A high school drama club is selling tickets to their show. There is a maximum of 400 tickets. The tickets cost $10 (if bought before day of show) and $15 (if bought day of show). Let x represent the number of tickets sold before the day of the show and y represents the number of tickets sold the day of the show. To meet the expenses of the show, the club must sell at least $3500 worth of tickets. The club sells 300 tickets before the day of the show. Is it possible to sell enough additional tickets on the day of the show to meet the expenses of the show? Justify your answer.

Answer:

Step-by-step explanation:

The probability of drawing two pink marbles from a bag with replacement can be calculated using the multiplication rule of independent events as P(pink and pink) = P(pink) x P(pink), where P(pink) is the probability of drawing a pink marble.

The subject of this problem is probability in Mathematics, particularly with replacement. The scenario involves drawing two pink marbles from a bag with replacement. This means that after the first marble is drawn, it is put back into the bag before the second one is drawn.

The probability of drawing a pink marble on the first draw is calculated by dividing the number of pink marbles by the total number of marbles. Similarly, the probability of drawing a pink marble on the second draw, with replacement, stays the same because the total number of marbles in the bag is the same as in the first draw.

To calculate the final probability, you use the multiplication rule of independent events (events where the outcome of the first event does not affect the outcome of the second event). According to this rule, the probability of both events happening is the product of the probabilities of each event. Hence, if P(pink) represents the probability of drawing a pink marble, the probability of drawing two pink marbles (with replacement) is P(pink and pink) = P(pink) x P(pink).

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Answer: The distance between A and B is 300 miles.

Step-by-step explanation:

Hi, to solve this problem we have to analyze the information given.

We know that when he was 15 miles away from point B, he was traveling at 60mph. if we apply the formula : time= distance /speed;

So, he traveled that distance in 15 minutes.

That means that he returned to point A in 3.75 hours (4 hours -15minutes) at a speed of 80 mph.

Applying the formula again to calculate the distance:

Answer:

The probability is 0.7946

Step-by-step explanation:

Let's call F the event that 16 of the 30 tortillas are failures, A the event that you choose a type 1 part and B the event that you choose a type 2 part.

So, the probability that you picked a Type 1 part given that 16 of the 30 tortillas are failures is calculated as:

P(A/F)=P(A∩F)/P(F)

Where P(F) = P(A∩F) + P(B∩F)

Then, the probability that a type 1 part created 16 failures can be calculated using the binomial distribution as:

[tex]P(x)=\frac{n!}{x!(n-x)!}*p^{x}*(1-p)^{n-x}[/tex]

Where x is the number of failures, n is the total number of tortillas and p is the failure rate, so:

[tex]P(16)=\frac{30!}{16!(30-16)!}*0.4^{16}*(1-0.4)^{30-16}=0.0489[/tex]

Therefore, The probability P(A∩F) that you choose a type 1 part and this part created 16 square tortillas is:

(0.3)(0.0489) = 0.0147

Because 0.3 is the probability to choose a type 1 part and 0.0489 is the probability that a type 1 part created 16 square tortillas.

At the same way, the probability that a type 2 part created 16 failures is:

[tex]P(16)=\frac{30!}{16!(30-16)!}*0.75^{16}*(1-0.75)^{30-16}=0.0054[/tex]

Therefore, P(B∩F) is:  (0.7)(0.0054) = 0.0038

Finally, P(F) and P(A/F) are equal to:

P(F) = 0.0147 + 0.0038 = 0.0185

P(A/F) = 0.0147/0.0185 = 0.7946

Answer:

1 foot

Step-by-step explanation:

Set equal:

4w² + 70w = 74

Move to one side:

4w² + 70w − 74 = 0

Simplify:

2w² + 35w − 37 = 0

Factor.  Using the AC method, ac = 2×-37 = -74.  Factors of -74 that add up to 35 are 37 and -2.  Dividing by a, the factors reduce to 37/2 and -1/1.

(w − 1) (2w + 37) = 0

Set each factor to 0 and solve:

w − 1 = 0

w = 1

2w + 37 = 0

w = -18.5

Since w must be positive, w = 1.  The width of the wood border is 1 foot.

Answer:

1 footsie

Step-by-step explanation:

the first thing ur gonna wanna do is set them equal:

4w^2 + 70w = 74

now in order to solve it you are goonna put it all onto one side:

4w^2+ 70w − 74 = 0

Now, you are gonna simplify this:

2w² + 35w − 37 = 0

(w − 1) (2w + 37) = 0

Set each factor to 0 and solve:

w − 1 = 0

w = 1

2w + 37 = 0

w = -18.5

Since w must be positive, w = 1.  so the width must be 1 foot

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:

(2.23, 7,57)

Step-by-step explanation:

equation of this circle is

x^2 + (y - 2)^2 = 36

y = 2.5x + 2

Substitute for y in the equation of the circle:-

x^2 + (2.5x + 2 - 2)^2 = 36

x^2 + 6.25x^2 = 36

x^2 = 36 / 7.25  

x = +/-   6  /  2.693  =  +/- 2.228

when x = 2.228 y = 2.5(2.228) + 2 =  7.57    to nearest hundredth

when x = -2.228 y = 2.5(-2.228) + 2 =   -3.57

So they intersect at 2 points but the intersect in the first quadrant is at (2.23, 7,57)        to nearest hundredth.

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]

Answer:

  11 miles

Step-by-step explanation:

After the first hour, they walk 3 more hours at 2 miles per hour. So, the total distance is ...

  5 mi + (3 h)(2 mi/h) = 5 mi + 6 mi = 11 mi

The blue team will walk 11 miles in 4 hours.

Answer:

B. One can be​ 90% confident that the mean additional tax owed is between the lower and upper bounds.

Step-by-step explanation:

Given:

n= 81

[tex]\bar{x}=3408[/tex]

[tex]\sigma= 2565[/tex]

Solution:

A confidence interval, in statistics, refers to the probability that a population parameter will fall between two set values for a certain proportion of times. Confidence intervals measure the degree of uncertainty or certainty in a sampling method. A confidence interval can take any number of probabilities, with the most common being a 95% or 99% confidence level.

Confidence interval = [tex]\bar{x} \pm z * \frac{\sigma}{\sqrt{n}}[/tex]

To Find the z  value:

Degree of freedom = n-1

=>81- 1

=> 80

Significance level = 1- confidence level

=>[tex]\frac{(1-\frac{90}{100})}{2}[/tex]

=>[tex]\frac{(1-0.90)}{2}[/tex]

=> [tex]\frac{0.1}{2}[/tex]

=>0.05

using this value In T- Distribution table we get

z =  1.645

Substituting the values  we have,

confidence interval = [tex]3408\pm 1.645 * \frac{2565}{\sqrt{81}}[/tex]

confidence interval = [tex]3408\pm 1.645 * \frac{2565}{\sqrt{9}}[/tex]

confidence interval = [tex]3408\pm 1.645 * 285[/tex]

confidence interval = [tex]3408\pm 468.825[/tex]

confidence interval= (2939.18, 3876.83)

To construct a 90% confidence interval for the mean additional amount of tax owed for estate tax returns, we need to use a z-score and the formula CI = x ± z * (σ/√n), where CI is the confidence interval, x is the sample mean, z is the z-score corresponding to the desired confidence level, σ is the population standard deviation, and n is the sample size. Plugging in the values from the problem, we can calculate the lower and upper bounds of the confidence interval.

To construct a 90% confidence interval for the mean additional amount of tax owed for estate tax returns, we can use the formula:

CI = x ± z * (σ/√n)

where CI is the confidence interval, x is the sample mean, z is the z-score corresponding to the desired confidence level, σ is the population standard deviation, and n is the sample size.

Plugging in the values from the problem, we have:

x = $3408

σ = $2565

n = 81

Using a z-score table or a statistical calculator, we can find that the z-score corresponding to a 90% confidence level is approximately 1.645.

Substituting these values into the formula, we get:

CI = $3408 ± 1.645 * ($2565/√81)

Simplifying the expression, we have:

CI = $3408 ± 1.645 * $285

Calculating the upper and lower bounds of the confidence interval, we get:

Lower bound = $3408 - 1.645 * $285 = $2977.82

Upper bound = $3408 + 1.645 * $285 = $3838.18

Therefore, the 90% confidence interval for the mean additional amount of tax owed for estate tax returns is approximately $2977.82 to $3838.18.

Interpreting the confidence interval, we can say that we are 90% confident that the true mean additional amount of tax owed for estate tax returns falls within this range.

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Option C

Domain: {-5, 1, 3, 7); Yes, it is a function

The given relation is :-

{(-5, 2), (7, 7), (3,6), (1, 7)}

It is of form (x, y)

The domain is the set of all the values of  "x" . The range is the set of all the values of  "y"

We need to find domain :-

The domain is the set of all possible x-values which will make the function "work", and will output real y-values.

Domain is the set of "x" values , in the given relation these are:-

Domain is :-  { -5, 7, 3, 1}

And Range is :- {2, 7, 6, 7}

Since there is one value of y for every value of  "x"

A relation from a set X to a set Y is called a function if each element of X is related to exactly one element in Y.

Hence, the relation is a function

The option C) is correct  

Answer:x

=

5

toppings

Step-by-step explanation:Total cost of cheese pizza:  

$

10.75

Any additional topping adds:  

+

$

1.25

So a cheese pizza with  

1

additional topping is:

$

10.75

+

$

1.25

=

$

12.00

A cheese pizza with  

2

additional toppings is:

$

10.75

+

$

1.25

+

$

1.25

=

$

10.75

+

2

×

$

1.25

=

$

13.25

A cheese pizza with  

3

additional toppings is:

$

10.75

+

$

1.25

+

$

1.25

+

$

1.25

$

10.75

+

3

×

$

1.25

=

$

14.50

If you pay attention to the pattern you can see that, for any number of toppings, say  

x

toppings, the price is going to be:

$

10.25

+

x

×

$

1.25

We are told the final cost is  

$

17.00

. That is

$

10.25

+

x

×

$

1.25

=

$

17.00

Subtract  

$

10.25

from both sides

$

10.25

−

$

10.25

+

x

×

$

1.25

=

$

17.00

−

$

10.25

x

×

$

1.25

=

$

6.25

Divide both sides by  

$

1.25

x

×

$

1.25

$

1.25

=

$

6.25

$

1.25

x

=

5

Answer:

Option C) The symbol y represents the predicted value of price.

Step-by-step explanation:

We are given the following in the question:

We find a regression equation with x representing the ratings and y representing price.

The equation has a slope of 130 and a y-intercept of 350.

Comparing with the slope intercept form:

[tex]y = mx + c\\\text{where m is the slope and c is the y intercept}[/tex]

Thus, we can write the equation as:

[tex]y = 130x + 350[/tex]

Here, y is the predicted variable that is the price, c is the price of hotel when a rating of 0 is given.

Thus, symbol y represents:

C) The symbol y represents the predicted value of price.

The equation of the regression line is y = 130x + 350. The symbol 'y' in this equation represents the predicted price of a hotel based on its rating.

The equation of a line in slope-intercept form is given by y = mx + b, where m is the slope of the line and b is the y-intercept. In this case, we have been provided with a slope of 130 and a y-intercept of 350. Therefore, the equation of the regression line is y = 130x + 350. This equation is the model, created using regression analysis, predicting the price of hotels based on their ratings.

The symbol y in this situation refers to the predicted value of price for a hotel depending on its rating. Hence, the correct answer for part (b) is 'C) The symbol y represents the predicted value of price'.

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Answer: 525 cycles per second.

Step-by-step explanation:

The equation for inverse variation between x and y is given by :-

[tex]x_1y_1=x_2y_2[/tex]       (1)

Given : The length of a violin string varies inversely with the frequency of its vibrations.

A violin string 14 inches long vibrates at a frequency of 450 cycles per second.

Let x =  length of a violin

y=  frequency of its vibrations

To find: The frequency of a 12 inch violin string.

Put [tex]x_1=14,\ x_2=12\\y_1=450,\ y_2=y[/tex] in equation (1) , we get

[tex](14)(450)=(12)(y)[/tex]  

Divide both sides by 12 , we get

[tex]y=\dfrac{(14)(450)}{12}=525[/tex]

Hence, the frequency of a 12 inch violin string = 525 cycles per second.

The frequency of a 12 inch violin string is approximately 388.57 cycles per second.

The length of a violin string varies inversely with the frequency of its vibrations. This means that as the length of the string decreases, the frequency of the vibrations increases, and vice versa. To find the frequency of a 12 inch violin string, we can set up the following proportion:

14 inches / 450 cycles per second = 12 inches / x cycles per second

To solve for x, we can cross multiply:

14 inches * x cycles per second = 12 inches * 450 cycles per second

x = (12 inches * 450 cycles per second) / 14 inches

Simplifying:

x = 388.57 cycles per second

Therefore, the frequency of a 12 inch violin string is approximately 388.57 cycles per second.

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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 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)

Tommy must clean at least 76 pools, in addition to mowing 41 lawns, to earn the $1500 he needs to buy a car.

Tommy has a summer job mowing lawns at $20 per lawn and cleaning pools at $9 per pool to save up for a car. To determine how many pools he needs to clean to reach his goal of $1500, we need to calculate his earnings from mowing lawns first and then see how much more he needs to earn from pool cleaning.

First, we calculate Tommy's lawn mowing earnings:
41 lawns × $20 per lawn = $820

After mowing lawns, Tommy will need an additional $1500 - $820 to buy the car. This difference is $680.

Next, to find out how many pools Tommy needs to clean, we divide the remaining amount by the amount he earns per pool:
$680 ÷ $9 per pool ≈ 75.56

Since Tommy can't clean a fraction of a pool, he will need to clean at least 76 pools to make enough money to buy the car. Therefore, the answer is 76 pools.

Answer:

What is the question?????

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.

(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:

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.

Answer:

x =  3.53 ft

y - 3.53 ft

z = 3.53 ft

Step-by-step explanation:

given details

volume = 44 ft^3

let cardboard dimension is x and y and height be z

we know that area of given cardboard without lid is given as

A = xy + 2xy + 2yz

xyz   = 44 ft^3

To minimize area we have

A = xy + 2x (44/xy) + 2y(44/xy)

A = xy + (44/y) + (44/x)

we have

[tex]Ax = y - \frac{44}{x^2}[/tex]

[tex]0 = yx^2 = 44[/tex]................1

[tex]Ay = x - \frac{44}{y^2}[/tex]

[tex]0 = x - \frac{44}{y^2}[/tex]

[tex]xy^2 = 44[/tex] ..............2

from 1 and 2

[tex]yx^2 = xy^2[/tex]

xy(y-x) = 0

xy = 0 or y = x

from geometry of probelem

x ≠ 0 and y ≠ 0

so y = x

x^3 = 44

x =  3.53 ft = y

z = 44/xy = 3.53

To find the dimensions of the box that requires the least amount of cardboard, we need to minimize the surface area of the box. Since it doesn't have a lid, the box will have an open top. Let's call the length of the box 'x' and the width and height 'y'. The dimensions of the box that requires the least amount of cardboard are x = 44 ft and y = 0 ft.

To find the dimensions of the box that requires the least amount of cardboard, we need to minimize the surface area of the box. Since it doesn't have a lid, the box will have an open top. Let's call the length of the box 'x' and the width and height 'y'.

The volume of the box is given as 44 ft3, so we have the equation x * y * y = 44.

To minimize the surface area, we can differentiate the surface area function with respect to x or y, set it equal to zero, and solve for the corresponding variable.

Let's differentiate the surface area function with respect to x to find the critical point:

0 = 2y2 + 2xy * dy/dx

Since the box has an open top, the length, x, cannot be zero. Therefore, we can solve the equation 2y2 + 2xy * dy/dx = 0 for dy/dx. This gives us:

dy/dx = -y/x

Now, we can substitute this into the equation for the surface area:

S = x * y2 + 2xy * dy/dx

Simplifying, we get:

S = x * y2 - 2y2

To find the critical point, we set the derivative equal to zero:

0 = y2 - 2y2

0 = -y2

Since y is squared, it cannot be negative. Therefore, the only possible critical point is when y is zero, which means the dimensions of the box are x = 44 ft and y = 0 ft.

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Answer:

m > 15

Step-by-step explanation:

Let m represent the number of miles ran each week by cross country team members

therefore, every week, m > 15.

Answer:

Part (A): it would take 2 seconds to reach maximum height of 264 foot.

Part (B): Ball will hit the ground in about 6.1 seconds

Part (C): S(0) represents the initial height of the base ball or the baseball was thrown  at a height of 200 ft.

Step-by-step explanation:

Consider the provided function.

[tex]s(t)=-16t^2+64t+200[/tex]

Part (A) After how many seconds does the ball reach it's maximum height? What is the maximum height?

The coefficient of t² is a negative number, so the graph of the above function is a downward parabola.

From the given function a=-16, b=64 and c=200

The downward parabola attain the maximum height at the x coordinate of the vertex. [tex]x=\frac{-b}{2a}[/tex]

Substitute the respectives.

[tex]x=\frac{-64}{2(-16)}=2[/tex]

Substitute x=2 in the provided equation.

[tex]s(t)=-16(2)^2+64(2)+200=264[/tex]

Hence, it would take 2 seconds to reach maximum height of 264 foot.

Part (B) How many seconds does it take until the ball finally hits the ground?

Substitute s(t)=0 in the provided equation.

[tex]-16t^2+64t+200=0[/tex]

Use the formula [tex]x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex] to find the solutions of the quadratic equation.

[tex]t=\frac{-64+\sqrt{64^2-4\left(-16\right)200}}{2\left(-16\right)}\\t=\pm\frac{4+\sqrt{66}}{2}\\t\approx-2.1\ or\ 6.1[/tex]

Reject the negative value as time can't be a negative number.

Hence, ball will hit the ground in about 6.1 seconds

Part (C) Find s(0) and describe what this means.

Substitute x=0 in the provide equation.

[tex]s(0)=-16(0)^2+64(0)+200[/tex]

[tex]s(0)=200[/tex]

S(0) represents the initial height of the base ball or the baseball was thrown  at a height of 200 ft.

Part (D) Use your results from parts (a) through (c) to graph the quadratic function.

Use the starting points (0,200), maximum point (2,264) and the end point (6.1,0) in order to draw the graph of the function.

Connect the points as shown in figure.

The required figure is shown below.

Answer:

E= 3/20

Step-by-step explanation:

Number of blue disk= 3

Number of green disk= 3

Number of orange disk= 4

Total number of disk = 3+3+34

= 10

Let B represent blue disk

Let G represent green disk

Let O represent orange disk

If three disk are selected at random, the possible outcome of one green disk and two orange disks are

OOG, OGO, GOO

Pr(OOG) = Pr(O1) * Pr(O2) * Pr(G3)

= 4/10 * 3/9 * 3/8

= 36/720

= 1/20

Pr(OGO) = Pr(O1) * Pr(G2) * Pr(O3)

= 4/10 * 3/9 * 3/8

= 36/720

= 1/20

Pr(GOOG) = Pr(G1) * Pr(O2) * Pr(O3)

= 4/10 * 4/9 * 3/8

= 36/720

= 1/20

Pr(total) = Pr(OOG) + Pr(OGO) + Pr(GOO)

= 1/20 + 1/20 + 1/20

= 3/20

Answer:

  v = 2√2i -2√2j

Step-by-step explanation:

  v = ||v||·cos(θ)i +||v||·sin(θ)j

  v = 4cos(315°)i +4sin(315°)j . . . . . . fill in the numbers

  v = 2√2i -2√2j . . . . . . . . . . . . . . . put in desired form

Answer:

a) d(x)=[tex]\sqrt{17x^{2} -12x+5}[/tex]

b)d'(x)=[tex]\frac{17x-6}{\sqrt{17x^{2} -12x+5} }[/tex]

c)The critical point is x=[tex]\frac{6}{17}[/tex]

d)Closest point is ([tex]\frac{6}{17}[/tex],[tex]\frac{7}{17}[/tex]

Step-by-step explanation:

We are given the line

[tex]y=4x-1[/tex]

Let a point Q([tex]x,y[/tex]) lie on the line.

Point P is given as P(2,0)

By distance formula, we have the distance D between any two points

A([tex]x_{1},y_{1}[/tex]) and B([tex]x_{2},y_{2}[/tex]) as

D=[tex]\sqrt{(x_{1}-x_{2})^2 + (y_{1}-y_2)^2}[/tex]

Thus,

d(x)=[tex]\sqrt{(x-2)^2+(y-0)^2}[/tex]

But we have, [tex]y=4x-1[/tex]

So,

d(x)=[tex]\sqrt{(x-2)^2+(4x-1)^2}[/tex]

Expanding,

d(x)=[tex]\sqrt{17x^2-12x+5}[/tex]  - - - (a)

Now,

d'(x)= [tex]\frac{\frac{d}{dx} (17x^2-12x+5)}{2(\sqrt{17x^2-12x+5}) }[/tex]

i.e.

d'(x)=[tex]\frac{17x-6}{\sqrt{17x^{2} -12x+5} }[/tex] - - - (b)

Now, the critical point is where d'(x)=0

⇒ [tex]\frac{17x-6}{\sqrt{17x^{2} -12x+5} }[/tex] =0

⇒[tex]x=\frac{6}{17}[/tex]    - - - (c)

Now,

The closest point on the given line to point P is the one for which d(x) is minimum i.e. d'(x)=0

⇒[tex]x=\frac{6}{17}[/tex]

as [tex]y=4x-1[/tex]

⇒y=[tex]\frac{7}{17}[/tex]

So, closest point is ([tex]\frac{6}{17},\frac{7}{17}[/tex])   - - -(d)

Answer:

x = 8; y = 2.5.

Step-by-step explanation:

As we know , when two complex numbers are equal their real as well as imaginary part are equal.

So comparing on both sides ,

2x = 16   and    5 = 2y

x = 8       and    y = 2.5.

So ,   x = 8; y = 2.5.

Answer: x = 2, y = -4 , z = 1

Step-by-step explanation:

-x + y + 3z = -3 - - - - - - - - - - 1

x - 2y - 2z = 8 - - - - - - -- - - - 2

3x - y - 4z = 6 - - - - - - - - - - - - - 3

Let us use the method of elimination

We would add equation 1 to equation 2. It becomes

-y+z= 5 - - - - - - - - - - - - - - -4

Multiply equation 2 by 3 and equation 3 by 1

3x - 6y -6z = 24- - - - - - - - - - 5

3x - y - 4z = 6 - - - - - - - - - - - - -6

Subtracting equation 6 from equation 5

-5y -2z = 18 - - - - - - - - - - 7

Substituting z = 5 + y into equation 7, it becomes

-5y -2(5+y) = 18

-5y -10-2y = 18

-5y -2y = 18+10

-7y = 28

y = 28/-7 = -4

z = 5 + y

z = 5 -4 = 1

We would substitute y = -4 and z = 1 into equation 2

It becomes

x - 2×-4 - 2×1 = 8

x+8-2 = 8

x +6 = 8

x = 8-6 = 2

x = 2, y = -4 , z = 1

Let us check by substituting the value into equation 1

-x + y + 3z = -3

-2-4+ 3= -3

-6 + 3 = -3

-3 = -3

Answer:

Option D.

Step-by-step explanation:

Given information: KLMN is parallelogram, K(7,7), L(5,3), M(1,1) and N(3,5).

Diagonals of a parallelogram bisect each other.

If diagonals of a parallelogram are perpendicular to each other then the parallelogram is a rhombus.

If a line passes through two points [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex], then the rate of change is

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

Slope of KM is

[tex]m_1=\frac{1-7}{1-7}=1[/tex]

Slope of LN is

[tex]m_2=\frac{5-3}{3-5}=-1[/tex]

The product of slopes of two perpendicular lines is -1.

Find the product of slopes.

[tex]m_1\cdot m_2=1\cdot (-1)=-1[/tex]

The product of slopes of KM and NL is -1. It means diagonals are perpendicular and KLMN is a rhombus.

Therefore, the correct option is D.