Answer:
a) [tex]P(\bar X>81.5)=1-0.933=0.067[/tex]
b) [tex]P(\bar X<69.5)=0.0062[/tex]
c) [tex]P(73.4<\bar X<84.05)=0.8755[/tex]
Step-by-step explanation:
1) Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Let X the random variable that represent interest on this case, and for this case we know the distribution for X is given by:
[tex]X \sim N(\mu=77,\sigma=27)[/tex]
And let [tex]\bar X[/tex] represent the sample mean, the distribution for the sample mean is given by:
[tex]\bar X \sim N(\mu,\frac{\sigma}{\sqrt{n}})[/tex]
On this case [tex]\bar X \sim N(77,\frac{27}{\sqrt{81}})[/tex]
Part a
We want this probability:
[tex]P(\bar X>81.5)=1-P(\bar X<81.5)[/tex]
The best way to solve this problem is using the normal standard distribution and the z score given by:
[tex]z=\frac{x-\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
If we apply this formula to our probability we got this:
[tex]P(\bar X >81.5)=1-P(Z<\frac{81.5-77}{\frac{27}{\sqrt{81}}})=1-P(Z<1.5)[/tex]
[tex]P(\bar X>81.5)=1-0.933=0.067[/tex]
Part b
We want this probability:
[tex]P(\bar X\leq 69.5)[/tex]
If we apply the formula for the z score to our probability we got this:
[tex]P(\bar X \leq 69.5)=P(Z\leq \frac{69.5-77}{\frac{27}{\sqrt{81}}})=P(Z<-2.5)[/tex]
[tex]P(\bar X\leq 69.5)=0.0062[/tex]
Part c
We are interested on this probability
[tex]P(73.4<\bar X<84.05)[/tex]
If we apply the Z score formula to our probability we got this:
[tex]P(73.4<\bar X<84.05)=P(\frac{73.4-\mu}{\frac{\sigma}{\sqrt{n}}}<\frac{X-\mu}{\frac{\sigma}{\sqrt{n}}}<\frac{84.05-\mu}{\frac{\sigma}{\sqrt{n}}})[/tex]
[tex]=P(\frac{73.4-77}{\frac{27}{\sqrt{81}}}<Z<\frac{84.05-77}{\frac{27}{\sqrt{81}}})=P(-1.2<z<2.35)[/tex]
And we can find this probability on this way:
[tex]P(-1.2<z<2.35)=P(z<2.35)-P(z<-1.2)[/tex]
And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.
[tex]P(-1.2<z<2.35)=P(z<2.35)-P(z<-1.2)=0.9906-0.1151=0.8755[/tex]
Write the vector v in terms of i and j whose magnitude ||v|| and direction 0 are given. ||v|| = 4, 0 = 315 degree v= (Type your answer in terms of i and j. Type an exact answer using radicals as needed.
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
Identify the values of x and y that make the equation 2x + 5i = 16 + (2y)i true. HELP ASAP!!
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.
A consumer products company relies on direct mail marketing pieces as a major component of its advertising campaigns. The company has three different designs for a new brochure and want to evaluate their effectiveness, as there are substantial differences in costs between the three designs. The company decides to test the three designs by mailing 5,000 samples of each to potential customers in four different regions of the country. Since there are known regional differences in the customer base, regions are considered as blocks.
Answer:
What is the question?????
Recall the equation for a circle with center ( h , k ) and radius r . At what point in the first quadrant does the line with equation y = 2.5 x + 5 intersect the circle with radius 5 and center (0, 5)?
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.
Evaluate the function f(x)=10-x for the domain {-2, 0, 2}
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]
For [tex]x = -2:[/tex][tex]f (-2) = 10 - (- 2) = 10 + 2 = 12[/tex]
For [tex]x = 0[/tex]:[tex]f (0) = 10-0 = 10[/tex]
For [tex]x = 2[/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]
A cardboard box without a lid is to be made with a volume of 44 ft3. Find the dimensions of the box that requires the least amount of cardboard.
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.
Explanation: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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I have a box of replacement parts that I need to choose one from and place into my tortilla making machine. These parts come in two types: Type 1 has a failure rate of .4, and Type 2 has a failure rate of .75. I also know that, in that box, 30% of the replacement parts are of Type 1. There's no other way to tell the two types apart from one another.
I choose a replacement part from the box at random, a place it into the machine and I use the machine to make 30 tortillas; of these, I find that 16 of the tortillas it created are square (failures).
Question: What is the probability that I picked a Type 1 part?
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
Tommy mows lawns and cleans pools during the summer he earns $20 per line and nine dollars per pool he needs $1500 to buy a car from his friend Tommy plans to mow 41 months this summer how many pools messy clean to earn at least $1500
Tommy must clean at least 76 pools, in addition to mowing 41 lawns, to earn the $1500 he needs to buy a car.
Explanation: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.
These marbles are placed in a bag and two of them are randomly drawn. What is the probability of drawing two pink marbles if the first one is placed back in the bag before the second draw? Give your answer as a ratio, reduced to simplest terms. [?] Hint: Multiply the probability of the 1st Event by the probability of the 2nd Event to get your answer. Enter Corporation. All Rights Reserved.
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.
Explanation: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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A bag contains only 3 blue disks, 3 green disks, and 4 orange disks. If 3 disks are selected at random from the bag, what is the probability that 1 of the disks will be green and 2 of the disks will be orangeA) 1/30
B) 1/20
C) 3/40
D) 1/10
E) 3/20
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
Every week, cross country team members run more than 15 miles. Write an inequality that represents this situation. Let m represent the number of miles ran each week by cross country team members.
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.
Need help answering this one! -geometry
Answer:
Step-by-step explanation:
A rectangle has sides in the ratio 1:3 a)Write down the length of the longer side of the length of the shorter side is: (i) 1 cm (ii) 2 cm (iii) 5 cm
(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:
The length of the shorter side is 1 cmThe length of the shorter side is 2 cmThe length of the shorter side is 5 cmLet 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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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. Find the frequency of a 12 inch violin string.
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.
Explanation: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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please help I already did the hint part but I’m not sure how to get the width or I don’t even know
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
In a random sample of 81 audited estate tax returns, it was determined that the mean amount of additional tax owed was $3408 with a standard deviation of $2565. Construct and interpret a 90% confidence interval for the mean additional amount of tax owed for estate tax returns. The lower bound is $ nothing. (Round to the nearest cent as needed.) The upper bound is $ nothing. (Round to the nearest cent as needed.) Interpret a 90% confidence interval for the mean additional amount of tax owed for estate tax returns. Choose the correct answer below.
a. One can be 90% confident that the mean additional tax owed is less than the lower bound
b. One can be 90% confident that the mean additional tax owed is between the lower and upper bounds
c. One can be 90% confident that the mean additional tax owed is greater than the upper bound.
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.
Explanation: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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Consider the line y=4 x -1 and the point P=(2,0). (a) Write the formula for a function d(x) that describes the distance between the point P and a point (x,y) on the line. You final answer should only involve the variable x. Then d(x) = √(4−x)2(4x−1)2 (b) d'(x)= (c) The critical number is x= . (d) The closest point on the line to P is ( , ).
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)
The blue team walks 5 miles the first hour and 2 miles an hour after that. How far will they walk in 4 hours?
11 miles
9 miles
7 miles
5 miles
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.
Let V denote the set of ordered triples (x, y, z) and define addition in V as in
R3.For each of the following definitions of scalar multiplication, decide whether V is a vector space.
(a) a(x,y,z)=(ax,y,az)
(b) a(x,y,z)=(ax,0,az)
(c) a(x,y,z)=(0,0,0)
(d) a(x,y,z)=(2ax,2ay,2az)
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.
On a coordinate plane, parallelogram K L M N shown. Point K is at (7, 7), point L is at (5, 3), point M is at (1, 1), and point N is at (3, 5). Which statement proves that parallelogram KLMN is a rhombus? a. The midpoint of both diagonals is (4, 4). b. The length of KM is [tex]\sqrt{72}[/tex] and the length of NL is [tex]\sqrt{8}[/tex]. c. The slopes of LM and KN are both One-half and NK = ML = [tex]\sqrt{20}[/tex]. d. The slope of KM is 1 and the slope of NL is –1.
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.
Kevin drove from A to B at a constant speed of 60 mph, turned immediately around, and returned at a constant speed of 80 mph. Exactly 4 hours before the end of his trip, he was still approaching B, only 15 miles away from it. What is the distance between A and B?
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;
Time: 15 miles / 60 miles per hour = 1/4 hour (15 minutes).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:
Distance: time x speed: 3.75 hours x 80 mph = 300 milesWhat is the domain of the relation below?
Is the relation a function?
{(-5, 2), (7, 7), (3,6), (1, 7)}
A domain: {-5, 1, 7}; Yes, it is a function
B domain: {-5, 1, 3, 7}; No, it is not a function
C domain: {-5, 1, 3, 7); Yes, it is a function
D domain: {2, 6, 7); Yes, it is a function
E domain: {2, 6, 7}; No, it is not a function
Option C
Domain: {-5, 1, 3, 7); Yes, it is a function
Solution: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
At Polynomials Pizza, a large cheese pizza costs $6.50. Each additional topping for the pizza costs $1.45. The total for a large pizza that Brandon ordered was $13.75. PART A Define the variable t = PART B Write the equation. PART C How many toppings did Brandon have on his pizza? Enter answer
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
Can someone answer this question correctly it's my test please don't answer if you don't understand please show work I need it today thank you
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)
The profit function p(x) of a tour operator is modeled by p(x) = −2x^2 + 700x − 10000, where x is the average number of tours he arranges per day. What is the range of the average number of tours he must arrange per day to earn a monthly profit of at least $50,000?
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.
Hey guys, how would i write this? Thank youuuu
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.
Different hotels in a certain area are randomly selected, and their ratings and prices were obtained online. Using technology, with x representing the ratings and y representing price, we find that the regression equation has a slope of 130 and a y-intercept of 350. Complete parts (a) and (b) below.
a. What is the equation of the regression line? Select the correct choice below and fill in the answer boxes to complete your choice.
A ○ C.
b. What does the symbol y represent?
A. The symbol y represents the average price of hotels in the area.
B. The symbol ý represents the amount that price increases with a 1-point increase in rating.
C) The symbol y represents the predicted value of price.
D. The symbol y represents the expected price when the hotel's rating is 0.
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.
Explanation: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'.
Learn more about Regression Analysis here:https://brainly.com/question/32707297
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Solve for x,z,y
-x + y + 3z = -3
X - 2y - 2z = 8
3x - y - 4z = 6
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
A restaurant prepared and served 1540 meals for students in the month of May. During May, the cooks worked 9 days (of 8 hours each). The restaurant used 13 direct labor employees. Determine the hourly labor productivity of the meals.
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
A person standing cloes to the edge on the top of a 200-foot building throws a baseball vertically upward. The quadratic functions(t)=-16t^2+64t+200models the ball's height above the ground, s(t), in feet, t seconds after it was thrown.A) After how many seconds does the ball reach it's maximum height? What is the maximum height?B) How many seconds does it take until the ball finally hits the ground?C) Find s(0) and describe what this means. D) Use your res ults from parts (a) through (c) to graph the quadratic function . Begin the graph with t = 0 and end with the value oft for which the ball hits the ground.
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.