

Therefore in one minutes, both boxes working together will produce

= 2x/5 + x/10

=5x/10

=x/2 boxes

To produce 3x boxes, the time required

= 3x/(x/2)

= 3 × 2

= 6

It take machines A and B, working simultaneously at their respective constant rates, to produce 3x boxes in 6 minutes

Answer:

Step-by-step explanation:

Answer:

About 1.65 meals per hour

Step-by-step explanation:

9 days of work in May * 8 hours per day = 72 hours of work in May per employee

72 hours * 13 employees = 936 hours worked for all employees in May

1540 meals in May/ 936 hours worked for all employees in May = about 1.65 meals per hour

Answer:

Let's x be the probability for 1, 3, 5 and 6.

The probability for 2 and 4 is going to be 3x.

The sum of the probabilities of all possible outcomes is always 1.

P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1

x + 3x + x + 3x + x + x = 1

10x = 1

x = 1/10

The probability of obtaining 1, 3, 5 or 6 is 1/10

The probability for 2 and 4 is 3/10

The probability of rolling a 2 or a 4 is [tex]$\frac{3}{14}$[/tex], and the probability of rolling any of the other numbers (1, 3, 5, or 6) is [tex]$\frac{1}{14}$.[/tex]

To solve this problem, we need to distribute the total probability of 1 (since the sum of all probabilities must equal 1) among the six outcomes of the die according to the given conditions.

 Let's denote the probability of rolling a 2 or a 4 as $p$. According to the problem, rolling a 2 or rolling a 4 is three times as likely as rolling each of the other four numbers. Therefore, the probability of rolling a 1, 3, 5, or 6 is [tex]$\frac{p}{3}$.[/tex]

 Since there are two outcomes with probability $p$ (rolling a 2 and rolling a 4) and four outcomes with probability $\frac{p}{3}$ (rolling a 1, 3, 5, or 6), we can set up the following equation to represent the total probability:

[tex]\[ 2p + 4\left(\frac{p}{3}\right) = 1 \][/tex]

 Now, let's solve for $p$:

[tex]\[ 2p + \frac{4p}{3} = 1 \][/tex]

[tex]\[ \frac{6p + 4p}{3} = 1 \][/tex]

[tex]\[ \frac{10p}{3} = 1 \][/tex]

[tex]\[ 10p = 3 \][/tex]

[tex]\[ p = \frac{3}{10} \][/tex]

 So, the probability of rolling a 2 or a 4 is[tex]$p = \frac{3}{10}$.[/tex]

 The probability of rolling a 1, 3, 5, or 6 is [tex]$\frac{p}{3} = \frac{3}{10} \times[/tex] [tex]\frac{1}{3} = \frac{1}{10}$.[/tex]

However, we must remember that the total probability must be distributed equally between rolling a 2 and rolling a 4. Since they are equally likely, each has a probability of half of $p$:

[tex]\[ p_{2} = p_{4} = \frac{p}{2} = \frac{3}{10} \times \frac{1}{2} = \frac{3}{20} \][/tex]

 Now, we can state the final probabilities for each outcome:

 - The probability of rolling a 1 is [tex]$\frac{1}{10}$.[/tex]

- The probability of rolling a 2 is [tex]$\frac{3}{20}$.[/tex]

- The probability of rolling a 3 is [tex]$\frac{1}{10}$.[/tex] - The probability of rolling a 4 is [tex]$\frac{3}{20}$.[/tex]

The probability of rolling a 5 is [tex]$\frac{1}{10}$.[/tex]

Answer:

Step-by-step explanation:

Gabe went out for lunch with his best friend the bill costs $16.40

This amount was before tax and the tip that he wanted to give.

He paid a 9 percent tax on the bill. The amount of the 9 percent tax is 9/100 × 16.40 = 0.09 × 16.40 = $1.476

He left a 20 percent tip. This means that amount left as tip is 20/100 × 16.40 = 0.2×16.40 = $3.28

The amount that Gabe paid would be the sum of the bill, the tip and the amount paid on tax. It becomes

16.40 + 1.476 + 3.28 = $21.156

Answer:

x = 36

Step-by-step explanation:

[tex] x - 12\sqrt{x} + 36 = 0 [/tex]

Subtract x and 36 from both sides.

[tex] -12\sqrt{x} = -x - 36 [/tex]

Divide both sides by -1.

[tex] 12\sqrt{x} = x + 36 [/tex]

Square both sides.

[tex] 144x = x^2 + 72x + 1296 [/tex]

Subtract 144x from both sides.

[tex] 0 = x^2 - 72x + 1296 [/tex]

Factor the right side.

[tex] 0 = (x - 36)^2 [/tex]

[tex] x - 36 = 0 [/tex]

[tex] x = 36 [/tex]

Since the solution of the equation involved squaring both sides, we musty check the answer for possible extraneous solutions.

Check x = 36:

[tex] x - 12\sqrt{x} + 36 = 0 [/tex]

[tex] 36 - 12\sqrt{36} + 36 = 0 [/tex]

[tex] 36 - 12\times 6 + 36 = 0 [/tex]

[tex] 36 - 72 + 36 = 0 [/tex]

[tex] 0 = 0 [/tex]

Since 0 = 0 is a true statement, the solution x = 36 is a valid solution.

Answer: 85.333 or 256 over 3.

Answer: x = 2

Step-by-step explanation:

The diagram of the kite is shown in the attached photo.

The perimeter of the kite is the distance around the kite.

The kite has 2 pairs of equal sides.

This means that

Side ML = side MN and side NO =

side LO

If Side MN = 8x – 3 and side NO = 2x + 1, then The perimeter of the kite is ML + MN + NO + LO

The perimeter of kite LMNO is given as 36 feet.

Therefore

ML + MN + NO + LO = 36

8x – 3 + 2x + 1 +8x – 3 + 2x + 1 = 36

8x + 8x + 2x+ 2x -3 +1 - 3 + 1

20x -4 = 36

20x = 40

x = 40/20 = 2

Answer: she runs 132 feets in each lap and 660 feets in 5 laps

Step-by-step explanation:

Stella runs laps around the edge of the yard. This means she runs round the entire shape of the yard.

Miss bridgeyard is 24 ft by 42 ft. This means that the length and width of Miss bridgeyard are not the same. Therefore, Miss bridgeyard has the shape of a rectangle. The distance that stella covers in one lap is the perimeter of the rectangular Miss bridgeyard.

Perimeter of a rectangle = 2( L + W )

If length,L = 42 feets and

Width ,W = 24 feets, the perimeter would be

2(42+24)/= 2×66 = 132 feets

She runs a distance of 132 feets in one lap.

Distance in 5 laps would be

132 × 5 = 660 feets

Answer:

y=525x+230

Step-by-step explanation:

525 is spent everyday. x is the number of days, so with each day $525 is spent.

$230 is a one time cost, regardless of how many days they stay on the trip.

Answer:

  f(0) = 1

Step-by-step explanation:

The ordered pair with first number 0 has second number 1. Each pair corresponds to (x, f(x)), so that pair has x=0 and f(0) = 1.

Answer:

[tex]f(0)=1[/tex]

Step-by-step explanation:

The given points are

[tex](-3,0), (-2,2),(0,1),(1,-2)[/tex]

Remember that each point has the form [tex](x,y)[/tex], where [tex]y=f(x)[/tex].

That means if we need to find [tex]f(0)[/tex], then we just need to look for the pair that belong to [tex]x=0[/tex].

If you observe, the pair we are looking for is [tex](0,1)[/tex], which relation is

[tex]f(0)=1[/tex].

Therefore, the answer is 1, that is, [tex]f(0)=1[/tex].

Answer:

a) No

b) No

c) No

d) No

Step-by-step explanation:

Remember, a set V wit the operations addition and scalar product is a vector space if the following conditions are valid for all u, v, w∈V and for all scalars c and d:

1. u+v∈V

2. u+v=v+u

3. (u+v)+w=u+(v+w).

4. Exist 0∈V such that u+0=u

5. For each u∈V exist −u∈V such that u+(−u)=0.

6. if c is an escalar and u∈V, then cu∈V

7. c(u+v)=cu+cv

8. (c+d)u=cu+du

9. c(du)=(cd)u

10. 1u=u

let's check each of the properties for the respective operations:

Let [tex]u=(u_1,u_2,u_3), v=(v_1,v_2,v_3)[/tex]

Observe that  

1. u+v∈V

2. u+v=v+u, because the adittion of reals is conmutative

3. (u+v)+w=u+(v+w). because the adittion of reals is associative

4. [tex](u_1,u_2,u_3)+(0,0,0)=(u_1+0,u_2+0,u_3+0)=(u_1,u_2,u_3)[/tex]

5. [tex](u_1,u_2,u_3)+(-u_1,-u_2,-u_3)=(0,0,0)[/tex]

then regardless of the escalar product, the first five properties are met for a), b), c) and d). Now let's verify that properties 6-10 are met.

a)

6. [tex]c(u_1,u_2,u_3)=(cu_1,u_2,cu_3)\in V[/tex]

7.

[tex]c(u+v)=c(u_1+v_1,u_2+v_2,u_3+v_3)=(c(u_1+v_1),u_2+v_2,c(u_3+v_3))\\=(cu_1+cv_1,u_2+v_2,cu_3+cv_3)=c(u_1,u_2,u_3)+c(v_1,v_2,v_3)=cu+cv[/tex]

8.

[tex](c+d)u=(c+d)(u_1,u_2,u_3)=((c+d)u_1,u_2,(c+d)u_3)=\\=(cu_1+du_1,u_2,cu_3+du_3)\neq (cu_1+du_1,2u_2,cu_3+du_3)=cu+du[/tex]

Since 8 isn't satify then V is not a vector space with the addition as in R^3 and the scalar product [tex]a(x,y,z)=(ax,y,az)[/tex]

b)  6. [tex]c(u_1,u_2,u_3)=(cu_1,0,cu_3)\in V[/tex]

7.

[tex]c(u+v)=c(u_1+v_1,u_2+v_2,u_3+v_3)=(c(u_1+v_1),0,c(u_3+v_3))\\=(cu_1+cv_1,0,cu_3+cv_3)=c(u_1,u_2,u_3)+c(v_1,v_2,v_3)=cu+cv[/tex]

8.

[tex](c+d)u=(c+d)(u_1,u_2,u_3)=((c+d)u_1,0,(c+d)u_3)=\\=(cu_1+du_1,0,cu_3+du_3)=(cu_1,0,cu_3)+(du_1,0,du_3) =cu+du[/tex]

9.

[tex]c(du)=c(d(u_,u_2,u_3))=c(du_1,0,du_3)=(cdu_1,0,cdu_3)=(cd)u[/tex]

10

[tex]1u=1(u_1,u_2,u3)=(1u_1,0,1u_3)=(u_1,0,u_3)\neq(u_1,u_2,u_3)[/tex]

Since 10 isn't satify then V is not a vector space with the addition as in R^3 and the scalar product [tex]a(x,y,z)=(ax,0,az)[/tex]

c) Observe that [tex]1u=1(u_1,u_2,u3)=(0,0,0)\neq(u_1,u_2,u_3)[/tex]

Since 10 isn't satify then V is not a vector space with the addition as in R^3 and the scalar product [tex]a(x,y,z)=(0,0,0)[/tex].

d)  Observe that [tex]1u=1(u_1,u_2,u3)=(2*1u_1,2*1u_2,2*1u_3)=(2u_1,2u_2,2u_3)\neq(u_1,u_2,u_3)=u[/tex]

Since 10 isn't satify then V is not a vector space with the addition as in R^3 and the scalar product [tex]a(x,y,z)=(2ax,2ay,2az)[/tex].

None of the given definitions make ( V ) a vector space because they fail to satisfy the necessary vector space axioms.

To determine whether ( V ) is a vector space under the given definitions of scalar multiplication, we need to check if each definition satisfies the vector space axioms.

Definition (a): [tex]\( a(x,y,z) = (ax,y,az) \)[/tex]

Additive Identity: Yes, [tex]\( 1(x,y,z) = (x,y,z) \)[/tex].

Scalar Distributive (over vectors): [tex]\( a((x_1,y_1,z_1)+(x_2,y_2,z_2)) = a(x_1+x_2, y_1+y_2, z_1+z_2) = (a(x_1+x_2), y_1+y_2, a(z_1+z_2)) \).[/tex]

Scalar Distributive (over scalars): [tex]\( (a+b)(x,y,z) = ((a+b)x,y,(a+b)z) = (ax+bx,y,az+bz) \).[/tex]

Associative: [tex]\( a(b(x,y,z)) = a(bx,y,bz) = (abx,y,abz) = (ab)(x,y,z) \).[/tex]

Conclusion: Does not satisfy scalar distributive over vectors.

Definition (b): [tex]\( a(x,y,z) = (ax,0,az) \)[/tex]

Additive Identity: Yes, \( 1(x,y,z) = (x,0,z) \).

Scalar Distributive (over vectors): [tex]\( a((x_1,y_1,z_1)+(x_2,y_2,z_2)) = a(x_1+x_2,y_1+y_2,z_1+z_2) = (a(x_1+x_2),0,a(z_1+z_2)) = (ax_1+ax_2,0,az_1+az_2) \)[/tex]

Scalar Distributive (over scalars): [tex]\( (a+b)(x,y,z) = ((a+b)x,0,(a+b)z) = (ax+bx,0,az+bz) \).[/tex]

Associative: [tex]\( a(b(x,y,z)) = a(bx,0,bz) = (abx,0,abz) = (ab)(x,y,z) \).[/tex]

Conclusion: Does not satisfy scalar distributive over vectors.

Definition (c): [tex]\( a(x,y,z) = (0,0,0) \)[/tex]

Additive Identity: Yes, [tex]\( 1(x,y,z) = (0,0,0) \).[/tex]

Scalar Distributive (over vectors): [tex]\( a((x_1,y_1,z_1)+(x_2,y_2,z_2)) = (0,0,0) \).[/tex]

Scalar Distributive (over scalars): [tex]\( (a+b)(x,y,z) = (0,0,0) \).[/tex]

Associative: [tex]\( a(b(x,y,z)) = (0,0,0) \).[/tex]

Conclusion: Does not satisfy any of the scalar distributive properties.

Definition (d): [tex]\( a(x,y,z) = (2ax,2ay,2az) \)[/tex]

Additive Identity: No, [tex]\( 1(x,y,z) = (2x,2y,2z) \).[/tex]

Scalar Distributive (over vectors): [tex]\( a((x_1,y_1,z_1)+(x_2,y_2,z_2)) = a(x_1+x_2, y_1+y_2, z_1+z_2) = (2a(x_1+x_2), 2a(y_1+y_2), 2a(z_1+z_2)) = (2ax_1+2ax_2, 2ay_1+2ay_2, 2az_1+2az_2) \).[/tex]

Scalar Distributive (over scalars): [tex]\( (a+b)(x,y,z) = (2(a+b)x, 2(a+b)y, 2(a+b)z) = (2ax+2bx, 2ay+2by, 2az+2bz) \).[/tex]

Associative: [tex]\( a(b(x,y,z)) = a(2bx,2by,2bz) = (4abx,4aby,4abz) \neq (2ab)(x,y,z) \).[/tex]

Conclusion: Does not satisfy scalar multiplication associativity.

For this case we have a function of the form [tex]y = f (x)[/tex], where:

[tex]f (x) = 10-x[/tex]

We must find the value of the function when:

[tex]x = -2,0,2[/tex]

[tex]f (-2) = 10 - (- 2) = 10 + 2 = 12[/tex]

[tex]f (0) = 10-0 = 10[/tex]

[tex]f (2) = 10-2 = 8[/tex]

Thus, we have that the function has a value of [tex]y = {12,10,8}[/tex] when [tex]x = {- 2,0,2}[/tex]

Answer:

[tex]y = {12,10,8}[/tex]