Step-by-step explanation:
Equation of a line is y = mx + c
where
m is the slope
c is the y intercept
To find the equation of a line given two points first find the slope of the line and use the formula
y - y1 = m( x - x1) to find the Equation of the line using any of the points given
Slope of the line using points
(3,2) and (4,6) is
[tex]m = \frac{6 - 2}{4 - 3} = \frac{4}{1} = 4[/tex]
So the equation of the line using point
( 3 , 2 ) and slope 4 is
y - 2 = 4( x - 3)Hope this helps you
Evaluate the expression: -(31 + 2) +7² - (-5²)
A) -9
B) -5
C) 41
OD -40
Answer: C. 41
Step-by-step explanation:
[tex]-\left(31+2\right)+7^2-\left(-5^2\right)[/tex]
[tex]=-33+7^2-\left(-5^2\right)[/tex]
[tex]\left(-5^2\right)=-25[/tex]
[tex]=-33+7^2-\left(-25\right)[/tex]
[tex]7^2=49[/tex]
[tex]=-33+49-\left(-25\right)[/tex]
[tex]-33+49=16[/tex]
[tex]=16-\left(-25\right)[/tex]
[tex]\mathrm{Apply\:rule\:}-\left(-a\right)\:=\:+a[/tex]
[tex]16+25=41[/tex]
Describe how the graph of y= x2 can be transformed to the graph of the given equation. y = (x - 6)2 Shift the graph of y = x2 down 6 units. Shift the graph of y = x2 right 6 units. Shift the graph of y = x2 up 6 units. Shift the graph of y = x2 left 6 units.
Answer:
Shift the graph of y = x2 right 6 units.
Transform the polar equation to a Cartesian (rectangular) equation: r= 4sinθ
options include:
x^2+y^2 = 4y
x^2+y^2 = -4
x^2+y^2 = 4
x^2+y^2 = -4y
Answer:
x^2 +y^2 = 4y
Step-by-step explanation:
Using the usual translation relations, we have ...
r^2 = x^2+y^2
x = r·cos(θ)
y = r·sin(θ)
Substituting for sin(θ) the equation becomes ...
r = 4sin(θ)
r = 4(y/r)
r^2 = 4y
Then, substituting for r^2 we get ...
x^2 +y^2 = 4y . . . . . matches the first choice
the difference of 8 and 2, added to x"
Answer:
see below
Step-by-step explanation:
Difference is subtract
(8-2)
Then add this to x
(8-2) +x
6+x
A population of bacteria P is changing at a rate of dP/dt = 3000/1+0.25t where t is the time in days. The initial population (when t=0) is 1000. Write an equation that gives the population at any time t. Then find the population when t = 3 days.
Answer:
- At any time t, the population is:
P = 375t² + 3000t + 1000
- At time t = 3 days, the population is:
P = 13,375
Step-by-step explanation:
Given the rate of change of the population of bacteria as:
dP/dt = 3000/(1 + 0.25t)
we need to rewrite the given differential equation, and solve.
Rewriting, we have:
dP/3000 = (1 + 0.25t)dt
Integrating both sides, we have
P/3000 = t + (0.25/2)t² + C
P/3000 = t + 0.125t² + C
When t = 0, P = 1000
So,
1000/3000 = C
C = 1/3
Therefore, at any time t, the population is:
P/3000 = 0.125t² + t + 1/3
P = 375t² + 3000t + 1000
At time t = 3 days, the population is :
P = 375(3²) + 3000(3) + 1000
= 3375 + 9000 + 1000
P = 13,375
Many elementary school students in a school district currently have ear infections. A random sample of children in two different schools found that 16 of 42 at one school and 18 of 34 at the other have ear infections. At the 0.05 level of significance, is there sufficient evidence to support the claim that a difference exists between the proportions of students who have ear infections at the two schools? Group of answer choices
Answer:
Step-by-step explanation:
The summary of the given data includes;
sample size for the first school [tex]n_1[/tex] = 42
sample size for the second school [tex]n_2[/tex] = 34
so 16 out of 42 i.e [tex]x_1[/tex] = 16 and 18 out of 34 i.e [tex]x_2[/tex] = 18 have ear infection.
the proportion of students with ear infection Is as follows:
[tex]\hat p_1 = \dfrac{16}{42}[/tex] = 0.38095
[tex]\hat p_2 = \dfrac{18}{34}[/tex] = 0.5294
Since this is a two tailed test , the null and the alternative hypothesis can be computed as :
[tex]H_0 :p_1 -p_2 = 0 \\ \\ H_1 : p_1 - p_2 \neq 0[/tex]
level of significance ∝ = 0.05,
Using the table of standard normal distribution, the value of z that corresponds to the two-tailed probability 0.05 is 1.96. Thus, we will reject the null hypothesis if the value of the test statistics is less than -1.96 or more than 1.96.
The test statistics for the difference in proportion can be achieved by using a pooled sample proportion.
[tex]\bar p = \dfrac{x_1 +x_2}{n_1 +n_2}[/tex]
[tex]\bar p = \dfrac{16 +18}{42 +34}[/tex]
[tex]\bar p = \dfrac{34}{76}[/tex]
[tex]\bar p = 0.447368[/tex]
[tex]\bar p + \bar q = 1 \\ \\ \bar q = 1 -\bar p \\ \\\bar q = 1 - 0.447368 \\ \\\bar q = 0.552632[/tex]
The pooled standard error can be computed by using the formula:
[tex]S.E = \sqrt{ \dfrac{ \bar p \bar q}{ n_1} + \dfrac{\bar p \bar p}{n_2} }[/tex]
[tex]S.E = \sqrt{ \dfrac{ 0.447368 * 0.552632}{ 42} + \dfrac{ 0.447368 * 0.447368}{34} }[/tex]
[tex]S.E = \sqrt{ \dfrac{ 0.2472298726}{ 42} + \dfrac{ 0.2001381274}{34} }[/tex]
[tex]S.E = \sqrt{ 0.01177284105}[/tex]
[tex]S.E = 0.1085[/tex]
The test statistics is ;
[tex]z = \dfrac{\hat p_1 - \hat p_2}{S.E}[/tex]
[tex]z = \dfrac{0.38095- 0.5294}{0.1085}[/tex]
[tex]z = \dfrac{-0.14845}{0.1085}[/tex]
z = - 1.368
Decision Rule: Since the test statistics is greater than the rejection region - 1.96 , we fail to reject the null hypothesis.
Conclusion: There is insufficient evidence to support the claim that a difference exists between the proportions of students who have ear infections at the two schools
A sample of a radioactive substance decayed 11% over the course of 3 weeks. How many grams were in the sample originally if 30.26 grams of the substance were remaining after the 3 weeks?
Answer:
34 grams
Step-by-step explanation:
If the remaining sample has 30.26 grams of radioactive substance, and 11% of it decayed, that means that 30.26 grams is 89% of the original. Let the original be x.
30.26=0.89x
Multiply both by one hundred
3026=89x
Divide both by 89
34=x
x=original, so the original was 34 grams.
Peter saved up $20,000 in an account earning a nominal 5% per year compounded continuously. How much was in the account at the end of two years? Round the answer to nearest dollar.
Answer: 22,103
Step-by-step explanation:
Compound interest is the interest calculated on the initial principal and the accumulated interest.
The amount in the account at the end of two years is $22,050.
What is compound interest?Compound interest is the interest calculated on the initial principal and the accumulated interest.
We have,
Principal = $20,000
Rate = r = 5%
It is compounded yearly.
Time = t = 2 years.
The formula for the amount having compound interest:
A = P [tex]( 1 + \frac{r}{n} )^{nt}[/tex]
A = 20,000 [tex](1 + \frac{5}{100\times1})^{2\times1}[/tex]
A = 20,000 ( 1 + 5/100 )²
A = 20,000 ( 105/100 )²
A = (20,000 x 105 x 105) / (100 x 100)
A = 2 x 105 x 105
A = $22,050
Thus the amount in the account at the end of two years is $22,050.
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Logan wants to mix an 18% acid solution with a 48% acid solution to get 15L of a 38% acid solution. How many liters of the 18% solution and how many liters of the 48% solution should be mixed?
Answer:
5 gallons of 18% solution
10 gallons of 48% solution
Step-by-step explanation:
x = gallons of 18% solution
y = gallons of 48% solution
Total volume is:
x + y = 15
Total amount of fertilizer is:
0.18 x + 0.48 y = 0.38 (15)
Solve by substitution.
0.18 x + 0.48 (15 − x) = 0.38 (15)
0.18 x + 7.2 − 0.48 x = 5.7
0.3 x = 1.5
x = 5
y = 10
x power 8 + x power 4 + 1
factorize
Answer:
[tex]1(x {}^{8} + x {}^{4} + 1)[/tex]
Step-by-step explanation:
[tex]x {}^{8} + {x}^{4} + 1 =1( x {}^{8} + x {}^{2} + 1)[/tex]
Hope this helps ;) ❤❤❤
Let me know if there is an error in my answer.
Subtract these polynomials.
(3x2 – 2x + 5) – (x2 + 3) =
Answer:
D. 2x^2 - 2x + 2
Step-by-step explanation:
(3x2 – 2x + 5) – (x2 + 3) add or subtract like terms
3x^2 - x^2 - 2x - 3 + 5 = 2x^2 - 2x + 2
Answer:
2x^2 - 2x + 2
Step-by-step explanation:
Ape-x
Given: 8(y + 2) = 48
Solve for “y.”
16
-6
20
4
The value of y will be equal to 4. The correct option is D.
What is an expression?Expression in maths is defined as the collection of the numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.
The given expression 8(y + 2) = 48 will be solved for y as below:-
8(y + 2) = 48
Divide both sides by 8 and solve.
[ 8 (y + 2) ] / 8 = 48 / 8
y + 2 = 6
Substract 2 from both the sides to get the value of y.
y + 2 - 2 = 6 -2
y = 4
Therefore, the value of y will be equal to 4. The correct option is D.
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5/3 x 6/7 real quick plz
Answer:
10/7 or 1 3/7. I hope this helps,
Step-by-step explanation:
Find the work W done by a force of 7pounds acting in the direction 30 degreesto the horizontal in moving an object 7feet from (0 comma 0 )to (7 comma 0 ).
Answer:
The work done by the force is 42.4 Joules
Step-by-step explanation:
The force F = 7 pounds
angle to the horizontal that the force acts ∅ = 30°
The object is moved a distance d = 7 feet
The coordinate (0 comma 0 )to (7 comma 0 ), indicates that the movement started from the origin, and is along the x-axis.
The work done by this force = F cos ∅ x d
==> 7 cos 30° x 7
==> 7 x 0.866 x 7 = 42.4 Joules
Please answer asap this person made a mistake what is the error and correct solution to this problem
Answer:
6
Step-by-step explanation:
Hello, please consider the following.
[tex](4+x)^2=4^2+2\cdot 4\cdot x+x^2=16+\boxed{8}x+x^2\\\\\text{ ... and not ...}\\\\16+\boxed{4}x+x^2[/tex]
So the correct equation becomes.
[tex]x^2+64=16+8x+x^2\\\\8x=64-16=48\\\\\text{ we divide by 8 both sides of the equation.}\\\\x=\dfrac{45}{8}=6[/tex]
Hope this helps.
Do not hesitate if you need further explanation.
Thank you
Answer:
Error : The expression ( 4 + x )² was expanded incorrectly.
Correct Solution : x = 6
Step-by-step explanation:
The planning of the solution is correct, by Pythagorean Theorem you can say that PQ² + QO² = PO², and hence through substitution x² + 8² = ( 4 + x )². Let's look into the calculations.
PQ² + QO² = PO²,
x² + 8² = ( 4 + x )²,
x² + 8² = 16 + 8x + x²,
64 = 16 + 8x,
48 = 8x,
x = 48 / 8 = 6, x = 6
As you can see, the only error in the calculations was expanding the expression ( 4 + x )². ( 4 + x )² = 4² + 2 [tex]*[/tex] 4 [tex]*[/tex] x + x² = 4² + 8x + x² = 16 + 8x + x², not 16 + 4x + x².
Step 1: Subtract 3 from both sides of the inequality
Step 2
Step 3: Divide both sides of the inequality by the
coefficient of x.
What is the missing step in solving the inequality 5 -
8x < 2x + 3?
O Add 2x to both sides of the inequality
O Subtract 8x from both sides of the inequality
O Subtract 2x from both sides of the inequality
Add 8x to both sides of the inequality.
Mark this and return
Save and Exit
Intext
Submit
Answer:
add 8x to both sides
Step-by-step explanation:
5-8x<2x+3
first step, subtract 3 from both sides:
2-8x<2x
second step,?
2<?x
so you need to add 8x first
According to the Census Bureau, 3.34 people reside in the typical American household. A sample of 26 households in Arizona retirement communities showed the mean number of residents per household was 2.70 residents. The standard deviation of this sample was 1.17 residents. At the .10 significance level, is it reasonable to conclude the mean number of residents in the retirement community household is less than 3.34 persons?
(a) State the null hypothesis and the alternate hypothesis. (Round your answer to 2 decimal places.)
H0: ? ?
H1: ? <
(b)
State the decision rule for .10 significance level. (Negative amount should be indicated by a minus sign. Round your answer to 3 decimal places.)
Reject H0 if t <
(c)
Compute the value of the test statistic. (Negative amount should be indicated by a minus sign. Round your answer to 3 decimal places.)
Value of the test statistic
(d)
Is it reasonable to conclude the mean number of residents in the retirement community household is less than 3.34 persons?
H0. Mean number of residents less than 3.34 persons.
Answer:
Step-by-step explanation:
Given that:
Mean = 3.34
sample size = 26
sample mean = 2.7
standard deviation = 1.17
level of significance = 0.10
The null hypothesis and the alternative hypothesis can be computed as follows:
[tex]\mathtt{H_o: \mu \geq 3.34} \\ \\ \mathtt{H_1: \mu < 3.34}[/tex]
degree of freedom = n - 1
degree of freedom = 26 -1
degree of freedom = 25
level of significance = 0.10
Since the alternative hypothesis contains <, then the test is left tailed
[tex]\mathtt{t_{\alpha, df} = t_{0.10, 25}}[/tex]
[tex]\mathtt{t_{0.10, 25}}[/tex] = - 1.316
The rejection region therefore consist of all values smaller than - 1.316, therefore ; reject [tex]H_o[/tex] if t < -1.316
The test statistics can be computed as follows:
[tex]t = \dfrac{X - \mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
[tex]t = \dfrac{2.7 - 3.34}{\dfrac{1.17}{\sqrt{26}}}[/tex]
[tex]t = \dfrac{-0.64}{\dfrac{1.17}{5.099}}[/tex]
t = - 2.789
Decision Rule: To reject the null hypothesis if the t test lies in the rejection region or less than the rejection region.
Conclusion: We reject the null hypothesis since t = (- 2.789) < -1.316. Then we conclude that the mean number of residents in the retirement community household is less than 3.34 persons.
What are the solutions of the equation x4 + 6x2 + 5 = 0? Use u substitution to solve.
x = i and x = i5
x=+ i and x
x= +115
O x=V-1 and x = = -5
x=+ -1 and x = = -5
Answer:
A; The first choice.
Step-by-step explanation:
We have the equation [tex]x^4+6x^2+5=0[/tex] and we want to solve using u-substitution.
When solving by u-substitution, we essentially want to turn our equation into quadratic form.
So, let [tex]u=x^2[/tex]. We can rewrite our equation as:
[tex](x^2)^2+6(x^2)+5=0[/tex]
Substitute:
[tex]u^2+6u+5=0[/tex]
Solve. We can factor:
[tex](u+5)(u+1)=0[/tex]
Zero Product Property:
[tex]u+5=0\text{ and } u+1=0[/tex]
Solve for each case:
[tex]u=-5\text{ and } u=-1[/tex]
Substitute back u:
[tex]x^2=-5\text{ and } x^2=-1[/tex]
Take the square root of both sides for each case. Since we are taking an even root, we need plus-minus. Thus:
[tex]x=\pm\sqrt{-5}\text{ and } x=\pm\sqrt{-1}[/tex]
Simplify:
[tex]x=\pm i\sqrt{5}\text{ and } x=\pm i[/tex]
Our answer is A.
find the h.c.f of 186,310,434
186|2
93|3
31|31
1
310|2
155|5
31|31
1
434|2
217|7
31|31
1
[tex]186=2\cdot3\cdot31\\310=2\cdot5\cdot31\\434=2\cdot7\cdot31\\\\\text{hcf}(186,310,434)=2\cdot31=62[/tex]
Molly’s house is located at point X. Molly wants Sophia and Cole to meet at her house because she thinks it is the same distance from Sophia’s house and Cole’s house. Which could prove that Molly’s house is the samedistance from Sophia’s and Cole’s houses?
Answer:
Cole's House
Step-by-step explanation:
Cole house is closer because molly and Sophia can go there together because there both girls
Sarah had a balance of $155 in her bank account at the start of the week. She withdrew $65.50 on Monday, $23.25 on Wednesday, and $26.45 on Thursday. On Friday she deposited $165.30. Write an expression that represents Sarah's spending.
Answer:
155 + 165.3 - 65.5 - 23.25 - 26.45
Step-by-step explanation:
She had $155 dollars in the starting = +155
She withdrew $65.5 = -65.5
She withdrew another $23.25 = -23.25
She withdrew another $26.45 = -26.45
The deposited $165.3 = +165.3
The expression looks like:
155 + 165.3 - 65.5 - 23.25 - 26.45
We could simplify the expression:
155 + 165.3 - 65.5 - 23.25 - 26.45
=> 320.3 - 88.75 - 26.45
=> 320.3 -115.2
=> 205.1
At the end of the week, she had a total of $205.10.
If the nth term is nn+1, then the (n+1)st term is:
Answer:
[tex]\large \boxed{\sf C. \ (n+1)^{n+1}+1}[/tex]
Step-by-step explanation:
[tex]n^n+1[/tex]
Plug in the value for n as n+1 in the nth term to find the (n+1)st term.
[tex](n+1)^{n+1}+1[/tex]
Answer:
[tex]\boxed{Option \ 3}[/tex]
Step-by-step explanation:
=> [tex]n^n+1[/tex]
Given that n = n+1
So,
=> [tex](n+1)^{n+1}+1[/tex]
What did this person do wrong? Honestly really stuck and do not remember geometry!
Answer:
See below.
Step-by-step explanation:
Using the right triangle altitude theorem, the correct proportions are:
[tex] \dfrac{AB}{AC} = \dfrac{AC}{AD} [/tex]
[tex] \dfrac{AB}{x} = \dfrac{x}{AD} [/tex]
[tex] \dfrac{25}{x} = \dfrac{x}{16} [/tex]
[tex] x^2 = 25 \times 16 [/tex]
[tex] x = 20 [/tex]
AC = 20 cm
[tex] \dfrac{AB}{CB} = \dfrac{CB}{DB} [/tex]
[tex] \dfrac{AB}{y} = \dfrac{y}{DB} [/tex]
[tex] \dfrac{25}{y} = \dfrac{y}{9} [/tex]
[tex]y^2 = 25 \times 9[/tex]
[tex]y = 15[/tex]
CB = 15 cm
An airplane travels 1200 miles in 4 hours with the wind. The same trip takes 5 hours against the wind. What is the speed of the plane in still air and what is the wind speed?
Answer:
Speed of plane in still air is 270 mph
Wind speed is 30 mph
Step-by-step explanation:
Check the picture.
The speed of the plane in still air is 270 mph and the speed of the wind will be 30 mph.
What is the distance formula?The distance traveled by an object is the product of the speed of an object and the time taken.
Distance = speed x time
An airplane travels 1200 miles in 4 hours with the wind. The same trip takes 5 hours against the wind.
Let the speed of the plane be x
The speed of wind be y
Distance covered with the wind = (x + y)t
1200 = (x + y)4
(x + y) = 1200/4
(x + y)= 300 .....(a)
Distance covered against the wind = (x - y)t
1200 = (x - y)5
(x - y) = 1200/5
(x - y) = 240 .......(b)
By solving both the equation
(x + y)= 300
(x - y) = 240
Therefore the values will be x= 270mph and y = 30 mph
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Traci Lynn currently walks to school from her apartment, which is 1.3 miles away from her first class. She typically walks at a speed of 3 miles per hour. She is considering buying a used bicycle from Deseret Industries to ride to campus. Traci Lynn assumes that if she were riding a bike, she could go about 5 miles per hour. How many minutes could Traci Lynn save getting to class each morning if she were to ride the bike?
5 minutes in bike and by walking 1 hours 3 minutes
Traci Lynn saves getting to class each morning if she were to ride 6.12 minute
What formula is used to find the time?The formula is used to find the value of time is;
Time = Distance/speed
Given that Traci Lynn currently walks to school from her apartment, which is 1.3 miles away from her first class.
If she goes to school from the apartment by walking, then it will take time is;
Time = 1.3/3
Time = 0.43 hours
If she goes to school from the apartment by bicycle, then it will take time is;
Time = 1.3/5
Times = 0.26 hours
Therefore,
The time in minutes could Traci Lynn save getting to class each morning if She ride the bike is;
0.43 - 0.26
= 0.17 hours
= 0.17 x 36 = 6.12 minute
Hence, Traci Lynn saves getting to class each morning if she were to ride 6.12 minute
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Chelsea played her tuba from 4:25 pm until 5:07
Answer:
42 minutes
Step-by-step explanation:
if you are asking how long it takes Chelsea to play her tuba then you do: 67 - 25 = 42
Answer:
Step-by-step explanation:
jhngjnh
Which expression is equal to (1+6i)−(7+3i) ?
Answer:
- 6+3iStep-by-step explanation:
[tex](1+6i)-(7+3i) ?\\Group\:the\:real\:part\:and\:the\:imaginary\\\:part\:of\:the\:complex\:number\\\left(a+bi\right)\pm \left(c+di\right)=\left(a\:\pm \:c\right)+\left(b\:\pm \:d\right)i\\=\left(1-7\right)+\left(6-3\right)i\\1-7=-6\\6-3=3\\=-6+3i[/tex]
A schoolteacher would like to know whether or not toothpaste brands are used differentially in her classroom (in other words, is one brand preferred over the others?). She asks her students to report which of three brands they use: Crest, Colgate, or Aquafresh. Below are the numbers of students who use each type of toothpaste (notice there are 45 students total in her class). Test her hypothesis using an alpha level of .05.
Crest Colgate Aquafresh
24 13 8
a. What test is appropriate for this analysis?
b. State the null hypothesis:
c. State the alternative hypothesis:
d. Find the critical value:
e. Calculate the test statistic:
f. Make a decision:
Answer:
Step-by-step explanation:
a. What test is appropriate for this analysis?
A Chi-square test of independence is appropriate for this analysis because it is used to compare two variables or testing relationship on categorical variables.
b. State the null hypothesis:
The null hypothesis is the default hypothesis
[tex]\mathtt{H_o:}[/tex] There is no particular preference for any brand of toothpaste among students.
c. State the alternative hypothesis:
The alternative hypothesis is the research hypothesis which comes in place to challenge the validity of the null hypothesis.
[tex]\mathtt{H_a:}[/tex] There is particular preference for brands of toothpaste among students.
d. Find the critical value:
degree of freedom = n-1
degree of freedom = 3 - 1
degree of freedom = 2
At the level of significance ∝ = 0.05
The confidence interval = 0.95 and degree of freedom = 2, the critical value from the chi-square distribution table = 5.991
e. Calculate the test statistic:
Using the chi square test statistics; we have the following:
Crest Colgate Aquafresh Total
24 13 8 45
Since we have three brands. Then, for each brand, the expected value
= Total /3
= 45/3
=15
Thus:
Chi -square [tex]\mathtt{X^2 = \dfrac{(observed \ value - expected \ value)^2}{expected \ value}}[/tex]
[tex]\mathtt{X^2 = \dfrac{(24 - 15)^2}{15} + \dfrac{(13 - 15)^2}{15} + \dfrac{(8 - 15)^2}{15} }[/tex]
[tex]\mathtt{X^2 = \dfrac{81}{15} + \dfrac{4}{15} + \dfrac{49}{15} }[/tex]
[tex]\mathtt{X^2 = \dfrac{81+4+49}{15}}[/tex]
[tex]\mathtt{X^2 = \dfrac{134}{15}}[/tex]
[tex]\mathtt{X^2 =8.93 }[/tex]
f. Make a decision:
Since the chi-square value is greater than the critical value , we reject the null hypothesis and conclude that the students have particular preference for brands of toothpaste.
You sell tickets at school for fundraisers. You sold car wash tickets, silly string fight tickets and dance tickets – for a total of 380 tickets sold. The car wash tickets were $5 each, the silly sting fight tickets were $3 each and the dance tickets were $10 each. If you sold twice as many silly string tickets as car wash tickets, and you have $1460 total. Write the matrix in the box below. Write the solution set for this system and include any necessary work.
Answer:
Matrix :
[tex]\begin{bmatrix}1&1&1&|&380\\ 5&3&10&|&1460\\ -2&1&0&|&0\end{bmatrix}[/tex]
Solution Set : { x = 123, y = 246, z = 11 }
Step-by-step explanation:
Let's say that x represents the number of car wash tickets, y represents the number of silly sting fight tickets, and z represents the number of dance tickets. We know that the total tickets = 380, so therefore,
x + y + z = 380,
And the car wash tickets were $5 each, the silly sting fight tickets were $3 each and the dance tickets were $10 each, the total cost being $1460.
5x + 3y + 10z = 1460
The silly string tickets were sold for twice as much as the car wash tickets.
y = 2x
Therefore, if we allign the co - efficients of the following system of equations, we get it's respective matrix.
System of Equations :
[tex]\begin{bmatrix}x+y+z=380\\ 5x+3y+10z=1460\\ y=2x\end{bmatrix}[/tex]
Matrix :
[tex]\begin{bmatrix}1&1&1&|&380\\ 5&3&10&|&1460\\ -2&1&0&|&0\end{bmatrix}[/tex]
Let's reduce this matrix to row - echelon form, receiving the number of car wash tickets, silly sting fight tickets, and dance tickets,
[tex]\begin{bmatrix}5&3&10&1460\\ 1&1&1&380\\ -2&1&0&0\end{bmatrix}[/tex] - Swap Matrix Rows
[tex]\begin{bmatrix}5&3&10&1460\\ 0&\frac{2}{5}&-1&88\\ -2&1&0&0\end{bmatrix}[/tex] - Cancel leading Co - efficient in second row
[tex]\begin{bmatrix}5&3&10&1460\\ 0&\frac{2}{5}&-1&88\\ 0&\frac{11}{5}&4&584\end{bmatrix}[/tex] - Cancel leading Co - efficient in third row
[tex]\begin{bmatrix}5&3&10&1460\\ 0&\frac{11}{5}&4&584\\ 0&\frac{2}{5}&-1&88\end{bmatrix}[/tex] - Swap second and third rows
[tex]\begin{bmatrix}5&3&10&1460\\ 0&\frac{11}{5}&4&584\\ 0&0&-\frac{19}{11}&-\frac{200}{11}\end{bmatrix}[/tex] - Cancel leading co - efficient in row three
And we can continue, canceling the leading co - efficient in each row until this matrix remains,
[tex]\begin{bmatrix}1&0&0&|&\frac{2340}{19}\\ 0&1&0&|&\frac{4680}{19}\\ 0&0&1&|&\frac{200}{19}\end{bmatrix}[/tex]
x = 2340 / 19 = ( About ) 123 car wash tickets sold, y= 4680 / 19 =( About ) 246 silly string fight tickets sold, z = 200 / 19 = ( About ) 11 tickets sold
A maker of microwave ovens advertises that no more than 10% of its microwaves need repair during the first 5 years of use. In a random sample of 50 microwaves that are 5 years old, 12% needed repairs at a=.04 can you reject the makers claim that no more than 10% of its microwaves need repair during the first five years of use?
Answer:
We conclude that no more than 10% of its microwaves need repair during the first five years of use.
Step-by-step explanation:
We are given that a maker of microwave ovens advertises that no more than 10% of its microwaves need repair during the first 5 years of use.
In a random sample of 50 microwaves that are 5 years old, 12% needed repairs.
Let p = population proportion of microwaves who need repair during the first five years of use.
So, Null Hypothesis, [tex]H_0[/tex] : p [tex]\leq[/tex] 10% {means that no more than 10% of its microwaves need repair during the first five years of use}
Alternate Hypothesis, [tex]H_A[/tex] : p > 10% {means that more than 10% of its microwaves need repair during the first five years of use}
The test statistics that will be used here is One-sample z-test for proportions;
T.S. = [tex]\frac{\hat p-p}{\sqrt{\frac{p(1-p)}{n} } }[/tex] ~ N(0,1)
where, [tex]\hat p[/tex] = sample proportion of microwaves who need repair during the first 5 years of use = 12%
n = sample of microwaves = 50
So, the test statistics = [tex]\frac{0.12-0.10}{\sqrt{\frac{0.10(1-0.10)}{50} } }[/tex]
= 0.471
The value of z-test statistics is 0.471.
Now, at a 0.04 level of significance, the z table gives a critical value of 1.751 for the right-tailed test.
Since the value of our test statistics is less than the critical value of z as 0.471 < 1.751, so we have insufficient evidence to reject our null hypothesis as the test statistics will not fall in the rejection region.
Therefore, we conclude that no more than 10% of its microwaves need repair during the first five years of use.
