Job has to pay $9.00 admission for the skating rink and $2 per to rent rollerblades how many hours can he skate for $19
Answer:
5 hours
Step-by-step explanation:
Here we can create an equation to solve for the number of hours (x) this man can skate for.
Since he has to pay a $9 flat fee, that will be a constant value or our y-intercept in this case.
Our equation will be y = 2x + 9
Since we know his budget is $19, we can set our equation equal to 19, or replace y in this case.
So now we have 19 = 2x +9
Now we need to subtract 9 from both sides to get 10 = 2x
Then we need to divide both sides by 2 to get 5 = x
So this man can skate for 5 hours considering his budget
what is the area of this circle?
Answer:
113.1
Step-by-step explanation:
d= 2r
r = 6
area = [tex]\pi r^{2}[/tex] = 36 x [tex]\pi[/tex] = 113.09
Kendra was asked to find the number of students who planned to attend the evening symphony performance. She was told that 40 percent of the 50 students planned to attend. Kendra’s work is shown below.
Answer:
Kendra should of divided by 2
Step-by-step explanation:
Answer: B
Step-by-step explanation:
Recommendations
Main
Science
ULUSULUI
Sixth grade
FF.21 Relate volume andsurface area HVU
The surface area of this cube is 54 square meters. What is the volume?
cubic meters
Submit pls help
Answer:
27 cubic meters
Step-by-step explanation:
The surface area of this cube is 54 square meters.
Given a cube of side length s,
Surface Area [tex]=6s^2[/tex]Therefore:
[tex]6s^2=54\\s^2=54/6\\s^2=9\\s=\sqrt{9}\\ s=3$ meters[/tex]
Volume of a Cube of side length s [tex]=s^3[/tex]
Volume of the cube [tex]=3^3[/tex]
=27 cubic meters
The volume of the cube is 27 cubic meters.
What’s the correct answer for this? Which value is equivalent to sin 20?
Answer:
So the value of sin20 = 0.342
Step-by-step explanation:
The value proportional or equivalent to sin20 can be gotten from calculator value directly or from a graph.
But right niw the most suitable answer is the value for the calculator because its difficulty and complications have been reduced
So the value of sin20 = 0.342
Which linear function has the same y-intercept as the one that is represented by the graph?
On a coordinate plane, a line goes through points (3, 4) and (5, 0).
A 2-column table with 4 rows. Column 1 is labeled x with entries negative 3, negative 1, 1, 3. Column 2 is labeled y with entries negative 4, 2, 8, 14.
A 2-column table with 4 rows. Column 1 is labeled x with entries negative 4, negative 2, 2, 4. Column 2 is labeled y with entries negative 26, negative 18, negative 2, 6.
A 2-column table with 4 rows. Column 1 is labeled x with entries negative 5, negative 3, 3, 5. Column 2 is labeled y with entries negative 15, negative 11, 1, 5.
A 2-column table with 4 rows. Column 1 is labeled x with entries negative 6, negative 4, 4, 6. Column 2 is labeled y with entries negative 26, negative 14, 34, 46.
The linear function y = 6x +10 has the same y intercept as the one that is represented by the graph. Option D is correct.
What are functions?Functions is the relationship between sets of values. And shows their characteristic curve as well.
Here, the curve shows in the graph as y intercept of 10 on y axis, is linear function. From options we have only one option that show the same y intercept y = 10.
That linear function ⇒ y =6x+18 (obtained from option D).
Thus, the linear function y = 6x +10 has the same y intercept as the one that is represented by the graph.
learn more about function here:
brainly.com/question/21145944
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Answer:
The linear function y = 6x +10 has the same y intercept as the one that is represented by the graph. Option D is correct.
Step-by-step explanation:
John borrowed $150,500 to start up his business. He borrowed the money at a 7.75% simple
interest rate for 2 years. How much does John have to pay back at the end of two years
Answer:
John have to pay back $23,327.50 at the end of two years
Step-by-step explanation:
Simple Interest (I) = (PRT) ÷ 100
where P = Principal, R = Rate, T= Time
I = ($150500 × 7.75 × 2) ÷ 100
I = ($2332750) ÷ 100
I = $23,327.50
Find the value of m.
C
1190
mº
A
B
A 61
B 30°
C 45
D 41°
Answer:
b
Step-by-step explanation:
Which account has the lowest minimum balance requirement?
Bank Account Terms and Conditions
A
B
C
D
$500
$250
$300
Minimum
Balance
Interest
$100
0%
0.02%
0%
Overdraft
0.2%
$0 if linked
with savings
account or
140 per
overdraft
$35 per
$25 per
$25 per
Fees
overdraft
Overdraft
overdraft
ATM
Us age/Fees
$0 for all
ATMs
$0 for ABC
bank ATMS
53 for non-
ABC bank
ATMs
$4 per
transaction
$3 per
transaction
Answer:
Bank account D
Step-by-step explanation:
In order to maintain bank account A,the minimum balance requirement is $500 which is the highest minimum balance requirement when compared to other bank accounts.
Also, in order to maintain bank account D,the least balance required is $100,this is the lowest when compared with other bank accounts available,hence it is the correct answer in this case since the requirement was about the bank account with the lowest minimum balance
Answer
Bank account D
what is the minimum amount of real solutions a quintic polynomial can have
Answer:
1
Step-by-step explanation:
I'm not sure if this is right, but I think it's 1. If you think about the simplest quintic polynomial, x^5, and set it to any real number, you'll always get 1 solution. I don't think it's possible to have 0 or no real solutions. Hope this helps!
C-Spec, Inc., is attempting to determine whether an existing machine is capable of milling an engine part that has a key specification of {eq}4 \pm 0.003 {/eq} inches. After a trial run on this machine, C-Spec has determined that the machine has a sample mean of 4.001 inches with a standard deviation of 0.002 inch.
a. Calculate the {eq}C_{pk} {/eq} for this machine
b. Should C-Spec use this machine to produce this part? Why?
Answer:
Step-by-step explanation:
Given that :
sample mean = 4.001 inches
sample standard deviation = 0.002 inch
a.
[tex]C_{pk} = min [\dfrac{(USL- \bar X)}{(3*std \ dev)} \ ; \ \dfrac{ (\bar X- LSL)}{(3*std \ dev)}][/tex]
specification = [tex]4 \pm 0.003[/tex]
Upper specification limit USL = 4 + 0.003 = 4.003
Lower specification limit LSL = 4 - 0.003 = 3.997
[tex]\dfrac{ (\bar X- LSL)}{(3*std \ dev)} = \dfrac{ (4.001-3.997)}{(3*0.002)}[/tex] [tex]= 0.667[/tex]
[tex]\dfrac{(USL- \bar X)}{(3*std \ dev)} =\dfrac{(4.003-4.001)}{(3*0.002)}[/tex][tex]= 0.333[/tex]
Thus ;
[tex]C_{pk} =[/tex][tex]min (0.333 , 0.667)[/tex] = 0.333
[tex]C_{pk}[/tex] is a measure of closeness to one's target and the consistency around the average performance.
b) No, C - spec should not use this machine to produce this part because [tex]C_{pk}[/tex] < 1.33 which typical means that the part is not fully capable of hitting the target specification on a consistent basis .
A movie theater has a seating capacity of 253. The theater charges $5.00 for children, $7.00 for students, and $12.00 of adults. There are half as many adults as there are children. If the total ticket sales was $ 1828 on a sold out night, how many children, students, and adults attended?
children attended=
adults attended=
students attended=
Answer:
children attended= 114
adults attended= 57
students attended= 82
Step-by-step explanation:
children attended= c
adults attended= a
students attended= s
c+s+a = 253
5c + 7s + 12a = 1828
c=2a
2a + s + a =253
5*2a + 7s + 12a = 1828
3a + s = 253
22a + 7s = 1828
s = 253 - 3a
22a + 7(253 - 3a) = 1828
22a - 21a + 7*253 = 1828
a = 1828 - 7*253
a = 57
s = 253 - 3a = 253 - 3*57 = 82
s = 82
c=2a=2*57= 114
c=114
Which two sentences contain incorrect punctuation?
One question that society finds problematic to answer is what energy sources will people utilize when the earth is drained of natural resources such as coal or oil.
Upon discovering the errors in the film —the abusive remarks—the movie release was immediately postponed and was sent back to the editors for correction.
Maintaining hygiene is a good way to keep illnesses at bay; this can be done by washing hands, brushing teeth, and bathing regularly.
The American writer and philosopher, Henry David Thoreau comments that by simplifying one's life, "the laws of the universe will appear less complex."
The adrenaline produced in our body during a confrontation leads to two possible responses by the person run away to safety, or stand one's ground and fight.
Reset
Answer:
D). The American writer and philosopher, Henry David Thoreau comments that by simplifying one's life, "the laws of the universe will appear less complex."
E). The adrenaline produced in our body during a confrontation leads to two possible responses by the person run away to safety, or stand one's ground and fight.
Explanation:
Punctuation marks are elucidated as the signs and symbols that function to enhance the readability of the sentences and indicate the grammatical construction of the readers. It assists them in understanding the meaning of the sentence more clearly and comprehensively.
In the given sentences, the last two sentence are incorrectly punctuated as they both wrongly employ comma(,) in it. Option D inappropriately punctuates comma(,) in the complex sentence while the last option falsely puts comma before conjunction 'or.' Thus, options D and E are the correct answers.
An advertising executive claims that there is a difference in the mean household income for credit cardholders of Visa Gold and of MasterCard Gold. A random survey of 1010 Visa Gold cardholders resulted in a mean household income of $50,930$ 50,930 with a standard deviation of $9500$ 9500. A random survey of 66 MasterCard Gold cardholders resulted in a mean household income of $45,720$ 45,720 with a standard deviation of $9800$ 9800. Is there enough evidence to support the executive's claim? Let μ1μ1 be the true mean household income for Visa Gold cardholders and μ2μ2 be the true mean household income for MasterCard Gold cardholders. Use a significance level of α=0.01α=0.01 for the test. Assume that the population variances are not equal and that the two populations are normally distributed.
Answer:
We conclude that there is no difference in the mean household income for credit cardholders of Visa Gold and of MasterCard Gold.
Step-by-step explanation:
We are given that a random survey of 10 Visa Gold cardholders resulted in a mean household income of $50,930 with a standard deviation of $9500.
A random survey of 66 MasterCard Gold cardholders resulted in a mean household income of $45,720 with a standard deviation of $9800.
Let [tex]\mu_1[/tex] = true mean household income for Visa Gold cardholders.
[tex]\mu_2[/tex] = true mean household income for MasterCard Gold cardholders.
So, Null Hypothesis, [tex]H_0[/tex] : [tex]\mu_1=\mu_2[/tex] {means that there is no difference in the mean household income for credit cardholders of Visa Gold and of MasterCard Gold}
Alternate Hypothesis, [tex]H_A[/tex] : [tex]\mu_1\neq \mu_2[/tex] {means that there is a difference in the mean household income for credit cardholders of Visa Gold and of MasterCard Gold}
The test statistics that would be used here Two-sample t test statistics as we don't know about the population standard deviations;
T.S. = [tex]\frac{(\bar X_1-\bar X_2)-(\mu_1-\mu_2)}{s_p\sqrt{\frac{1}{n_1} +\frac{1}{n_2} } }[/tex] ~ [tex]t__n__1-_n__2-2[/tex]
where, [tex]\bar X_1[/tex] = sample mean household income for Visa Gold cardholders = $50,930
[tex]\bar X_2[/tex] = sample mean household income for MasterCard Gold cardholders = $45,720
[tex]s_1[/tex] = sample standard deviation for Visa Gold cardholders = $9,500
[tex]s_2[/tex] = sample standard deviation for MasterCard Gold cardholders = $9,800
[tex]n_1[/tex] = sample of Visa Gold cardholders = 10
[tex]n_2[/tex] = sample of MasterCard Gold cardholders = 6
Also, [tex]s_p=\sqrt{\frac{(n_1-1)s_1^{2} +(n_2-1)s_2^{2} }{n_1+n_2-2} }[/tex] = [tex]\sqrt{\frac{(10-1)\times 9,500^{2} +(6-1)\times 9.800^{2} }{10+6-2} }[/tex] = 9,608.22
So, the test statistics = [tex]\frac{(50,930-45,720)-(0)}{9,608.22 \times \sqrt{\frac{1}{10} +\frac{1}{6} } }[/tex] ~ [tex]t_1_4[/tex]
= 1.05
The value of t test statistics is 1.05.
Now, at 0.01 significance level the t table gives critical values of -2.977 and 2.977 at 14 degree of freedom for two-tailed test.
Since our test statistic lies within the range of critical values of t, so we have insufficient evidence to reject our null hypothesis as it will not fall in the rejection region due to which we fail to reject our null hypothesis.
Therefore, we conclude that there is no difference in the mean household income for credit cardholders of Visa Gold and of MasterCard Gold.
As quality control manager at a raisin manufacturing and packaging plant, you want to ensure that all the boxes of raisins you sell are comparable, with 30 raisins in each box (assume the population average number of raisins per box is 30). In the plant, raisins are poured into boxes until the box reaches its sale weight. To determine whether a similar number of raisins are poured into each box, you randomly sample 36 boxes about to leave the plant and count the number of raisins in each. In this sample, you find the mean number of raisins in each box to be 28.05, with a standard deviation of 3.87.
1. Perform a two-tailed, single sample t test (p<.05) to determine whether the average number of raisins per box differs from the expected 30.
2. What type of statistical analysis should be run to address this question?3. What are the assumptions of your chosen statistical test?4. Identify the null and alternative/research hypotheses in both words and symbols.
Answer:
Step-by-step explanation:
The population average number of raisins per box is 30.
The type of statistical analysis to be run here is the inferential statistical analysis and its assumptions include normality, linearity and equality of variance.
Null hypothesis: Average number of raisins per box is 30: u = 30
Alternative hypothesis: Average number of raisins per box is not 30: u ≠ 30
To perform a two tailed single sample test, we have
z statistic = (the sample mean - hypothesized mean) / (SD/√n)
sample mean = 28.05, SD = 3.87 hypothesized mean = 30 n = 36
Thus, we have 28.05 - 30 / (3.87/√36)
= -1.95 / (3.87/6)
= -1.95 / 0.645
= -3.023
we say this is a left tailed test and use the negative z table
where -3.02 = 0.0013
To determine whether the average number of raisins per box differs from the expected 30.
Using p < 0.05, since the observed value 0.0013 is less than 0.05, we conclude the the average number of raisins per box differs from the expected 30.
Write what the expressions below best represent within the context of the word problem.
Mark had 3 times as many quarters as nickels. He had $1.60 in all. How many nickels and how many quarters did Mark
have?
x represents
3x represents
TURN IT IN
NEXT QUESTION
ASK FOR HELP
Answer:
2 nickels
6 quarters
Step-by-step explanation:
Let quarters be 3x and nickels be $x
if Mark had 3 times as many quarters as nickels, and if he had $1.60 in all, than
( 0.25)3x + 0.05x=1.6
0.75x+ 0.05x=1.6
0.8x=1.6
x=1.6/0.8
x=2 nickels
and 3 (2)=6 quarters
Answer:
There are 4 quarters and 12 nickels.
Step-by-step explanation:
A quarter is equal to $0.25.
A nickel is equal to $0.05.
Let the number of nickels be y.
Let the number of quarters be x.
Mark had 3 times as many quarters as nickels. This means that:
y = 3x
3x represents the number of nickels.
We are told that he had $1.60 in all. This means that:
0.25x + 0.05y = 1.60
=> 0.25x + 0.05 * 3x = 1.60
0.25x + 0.15x = 1.60
0.40x = 1.60
=> x = 1.6 / 0.4 = 4
Therefore:
y = 3 * 4 = 12
There are 4 quarters and 12 nickels.
Leg of the right triangle is three units in the hypotensive is seven units what is the length in units of the other leg of the triangle
Answer:
2[tex]\sqrt{10}[/tex]
Step-by-step explanation:
You can get this by using the Pythagorean Theorem
a^2 + b^2 = c^2
where a and b are the legs and c is the hypotenuse
3^2 + b^2 = 7^2
9 + b^2 = 49
b^2 = 40
sqrt of both sides
b = [tex]\sqrt{40}[/tex]
b = 2[tex]\sqrt{10}[/tex]
if this helps please give me brainliest
V3
The angle 01 is located in Quadrant III, and sin(01) =
2
What is the value of cos(01)?
Express your answer exactly.
Question:
The angle [tex] \theta_1[/tex] is located in Quadrant III, and [tex] sin(\theta_1) = - \sqrt{\frac{3}{2}} [/tex].
What is the value of [tex] cos(\theta_1)[/tex] ?
Express your answer exactly. [tex] cos(\theta1) = [/tex] ________
Answer:
[tex] cos(\theta_1) = - \frac{1}{2} [/tex]
Step-by-step explanation:
Given:
[tex] sin(\theta_1) = - \sqrt{\frac{3}{2}} [/tex]
Since the angle [tex] \theta_1[/tex] is located in Quadrant III, we have:
[tex] sin(\theta_1) = sin(\pi + \frac{\pi}{3}) [/tex]
[tex] \theta_1 = \pi + \frac{\pi}{3}[/tex]
[tex] \theta_1 = \frac{4 \pi}{3}[/tex]
Thus, [tex] cos(\theta_1) [/tex] =
[tex] cos(\theta_1) = cos(\pi + \frac{\pi}{3}) [/tex]
[tex] cos(\theta_1) = -cos(\frac{\pi}{3}) [/tex]
[tex] cos(\theta_1) = - \frac{1}{2} [/tex]
The scale for a drawing is 1 centimeter to 5 meters. If the actual object is 35 meters, the drawing is _____ centimeters long.
30
175
7
40
Answer:
7
Step-by-step explanation:
Answer:
the answer is 7
Step-by-step explanation:
Which number is the largest
Step-by-step explanation:
0.153 is the largest number
Answer:
I believe 0.153 to be the greatest
Step-by-step explanation:
0.153
15%= 0.15
1/12= 0.0833333
1/7= 0.1428571429
Which game isn’t fair for two players to play?
Answer:
cube game
Step-by-step explanation:
one of them wins every time
A number cube is rolled. Player 1 gets a point for any number that is 3 or higher, and player 2 gets a point for any number less than 3
If a number cube is rolled and Player 1 gets a point for any number that is 3 or higher and Player 2 gets a point for any number less than 3 it wouldn't be fair to Player 2 because Player 1 has a higher chance of getting points because he/she has more opportunities.
I hope this is correct
What’s the correct answer for this?
Answer:
Step-by-step explanation:
The drama club is selling short-sleeved shirts for $5 each, and long-sleeved shirts for $10 each. They hope to sell all of the shirts they ordered, to earn a total of $1,750. After the first week of the fundraiser, they sold StartFraction one-third EndFraction of the short-sleeved shirts and StartFraction one-half EndFraction of the long-sleeved shirts, for a total of 100 shirts. How many short-sleeved shirts were ordered?
How many long-sleeved shirts were ordered?
Answer:
150 short-sleeved shirts100 long-sleeved shirtsStep-by-step explanation:
Let S and L represent the numbers of short- and long-sleeved shirts ordered.
The expected earnings were ...
5S +10L = 1750
The number sold was ...
(1/3)S + (1/2)L = 100
We can multiply the first equation by 1/5 and the second equation by 6 to put the equations into standard form:
S +2L = 3502S +3L = 600Subtracting the second equation from twice the first gives ...
2(S +2L) -(2S +3L) = 2(350) -(600)
L = 100 . . . . . . simplify
2S +3(100) = 600
S = 150
150 short-sleeved and 100 long-sleeved shirts were ordered.
Answer:
b
Step-by-step explanation:
i did it on edge
Pythagorean Theorem Word
Problems
You start driving north for 32 miles, turn
right, and drive east for another 24 miles,
At the end of driving, what is your straight
line distance from your starting point?
Answer:
40 miles
Step-by-step explanation:
In the attached diagram, Point A is the starting point and C is the end point. We want to determine the distance from A to C.
The path driven forms a right triangle in which AC is the hypotenuse.
We therefore use the Pythagorean Theorem to solve for the AC.
Pythagorean Theorem: [tex]Hypotenuse^2=Opposite^2+Adjacent^2[/tex]
[tex]|AC|^2=32^2+24^2\\|AC|^2=1600\\|AC|=\sqrt{1600}\\ |AC|=40$ miles[/tex]
The straight line distance from the starting point is 40 miles.
tbe ages of jerry mile and rawl total 80 years jerry is twice as old as mile and mile is three times as old as rawl how old is jerry????
Answer:
rawl- 8 years old
mile- 24 years old
jerry- years old
Step-by-step explanation:
If the gardener can fill the border with 30 flowers planted 12 inches apart, how far apart should the gardener plant 36 flowers instead?
Answer:
10 inches
Step-by-step explanation:
This inversely proportional function, the more flowers - the less distance.
Flowers = k/distance
30 = k/12
k = 30*12 = 360
Flowers = 360/distance
New number of flowers and new distance:
Flowers = 360/distance
36 = 360/distance
distance = 360/36= 10 (inches)
The owner of a small deli is trying to decide whether to discontinue selling magazines. He suspects that only 10% of his customers buy a magazine and he thinks that he might be able to use the display space to sell something more profitable. Before making a final decision, he decides that for one day he will keep track of the number of customers that buy a magazine. Assuming his suspicion that 10% of his customers buy a magazine is correct, what is the probability that exactly 5 out of the first 13 customers buy a magazine?
Answer:
0.55% probability that exactly 5 out of the first 13 customers buy a magazine
Step-by-step explanation:
For each customer, there are only two possible outcomes. Either they buy a magazine, or they do not. The probability of a customer buying a magazine is independent of other customers. So we use the binomial probability distribution to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
10% of his customers buy a magazine
This means that [tex]P = 0.1[/tex]
What is the probability that exactly 5 out of the first 13 customers buy a magazine?
This is P(X = 5) when n = 13. So
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 5) = C_{13,5}.(0.1)^{5}.(0.9)^{8} = 0.0055[/tex]
0.55% probability that exactly 5 out of the first 13 customers buy a magazine
Answer:
[tex] P(X=5) = (13C5) (0.1)^5 (1-0.1)^{13-5}= 0.00554[/tex]
So then he probability that exactly 5 out of the first 13 customers buy a magazine is 0.0554
Step-by-step explanation:
Let X the random variable of interest "number of customers that buy a magazine", on this case we can model the variable of interest with this distribution:
[tex]X \sim Binom(n=13, p=0.1)[/tex]
The probability mass function for the Binomial distribution is given as:
[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]
Where (nCx) means combinatory and it's given by this formula:
[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]
We want to find this probability:
[tex] P(X=5)[/tex]
And using the probability mass function we got:
[tex] P(X=5) = (13C5) (0.1)^5 (1-0.1)^{13-5}= 0.00554[/tex]
So then he probability that exactly 5 out of the first 13 customers buy a magazine is 0.0554
What is the value of x in the product of powers below? 6 Superscript 9 times 6 Superscript x = 6 squared Negative 11 negative 7 7 11 NEED IT QUICK
Answer:
-7
Step-by-step explanation:
(6^9)(6^x) = 6^2
Taking logarithms to the base 6, we have ...
9 + x = 2
x = -7 . . . . . subtract 9
Answer:
-7
Step-by-step explanation:
Please help. I will make brainlest award for the correct answers
Step-by-step explanation:
HOPE THIS HELP YOU !!! :))) y = [ 1 and u = [ 6
3 ] 2 ]
Write y as the sum of a vector in Span {u} and a vector orthogonal to u.
y = z = ___ + ___. (Type an integer or simplified fraction for each matrix element. List the terms in the same order as they appear in the original list.)
Answer:
Take [tex]b = \frac{3}{10}(6,2)[/tex] as the vector in Span(u) and [tex]w = \frac{4}{5}(-1,3)[/tex] as the orthogonal vector to u
Step-by-step explanation:
Given vectors a,b we can calculate the projection of a onto b by the formula
[tex] \frac{a\cdot b}{b\cdot b}b[/tex]. where [tex]\cdot[/tex] is the dot product of vectors.
For this example, we will take y = [1,3] and u = [6,2].
Let b be the projection of y onto u. That is
[tex] b= \frac{y\cdot u}{u\cdot u}u = \frac{1\cdot 6+ 3\cdot 2}{6\cdot 6+2\cdot 2}(6,2) = \frac{3}{10}(6,2)[/tex]
Now, we want to find an orthogonal vector w such that b+w = y. From this equation, we can find w by taking w = y-b. Then
[tex] w = (1,3)-\frac{3}{10}(6,2) = \frac{4}{5}(-1,3)[/tex]
We must check that w and u are orthogonal. Note that if the vectors a,b are orthogonal, then the vectors a, kb are also orthogonal for every non-zero value of k. So, to check that a,kb are orthogonal, it suffices to check that a,b are orthogonal. We know that two vectors are orthogonal if and only if their dot product is zero. In our case, take the vector (6,2) and (-1,3). Note that
[tex](6,2)\cdot(-1,3) = -6+2\cdot3 = 0[/tex]
So, by the previous analysis, w and u are orthogonal.