The basis for W is {x - 1, x^2 - 1}, and since there are two linearly independent polynomials, the dimension of W is 2, which confirms our conjecture.
a) To show that the set W of polynomials in P2 such that p(1) = 0 is a subspace of P2, we need to verify the three conditions for a subset to be a subspace:
The zero polynomial, denoted as 0, must be in W:
Let p(x) = ax^2 + bx + c be the zero polynomial. For p(1) = 0 to hold, we have:
p(1) = a(1)^2 + b(1) + c = a + b + c = 0.
Since a, b, and c are arbitrary coefficients, we can choose them such that a + b + c = 0. Thus, the zero polynomial is in W.
W must be closed under addition:
Let p(x) and q(x) be polynomials in W. We need to show that their sum, p(x) + q(x), is also in W.
Since p(1) = q(1) = 0, we have:
(p + q)(1) = p(1) + q(1) = 0 + 0 = 0.
Therefore, p(x) + q(x) satisfies the condition p(1) = 0 and is in W.
W must be closed under scalar multiplication:
Let p(x) be a polynomial in W and c be a scalar. We need to show that the scalar multiple, cp(x), is also in W.
Since p(1) = 0, we have:
(cp)(1) = c * p(1) = c * 0 = 0.
Thus, cp(x) satisfies the condition p(1) = 0 and is in W.
Since W satisfies all three conditions, it is indeed a subspace of P2.
b) Conjecture about the dimension of W:
The dimension of W can be conjectured by considering the degree of freedom available in constructing polynomials that satisfy p(1) = 0. Since p(1) = 0 implies that the constant term of the polynomial is zero, we have one degree of freedom for choosing the coefficients of x and x^2. Therefore, we can conjecture that the dimension of W is 2.
c) Confirming the conjecture by finding the basis for W:
To find the basis for W, we need to determine two linearly independent polynomials in W. We can construct polynomials as follows:
Let p1(x) = x - 1.
Let p2(x) = x^2 - 1.
To confirm that they are in W, we evaluate them at x = 1:
p1(1) = (1) - 1 = 0.
p2(1) = (1)^2 - 1 = 0.
Both p1(x) and p2(x) satisfy the condition p(1) = 0, and they are linearly independent because they have different powers of x.
Therefore, the basis for W is {x - 1, x^2 - 1}, and since there are two linearly independent polynomials, the dimension of W is 2, which confirms our conjecture.
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On their website, a property management company states that the mean monthly rent for apartments on the east side of town is $700. A researcher for a consumer advocacy group believes that, due to the construction of newer apartment complexes in neighboring towns, the mean monthly rent on the east side, , is now lower. A random sample of 12 monthly rents for apartments on the east side has a mean of $687, with a standard deviation of $17. Assume that current monthly rents for apartments on the east side are approximately normally distributed. Based on the sample, is there enough evidence to conclude, at the 0.05 level of significance, that the population mean monthly rent is now less than what is stated on the website?
Perform a one-tailed test. Then complete the parts below. Carry your intermediate computations to three or more decimal places.
(a) State the null hypothesis and the alternative hypothesis . (b) Determine the type of test statistic to use. (c) Find the value of the test statistic. (Round to three or more decimal places.) (d) Find the critical value. (Round to three or more decimal places.) (e) Can we conclude that the mean monthly rent for apartments on the east side is less than what is stated on the website?
The null hypothesis is μ≥700 and alternative hypothesis is μ<700, the type of test statistic is t statistic, the statistical test value is -2.650, the critical value is -1.796 and the null hypothesis is reject.
Given the average monthly rent for apartments on the east side of the cities is $700, and the mean for a random sample of 12-month rents for apartments on the east side is $687, with a standard deviation of $17.
Statistical hypothesis testing is an important technique in inferential statistics. Used to test hypotheses about population parameters. Statistical tests are used to compute test statistics and p-values. Finally, use this test statistic and the p-value to make a decision about rejecting the null hypothesis.
(a) Let μ be the population and monthly rent. the null and alternative hypotheses are given below.
Null Hypothesis: μ≥700
Alternative hypothesis: μ< 700 This is a left-sided hypothesis test.
(b) Here, the sample size is less than 30 and the population mean is unknown. Therefore, a one-sample t-test can be used here. Therefore, the t statistic is the test statistic.
(c) Given [tex]\bar{x}[/tex]=687, s=17 and n=12, the test statistic is computed as
[tex]\begin{aligned}t&=\frac{\bar{x}-\mu_{0}}{s/\sqrt{n}}\\ &=\frac{687-700}{17/\sqrt{12}}\\ &=-2.64901888216\\ &=-2.650\end[/tex]
(d) For n-1=12-1=11 degrees of freedom
The critical value for a significance level of 0.05 with 11 degrees of freedom is
[tex]\begin{aligned}t_{crit}&=-1.7959\\ &=-1.796\end[/tex]
(e) Reject the null hypothesis because the statistical test value (-2.650) is less than the critical value (-1.796). There is ample evidence to support the claim that the average monthly rent for East Side apartments is lower than stated on the website.
Hence, The value of test statistic and critical value has a mean of $687 and a standard deviation of $17, which are -2.650 and -1.796.
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Lena knows the following information about her box of 181818 candies:
10 candies contain both chocolate and caramel.3 candies contain neither chocolate nor caramel.12 candies in total contain chocolate. Can you help Lena organize the results into a two-way frequency table?
See below for the organized result in a two-way frequency table
How to organize the results?The given parameters are:
Candies = 18
Chocolate and Caramel = 10
None = 3
Chocolate = 12
Using the above parameters, we have the following rows and columns:
Chocolate
Present Absent Total
Caramel Present
Absent
Total
Next, we populate the table as follows:
Chocolate
Present Absent Total
Caramel Present 10 3
Absent
Total 12
The total number of candies is 18.
So, we have:
Chocolate
Present Absent Total
Caramel Present 10 3 13
Absent
Total 12 6 18
Subtract the cells from the column total
Chocolate
Present Absent Total
Caramel Present 10 3 13
Absent 2 3 5
Total 12 6 18
The above represents the organized result
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Six cards with the numbers 1 to 6 are randomly placed face down on a table. what is the probability that a card with a 4 is chosen?
Answer:
P(draw 4) = 1/6
Step-by-step explanation:
Six cards means: n(s) = 6
Joey hid them in bunches of 24 and in all he hid 503 bunches how many did he hide in all
Given the equation, y=2x+6, what is its slope?
Select one:
O a. 6
Ob. 2
O c. 3
O d. 1
Answer:
B, 2
Step-by-step explanation:
The slope in a y=mx+b equation is m. The m in your equation is 2.
Therfore, your slope is 2.
The graph shown here is the graph of which of the following rational
functions?
Considering it's asymptotes, the rational function shown in the graph is:
C. [tex]F(x) = \frac{1}{(x + 2)(x - 1)}[/tex]
What are the asymptotes of a function f(x)?The vertical asymptotes are the values of x which are outside the domain, which in a fraction are the zeroes of the denominator.The horizontal asymptote is the value of f(x) as x goes to infinity, as long as this value is different of infinity.In this graph, there are vertical asymptotes at x = -2 and x = 1, hence the denominator of the function is:
(x + 2)(x - 1).
The horizontal asymptote is of y = 0, hence the function can be given by:
C. [tex]F(x) = \frac{1}{(x + 2)(x - 1)}[/tex]
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What is the value of the rational expression
2x- 1
x+5
when x = 0?
Answer:
5
Step-by-step explanation
becasue 2x0=0 and 1x0=0
and when you add 5 to 0 you get five
Pls help with b
The diameters of two circular pulleys are 6cm and 12 cm, and their centres
are 10cm apart.
a. Angle a = 72.54 degrees
b. Hence find, in centimetres correct to one decimal place, the length of a
taut belt around the two pulleys
Which equation represents x, the side length of the greater square? x2 (x – 2)2 = 10 x2 2x2 = 10 x2 (x – 2)2 = 100 x2 2x2 = 100
Step-by-step explanation:
10x2(x-2)2=100x2x2 2x2
Given the following table with selected values of f (x) and g(x), evaluate f (g(4)).
x –6 –4 1 3 4
f (x) 4 –1 –6 1 3
g(x) 1 4 3 –4 –6
–4
–1
1
4
The value of f(x) when x is -6 is 1.
Composite functionsComposite functions are also known as function of a function. In order to get a composite functions f(g(x)).
g(4) = -6 (the value of g(x) when x is equivalent to 4 is -6)
f(g(x))= f(g(4)) = f(-6)
f(-6) = 1
The value of f(x) when x is -6 is 1.
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Select the equivalent expression. (Please Help!!!)
(a/b^3)^4 = ?
Choose 1 answer:
A. a^8/b^8
B. a^4/b^12
C. a/b^81
D. a^5/b^7
Answer: B. a^4/b^12
Step-by-step explanation:
[tex]\large\displaystyle\text{$\begin{gathered}\sf \left(\frac{a}{b^{3} }\right)^{4} \end{gathered}$}[/tex]
Use the rules of exponents to simplify the expression.
[tex]\large\displaystyle\text{$\begin{gathered}\sf \left(\frac{a^{1} }{b^{3}}\right)^{4} \end{gathered}$}[/tex]To raise the quotient of two numbers to a power, raise each power to the power and then divide it.
[tex]\large\displaystyle\text{$\begin{gathered}\sf \frac{(a^{1})^{4} }{(b^{3})^{4} } \end{gathered}$}[/tex]To raise a power to another power, multiply the exponents.
[tex]\large\displaystyle\text{$\begin{gathered}\sf \frac{a^{4} }{b^{3\times4} } \end{gathered}$}[/tex]Multiply 3 by 4.
[tex]\boxed{\large\displaystyle\text{$\begin{gathered}\sf \frac{a^{4} }{b^{12} } \end{gathered}$}}[/tex]We conclude that: the correct option is "B".
Which graph shows the system
x²+y=2
x²+y²=9?
Green’s Sport Shop offers its salespeople an annual salary of $10,000 plus a 6% commission on all sales above $20,000. Every employee receives a Christmas bonus of $500. What are Mr. Cahn’s total earnings in a year in which his sales amounted to $160,000?
(A) $18,900
(B) $18,400
(C) $19,600
(D) $20,100
Mr. Cahn’s total earnings in a year is $18,900.
What is the total earnings?The total earnings is a function of the annual salary, commission and Christmas bonus.
Total earnings = annual salary + commission + Christmas bonus
Commission = 6% x (160,000 - 20,000)
6% x $140,000
0.06 x 140,000 = $8,400
Total earnings = $8,400 + 10,000 + $500 = $18,900
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On a hike, Maria walked up a hill in 30 minutes, and then she ran back down the hill in 10 minutes. Which of the following graphs could model the total distance Maria hiked?
The graph that describes the motion of Maria is the graph in Option B.
What is displacement?Displacement is the term that is used to describe the distance covered in a specified direction.
We are told that Maria walked up a hill in 30 minutes, and then she ran back down the hill in 10 minutes. The graph that describes this motion is the graph in Option B.
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8.
P₁V₁ = P₂V₂
T₁ T₂
Rewrite this expression to find T₂.
Answer:
[tex]T_{2} = \frac{P_{2} V_{2} T_{1}} {P_{1} V_{1}}[/tex]
Step-by-step explanation:
The Combined Gas Law expression is:
P₁V₁ / T₁ = P₂V₂ / T₂
To solve for T₂, you need to.....
[tex]\frac{P_{1} V_{1} }{T_{1} } = \frac{P_{2} V_{2} }{T_{2} }[/tex] <------ Original expression
[tex]\frac{T_{2} P_{1} V_{1} }{T_{1} } = {P_{2} V_{2} }[/tex] <----- Multiply both sides by T₂
[tex]{T_{2} P_{1} V_{1} } = {P_{2} V_{2} T_{1}}[/tex] <----- Multiply both sides by T₁
[tex]T_{2} = \frac{P_{2} V_{2} T_{1}} {P_{1} V_{1}}[/tex] <----- Divide both sides by P₁V₁
4+5(7-1)+8/2
Simplify
Answer:38
Step-by-step explanation:4+5×6+4=4+30+4=38
please help with question!!
A. 6x²-2x+5, is the quotient of the polynomial.
ILL GIVE BRAINLIEST
1 x/3 - 3/4= 5/12
solve for x
eli ate one-third of a pie on friday, one-fourth of the pie on saturday, and the final 50 ounces of the pie on sunday. how many ounces of pie did eli eat on saturday
If Eli ate one third of a pie on friday ,one fourth proportion of the pie on saturday and 50 ounces of the pie on sunday then it is found that she ate 30 ounces of pie.
Given that Eli ate one third of a pie on friday, one fourth of the pie on saturday, and final 50 ounces of the pie on sunday which is left out.
We have to find the amount of pie that she ate one saturday.
We are given fraction and we know that sum of all fractions of something is equal to 1.
Let the proportion that she ate on sunday be x.
1/3+1/4+x=1
Taking LCM.
(4+3+12x)/12=1
7+12x=12
12x=12-7
12x=5
x=5/12
So the left out out portion that she ate on sunday is 5/12 of the pie.
We are given that she ate 50 ounces on sunday so the total pie will be 12/5*50=120 ounces.
Portion eaten by Eli on saturday =1/4
Pie that she ate on saturday=1/4*120
=30 ounces.
Hence Eli ate 30 ounces of pie on saturday.
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Can anyone solve question 2 and 10 with step by step explaination??
Answer:
Question 2: 4 1/2
Question 10: $7.24
Step-by-step explanation:
Question 2:
1 1/2/x = 3/9
x: total pages read3: minutes first spent9: minutes spent total1 1/2: pages read in 3 minutesto solve for x, we need to use 1 1/2/x and multiply it by x:
1 1/2/x * x
then we need to have 3/9 multiplied by x:
3/9 * x
this is equal to 1 1/2 = 3/9 * x
now you need to multiply both sides by the opposite of 3/9 = 9/3
3/2 * 9/3 = 3/9 * 9/3 * x which equals 9/2 = 4 1/2
final answer: 4 1/2
Question 10:
Jeff bought 3 boxes of cereal $2.99 each which comes up to $8.97,
he bought one gallon of milk for $4.98,
and 4 bananas for $0.50 each which in total is $2.00.
total amount spent: 8.97 + 4.98 + 2.00 = $15.95
We know there is a 20% off discount so what we have to do is take 20% off $15.95!
20%= $3.19
15.95 - 3.19 = 12.76
The total is now $12.76. But, Jeff payed with a $20.00 bill. Because of this, he will get $7.24 back. 12.76 + 7.24 = 20
Final answer: $7.24
A pendulum is swinging next to a wall. the distance d(t) (in cm) between the of the pendulum and the wall as a function of time t (in seconds) can be modeled by a sinusoidal expression of the form a * sin (b * t) + d at t = 0, when the endulum is exactly in the middle of its swing, the bob is 5 cm away from the wall. the bob reaches the closest point to the wall, which is 3 cm from the wall, 1 second later. find d(t). (t should be in radians)
The equation of the function d(t) is d(t) = -2sin(π/2 t) + 5
How to determine the function d(t)?The function is given as:
d(t) = a * sin(b * t) + d
The bob is 5 cm away from the wall implies that the vertical shift is 5.
This means that
d = 5
So, we have
d(t) = a * sin(b * t) + 5
The bob reaches the closest point to the wall, which is 3 cm from the wall.
So, the amplitude is
A = 3 - 5
Evaluate
A= -2
So, we have:
d(t) = -2 * sin(b * t) + 5
The period is calculated as:
B = 2π/t
Where
t = (5 + 3)/2
This gives
t = 4
Substitute t = 4 in B = 2π/t
B = 2π/4
Evaluate
B = π/2
So, we have:
d(t) = -2sin(π/2 t) + 5
Hence, the equation of the function d(t) is d(t) = -2sin(π/2 t) + 5
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Which of the following is characterized by drawing the
boundaries of legislative districts in bizarre or unusual ways
to favor one party?
Select one:
O a. Gerrymandering
O b. Meandering
O c. Facilitating
Od. Snaking
We can actually deduce here that the following that is characterized by drawing the boundaries of legislative districts in bizarre or unusual ways to favor one party is: A. Gerrymandering.
What is gerrymandering?Gerrymandering is actually known to be characterized by the practice of setting electoral district boundaries in order to favour some specific political interest in some legislative bodies and district.
Gerrymandering usually leads to the creation of convoluted districts rather than compact areas. The gerrymandering is seen in the United States.
The word "gerrymandering" was actually coined when Massachusetts's redistricting maps of 1812 was reviewed by Governor Elbridge Gerry. This was done when it was noted that one of the districts resembled a salamander.
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Instructions: Find the length of the missing sides. Round your
answers to the nearest tenth. (Hint: Always make sure to label
your triangle first with your reference angle)
Degree 54
Hypotenuse 24
How do i find x and y?
The values of x and y are 19 and 14 respectively.
How to determine the lengthGiven the hypotenuse = 24
Opposite = x
Adjacent = y
Let's use the sine ratio to find the opposite side
sin of angle = opposite/hypotenuse
sin 54 = x/24
Cross multiplysin 54 × 24 ,= x
x = 19.42
x = 19
To find the adjacent side, we use cosine ratio
cos = adjacent/hypotenuse
cos 54 = y/25
Cross multiply
cos 54 × 24 = y
y = 14
Thus, the values of x and y are 19 and 14 respectively.
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Which factorization could represent the number of water bottles and weight of each water bottle?
The factorization that could represent the number of water bottles and weight of each water bottle is 12(5x^2 + 4x + 2). Option B
What is factorization?The term factorization has to do with the process of obtaining common factors in an expression. It involves dividing each term in the expression with a factor that is common to all the terms in the expression.
The factorization that could represent the number of water bottles and weight of each water bottle is 12(5x^2 + 4x + 2).
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Missing parts;
Mara carried water bottles to the field to share with her team at halftime. The water bottles weighed a total of 60x2 + 48x + 24 ounces. Which factorization could represent the number of water bottles and weight of each water bottle? 6(10x2 + 8x + 2) 12(5x2 + 4x + 2) 6x(10x2 + 8x + 2) 12x(5x2 + 4x + 2)
please help me 20 points
Add up the number of students within the random sample data:
32+26+28+22 = 108
We can then use proportional fractions / cross-multiply to answer the question.
In a sample size of 108 students, 32 of them preferred Movie A.
There are 900 students in total.
[tex]\frac{32}{108} =\frac{x}{900}[/tex]
32*900 = 28800
108*x = 108x
108x = 28800
x = [tex]266\frac{2}{3}[/tex] or [tex]\frac{800}{3}[/tex]
In decimal form that is approximately 266.667 and round that to the nearest whole number, you get 267 students who would prefer Movie A.
Make x the subject of the formula.
y = √x² +1
Answer: x = [tex]\sqrt{y^2 -1}[/tex]
Step-by-step explanation:
For the following sample, calculate ss. then, compute the sample variance and standard deviation. scores: 5, 2, 2, 7, 9
The sample standard deviation is 7.6 and the sample variance is 2.76.
for the scores 5, 2, 2, 7, 9.
According to the question,
Sample size n = 5
Sample Mean = [5+2+2+7+9]/5
= 5
Formula for sample standard deviation = (x-mean)²/n
= [(5-5)²+(2-5)²+(2-5)²+(7-5)²+(9-5)²]/5
= [0+9+9+4+16]/5
= 38/5
= 7.6
Formula for sample variance = √sample standard deviation
=√7.6
=2.76.
Hence, the sample standard deviation is 7.6 and the sample variance is 2.76.
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Question 9 of 10
What is the solution to this equation?
x+12= -5
O A. x = 17
O B. x= -7
O C. x = 7
O D. x= -17
Answer:its b
Step-by-step explanation:
trust me its b
A relatively prime date is a date for which the number of the month and the number of the day are relatively prime. For example, June 17 is a relatively prime date because the greatest common factor of 6 and 17 is 1. How many relatively prime dates are in the month with the fewest relatively prime dates
There are 11 prime dates in it.
The prime numbers between 1 and 30 are 1, 2,3,5,7,11,13,17,19,23,29
Relative prime:
Two integers are relatively prime (or coprime) if there is no integer greater than one that divides them both (that is, their greatest common divisor is one). For example, 12 and 13 are relatively prime, but 12 and 14 are not.
Hence There are 11 prime dates in the month of June.
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Answer: 10
Step-by-step explanation:
Since exactly 1 in every $n$ consecutive dates is divisible by $n$, the month with the fewest relatively prime days is the month with the greatest number of distinct small prime divisors. This reasoning gives us June ($6=2\cdot3$) and December ($12=2^2\cdot3$). December, however, has one more relatively prime day, namely December 31, than does June, which has only 30 days. Therefore, June has the fewest relatively prime days. To count how many relatively prime days June has, we must count the number of days that are divisible neither by 2 nor by 3. Out of its 30 days, $\frac{30}{2}=15$ are divisible by 2 and $\frac{30}{3}=10$ are divisible by 3. We are double counting the number of days that are divisible by 6, $\frac{30}{6}=5$ days. Thus, June has $30-(15+10-5)=30-20=\boxed{10}$ relatively prime days.
Determine the x-intercept(s) of the rational function:
f(x) = (x ^ 2 - 16)/(x ^ 2 - 2x - 3)
Answer:
(-4,0) and (4,0)
Step-by-step explanation:
The x-intercepts occur for values of x where f(x) = 0.
f(x) = 0 when the nominator ([tex]x^{2} - 16[/tex]) = 0.
[tex]x^{2} -16[/tex] is a difference of squares,
so you can factor it as (x + 4)(x – 4)
When x = –4 or x = 4, the nominator is 0, and f(x) = 0