[tex]\bf 12~~,~~\stackrel{12+3}{15}~~,~~\stackrel{15+3}{18}~~,~~\stackrel{18+3}{21}~\hspace{10em}\stackrel{\textit{common difference}}{d=3} \\\\[-0.35em] ~\dotfill\\\\ n^{th}\textit{ term of an arithmetic sequence} \\\\ a_n=a_1+(n-1)d\qquad \begin{cases} a_n=n^{th}\ term\\ n=\textit{term position}\\ a_1=\textit{first term}\\ d=\textit{common difference}\\ \cline{1-1} a_1=12\\ d=3 \end{cases} \\\\\\ a_n=12+(n-1)3\implies a_n=12+3n-3\implies a_n=3n+9[/tex]
Answer:
tₙ = 3(3 + n)
Step-by-step explanation:
Points to remember
nth term of an AP
tₙ = a + (n - 1)d
Where a - first term of AP
d - Common difference of AP
To find the nth term
The given series is,
12,15,18,21 .....
Here a = 12 and d = 15 - 12 = 3
tₙ = a + (n - 1)d
= 12 + (n - 1)3
=12 + 3n - 3
= 9 + 3n
= 3(3 + n)
Therefore tₙ = 3(3 + n)
Find the union C1 ∪ C2 and the intersection C1 ∩ C2 of the two sets C1 and C2, where (a) C1 = {0, 1, 2, }, C2 = {2, 3, 4}. (b) C1 = {x : 0
Answer:
See below.
Step-by-step explanation:
C1 = {0, 1, 2} and C2 = {2, 3, 4}.
C1 ∪ C2 = {0, 1, 2, 3, 4} (Note: the 2 is not repeated in the result}.
C1 ∩ C2 = {2}.
In triangle ABC, an altitude is drawn from vertex C to the line containing AB. The length of this altitude is h and h=AB. Which of the following is true?
I. Triangle ABC could be a right triangle.
II. Angle C cannot be a right angle.
III. Angle C could be less than 45 degrees.
Answer:
II. Angle C cannot be a right angle.
III. Angle C could be less than 45 degrees.
The given altitude of triangle ABC is h, is located inside the triangle and
extends from side AB to the vertex C.
The true statements are;
I. Triangle ABC could be a right triangle
II. Angle C cannot be a right angle
Reasons:
I. Triangle ABC could be a right triangle
The altitude drawn from the vertex C to the line AB = h
The length of h = AB
Where, triangle ABC is a right triangle, we have;
The legs of the right triangle are; h and AB
The triangle ABC formed is an isosceles right triangle
Therefore, triangle ABC could be an isosceles right triangle; True
II. Angle C cannot be a right angle: True
If angle ∠C is a right angle, we have;
AB = The hypotenuse (longest side) of ΔABC
Line h = AB is an altitude, therefore, one of the sides of ΔABC is hypotenuse to h, and therefore, longer than h and AB, which is false
Therefore, ∠C cannot be a right angle
III. Angle C could be less than 45 degrees; False
The minimum value of angle C is given by when triangle ABC is an isosceles right triangle. As the position of h shifts between AB, the lengths of one of the sides of ΔABC increases, and therefore, ∠C, increases
Therefore, ∠C cannot be less than 45°
The true statements are I and II
Learn more here:
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Select all the correct locations on the image. Select all the expressions that result in a product that is a rational number. MULITIPLE CHOICE
4/3 x 12/3
32/4 x 15/4
[tex]\sqrt{\frac{3}{2} }[/tex] x 22/7
[tex]\sqrt{11}[/tex] x 2/3
Answer:
The 1st & 2nd option
Step-by-step explanation:
Ans1: 16/3
Ans2: 30
Ans3: 11/7×surd6
Ans4: 2/3×surd11
Rational number is a number that can be expressed in ratio (quotient)
It can be expressed in the form of repeating or terminating decimal
Example:
16/3 is equal to
5.3333333333....(repeating decimal)
thus it is a rational number
1/4 is equal to
0.25 (terminating decimal)
thus it is a rational number
In a Power Ball lottery, 5 numbers between 1 and 12 inclusive are drawn. These are the winning numbers. How many different selections are possible? Assume that the order in which the numbers are drawn is not important.
Answer:
792
Step-by-step explanation:
₁₂C₅ = (12!) / (5! (12-5)!)
₁₂C₅ = (12!) / (5! 7!)
₁₂C₅ = 792
There are 792 different selections are possible.
What is the combination?Each of the different groups or selections can be formed by taking some or all of a number of objects, irrespective of their arrangments is called a combination.
We are given that in a Power Ball lottery, 5 numbers between 1 and 12 inclusive are drawn. These are the winning numbers.
Therefore, the selections are possible as;
₁₂C₅ = (12!) / (5! (12-5)!)
₁₂C₅ = (12!) / (5! 7!)
₁₂C₅ = 792
Hence, there are 792 different selections are possible.
Learn more about combinations and permutations here:
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According to a recent study, 9.2% of high school dropouts are 16- to 17-year-olds. In addition, 6.2% of high school dropouts are white 16- to 17-year-olds. What is the probability that a randomly selected dropout is white, given that he or she is 16 to 17 years old?
The probability that a randomly selected dropout aged 16 to 17 is white, given the provided statistics, is 67.39%.
Explanation:The student is asking a question related to conditional probability in the field of Mathematics. The question prompts us to find out the probability that a randomly selected high school dropout in the age range of 16 to 17 is white. To find the answer, we use the following formula:
P(A|B) = P(A ∩ B) / P(B)
Where:
P(A|B) is the probability of event A happening given that event B has occurred.
P(A ∩ B) is the probability of both event A and event B happening together.
P(B) is the probability of event B happening.
From the problem statement, we know that P(B), the percentage of dropouts who are 16-17 years old, is 9.2%. Also, P(A ∩ B), the percent of dropouts who are both white and 16-17 years old, is given as 6.2%. We are supposed to find P(A|B), the probability that a dropout is white given that they are 16-17 years old.
Therefore, by substituting these values into the formula, we get:
P(A|B) = 6.2% / 9.2% = 67.39%
Rounded to two decimal places, the answer is 67.39%. So, there is approximately a 67.39% chance that a random high school dropout aged 16-17 is white.
Learn more about Conditional Probability here:https://brainly.com/question/32171649
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The spread of a virus can be modeled by exponential growth, but its growth is limited by the number of individuals that can be infected. For such situations, the function P(t) = ((Kpe)^rt)/K+p(e^rt - 1) can be used, where P(t) is the infected population t days after the first infection, p is the initial infected population, K is the total population that can be infected, and r is the rate the virus spreads, written as a decimal.
a. A town of 10,000 people starts with 2 infected people and a virus growth rate of 20%. When will the growth of the infected population start to level off, and how many people will be infected at that point? Explain your reasoning, and include any graphs you draw, with or without technology.
b. When will the infected population equal to the uninfected population?
Answer:
a) growth will reach a peak and begin declining after about 42.6 days. 5000 people will be infected at that point
b) the infected an uninfected populations will be the same after about 42.6 days
Step-by-step explanation:
We have assumed you intend the function to match the form of a logistic function:
[tex]P(t)=\dfrac{Kpe^{rt}}{K+p(e^{rt}-1}[/tex]
This function is symmetrical about its point of inflection, when half the population is infected. That is, up to that point, it is concave upward, increasing at an increasing rate. After that point, it is concave downward, decreasing at a decreasing rate.
a) The growth rate starts to decline at the point of inflection, when half the population is infected. That time is about 42.6 days after the start of the infection. 5000 people will be infected at that point
b) The infected and uninfected populations will be equal about 42.6 days after the start of the infection.
Find the length of AC express your answer in terms of pie
Answer:
The length of arc AC is [tex]15\pi\ cm[/tex]
Step-by-step explanation:
step 1
Find the circumference of the circle
The circumference is equal to
[tex]C=\pi D[/tex]
we have
[tex]D=36\ cm[/tex]
substitute
[tex]C=\pi (36)[/tex]
[tex]C=36\pi\ cm[/tex]
step 2
Find the length of arc AC
Remember that the circumference subtends a central angle of 360 degrees
The measure of arc AC is equal to
arc AC+30°=180° ----> because the diameter divide the circle into two equal parts
arc AC=180°-30°=150°
using proportion
[tex]\frac{36\pi}{360}=\frac{x}{150}\\ \\x=36\pi*150/360\\ \\x=15\pi\ cm[/tex]
In a large school, it was found that 71% of students are taking a math class, 77% of student are taking an English class, and 58% of students are taking both. Find the probability that a randomly selected student is taking a math class or an English class. Write your answer as a decimal, and round to 2 decimal places if necessary.
Answer:
P (Math or English) = 0.90
Step-by-step explanation:
* Lets study the meaning of or , and on probability
- The use of the word or means that you are calculating the probability
that either event A or event B happened
- Both events do not have to happen
- The use of the word and, means that both event A and B have to
happened
* The addition rules are:
# P(A or B) = P(A) + P(B) ⇒ mutually exclusive (events cannot happen
at the same time)
# P(A or B) = P(A) + P(B) - P(A and B) ⇒ non-mutually exclusive (if they
have at least one outcome in common)
- The union is written as A ∪ B or “A or B”.
- The Both is written as A ∩ B or “A and B”
* Lets solve the question
- The probability of taking Math class 71%
- The probability of taking English class 77%
- The probability of taking both classes is 58%
∵ P(Math) = 71% = 0.71
∵ P(English) = 77% = 0.77
∵ P(Math and English) = 58% = 0.58
- To find P(Math or English) use the rule of non-mutually exclusive
∵ P(A or B) = P(A) + P(B) - P(A and B)
∴ P(Math or English) = P(Math) + P(English) - P(Math and English)
- Lets substitute the values of P(Math) , P(English) , P(Math and English)
in the rule
∵ P(Math or English) = 0.71 + 0.77 - 0.58 ⇒ simplify
∴ P(Math or English) = 0.90
* P(Math or English) = 0.90
To find the probability that a randomly selected student is taking a math class or an English class, use the principle of inclusion-exclusion. The probability is 90%.
Explanation:To find the probability that a randomly selected student is taking a math class or an English class, we can use the principle of inclusion-exclusion. We know that 71% of students are taking a math class, 77% are taking an English class, and 58% are taking both.
To find the probability of taking either math or English, we add the probabilities of taking math and English, and then subtract the probability of taking both:
P(Math or English) = P(Math) + P(English) - P(Math and English)
= 71% + 77% - 58%
= 90%
Therefore, the probability that a randomly selected student is taking a math class or an English class is 90%.
What is 3 root 17 in a decimal
Answer:
[tex]\sqrt[3]{17}= 2.57128=257128[/tex]×[tex]10^{-5}[/tex]
Step-by-step explanation:
We need to find: [tex]\sqrt[3]{17}[/tex] in decimals. To do this, we are going to need the help of a calculator. After plugging the values, we get that:
[tex]\sqrt[3]{17}= 2.57128 = 257128[/tex]×[tex]10^{-5}[/tex]
In this case, I just considered 5 significant figures!
Raise the quality in parentheses to the indicated exponent, and simplify the resulting expression with positive exponents.
For this case we have the following expression:
[tex](\frac {-27x ^ 0 * y ^ {- 2}} {54x ^ {- 5} * y ^ {- 4}}) ^ {- 2} =[/tex]
By definition we have to:
[tex]a^0 = 1[/tex]
So:
[tex](\frac {-27y ^ {- 2}} {54x ^ {- 5} * y ^ {- 4}}) ^ {- 2} =[/tex]
Simplifying:
[tex](\frac {-y ^ {- 2}} {2x ^ {- 5} * y ^ {- 4}}) ^ {- 2} =[/tex]
By definition of power properties we have to:
[tex](a ^ n) ^ m = a ^ {n * m}[/tex]
So, rewriting the expression we have:
[tex](-1)^{-2}\frac{-y^{-2*-2}}{2^{-2}*x^{-5*-2}*y^{-4*-2}}=\\\frac{1}{(-1)^2}*\frac{-y^{4}}{2^{-2}x^{10}*y^{8}}=[/tex]
SImplifying:
[tex]+1*\frac{y^{4-8}}{2^{-2}x^{10}}=\\\frac{y^{-4}}{2^{-2}x^{10}}=\\\frac{2^2}{x^{10}y^{4}}[/tex]
Answer:
[tex]\frac{4}{x^{10}y^{4}}[/tex]
eight times the sum of 5 and some number is 104. What is the number
Answer:
8 is the number.
Step-by-step explanation:
We are given the following expression in words which we are to translate into mathematical expression and tell the number:
'eight times the sum of 5 and some number is 104'
Assuming the number to be [tex]x[/tex], we can write it as:
[tex] 8 ( 5 + x ) = 1 0 4 [/tex]
[tex] 5 + x = \frac { 1 0 4 } { 8 } [/tex]
[tex] 5 + x = 1 3 [/tex]
[tex]x=13-5[/tex]
x = 8
ANSWER
[tex]8[/tex]
EXPLANATION
Let the number be y.
Eight times the sum of the number and 5 is written as:
[tex]8(5 + y)[/tex]
From the question, this expression must give us 104.
This implies that:
[tex]8(5 + y) = 104[/tex]
Expand the parenthesis to get;
[tex]40 + 8y = 104[/tex]
Group similar terms to get:
[tex]8y = 104 - 40[/tex]
Simplify:
[tex]8y = 64[/tex]
[tex]y = \frac{64}{8} [/tex]
This finally evaluates to
[tex]y=8[/tex]
Hence the number is 8
6x - 8 = 16
Solve the following equation. Then place the correct number in the box provided.
Answer:
x = 4
Step-by-step explanation:
Add 8 to both sides of the equation:
6x -8 +8 = 16 +8
6x = 24
Divide both sides of the equation by 6:
6x/6 = 24/6
x = 4
For this case we have the following equation:
[tex]6x-8 = 16[/tex]
We must find the value of the variable "x":
Adding 8 to both sides of the equation we have:
[tex]6x = 16 + 8\\6x = 24[/tex]
Dividing between 6 on both sides of the equation we have:
[tex]x = \frac {24} {6}\\x = 4[/tex]
Thus, the solution of the equation is[tex]x = 4[/tex]
Answer:
[tex]x = 4[/tex]
In a certain country, the true probability of a baby being a boy is 0.527. Among the next seven randomly selected births in the country, what is the probability that at least one of them is a girl?
Answer: 0.9887
Step-by-step explanation:
Given : The true probability of a baby being a boy : [tex]0.527[/tex]
The number of selected births : [tex]7[/tex]
Now, the when next seven randomly selected births in the country, then the probability that at least one of them is a girl is given by :_
[tex]\text{P(at least one girl)=1-P(none of them girl)}\\\\=1-(0.527)^7=0.988710510435\approx0.9887[/tex]
Hence, the probability that at least one of them is a girl =0.9887
Final answer:
The probability of at least one girl being born among seven births in a country where a boy has a 0.527 chance of being born is calculated by subtracting the probability of all seven being boys from 1.
Explanation:
The question asks about the probability of at least one girl being born among the next seven randomly selected births in a country where the true probability of a baby being a boy is 0.527. To find this, we can use the complement rule, which focuses on the probability that all seven births will be boys, the opposite of what we want to find.
First, we find the probability of all seven being boys:
P(all boys) = (0.527)⁷
Next, we subtract this from 1 to get the probability of at least one girl:
P(at least one girl) = 1 - P(all boys)
This gives us the probability that at least one of the next seven births will be a girl.
Suppose a batch of steel rods produced at a steel plant have a mean length of 150 millimeters, and a standard deviation of 12. If 100 rods are sampled at random from the batch, what is the probability that the mean length of the sample rods would differ from the population mean by less than 0.5 millimeters?
Answer:
whatever
Step-by-step explanation:
-789237
Which value below is included in the solution set for the inequality statement?
-3(x - 4) > 6(x - 1)
3
-1
7
2
Answer:
Step-by-step explanation:
Anwer -1
Answer:
-1
Step-by-step explanation:
You could solve for x... or just plug those values in to see which would make the inequality true
Let's check x=3
-3(3-4)>6(3-1)
-3(-1)>6(2)
3>12 this is false so not x cannot be 3
Let's check x=-1
-3(-1-4)>6(-1-1)
-3(-5)>6(-2)
15>-12 this is true so x can take on the value -1
Let's check x=7
-3(7-4)>6(7-1)
-3(3)>6(6)
-9>36 is false so x cannot be 7
Let's check x=2
-3(2-4)>6(2-1)
-3(-2)>6(1)
6>6 is false so x cannot be 2
factor this trinomial x^2+x-2
Answer:
(x+2) (x-1)
Step-by-step explanation:
x^2+x-2
What 2 numbers multiply together to give -2 and add together to give 1
2 * -1 = -2
2 + -1 = 1
(x+2) (x-1)
Answer:
(x+2) (x-1)
Step-by-step explanation:
Need help on these, thank you!
Answer:
[tex]5x+2 , 2x^2+5x-2,\frac{x^2+5x}{2-x^2}[/tex]
Step-by-step explanation:
We are given f(x) and g(x)
1. (f+g)(x)
(f+g)(x) = f(x) + g(x)
= [tex]x^2+5x+2-x^2[/tex]
= [tex]5x+2[/tex]
Domain : All real numbers as it there exists a value of (f+g)(x) f every x .
2. (f-g)(x)
(f-g)(x) = f(x)-g(x)
= [tex]x^2+5x-2+x^2[/tex]
=[tex]2x^2+5x-2[/tex]
Domain : All real numbers as it there exists a value of (f-g)(x) f every x .
Part 3 .
[tex](\frac{f}{g})(x)\\(\frac{f}{g})(x) = \frac{f(x)}{g(x)}\\=\frac{x^2+5x}{2-x^2}[/tex]
Domain : In this case we see that the function is not defined for values of x for which the denominator becomes 0 or less than zero . Hence only those values of x are defined for which
[tex]2-x^2>0[/tex]
or [tex]2>x^2[/tex]
Hence taking square roots on both sides and solving inequality we get.
[tex]-\sqrt{2} <x<\sqrt{2}[/tex]
If f(x) = √2x + 3 and g(x) = x^2, for what value(s) of x does f(g(x)) = g(f(x))? (approximate when needed). Please give an explanation with your answer!
Answer:
[tex]\large\boxed{x=\sqrt6-3}[/tex]
Step-by-step explanation:
[tex]Domain:\\2x+3\geq0\to x\geq-1.5[/tex]
[tex]f(x)=\sqrt{2x+3},\ g(x)=x^2\\\\f(g(x))-\text{substitute x = g(x) in}\ f(x):\\\\f(g(x))=f(x^2)=\sqrt{2x^2+3}\\\\g(f(x))-\text{substitute x = f(x) in}\ g(x):\\\\g(f(x))=g(\sqrt{2x+3})=(\sqrt{2x+3})^2=2x+3\\\\f(g(x))=g(f(x))\iff\sqrt{2x^2+3}=2x+3\qquad\text{square of both sides}\\\\(\sqrt{2x^2+3})^2=(2x+3)^2}\qquad\text{use}\ (\sqrt{a})^2=a\ \text{and}\ (a+b)^2=a^2+2ab+b^2\\\\2x^2+3=(2x)^2+2(2x)(3)+3^2\\\\2x^2+3=4x^2+12x+9\qquad\text{subtract}\ 2x^2\ \text{and 3 from both sides}[/tex]
[tex]0=2x^2+12x+6\qquad\text{divide both sides by 2}\\\\x^2+6x+3=0\qquad\text{add 6 to both sides}\\\\x^2+6x+9=6\\\\x^2+2(x)(3)+3^2=6\qquad\text{use}\ (a+b)^2=a^2+2ab+b^2\\\\(x+3)^2=6\iff x+3=\pm\sqrt6\qquad\text{subtract 3 from both sides}\\\\x=-3-\sqrt6\notin D\ \vee\ x=-3+\sqrt6\in D[/tex]
new vintage 1965 convertible requires an ol change every 400 miles and replacement of all fBuids every 13,000 mles If these services have pust been performed by the dealer, how many mles from now will both be due at the same tme? The services will both be due at the same ime agan in mdes (Type a whole rumber)
Answer: There are 26000 miles that will both be due at the same time.
Step-by-step explanation:
Since we have given that
Number of miles required by new vintage = 400
Number of miles if these services have must been performed by the dealer = 13000
We need to find the number of miles from now that will both be due at the same time.
We would use "LCM of 400 and 13000":
As we know that LCM of 400 and 13000 is 26000.
So, there are 26000 miles that will both be due at the same time.
Please Help!
Rewrite each expression in terms of sinθ, and simplify.
6. [tex]\frac{tan(x)}{cot(x)}[/tex]
7. [tex]cos(x)cot(x)+sin(x)[/tex]
Also, please show your work if you can!
Thanks in advance.
Step-by-step explanation:
[tex]6.\\\\\dfrac{\tan x}{\cot x}\qquad\text{use}\ \cot x=\dfrac{1}{\tan x}\\\\=\dfrac{\tan x}{\frac{1}{\tan x}}=\tan x\cdot\dfrac{\tan x}{1}=\tan x\cdot\tan x=\left(\tan x\right)^2\qquad\text{use}\ \tan x=\dfrac{\sin x}{\cos x}\\\\=\left(\dfrac{\sin x}{\cos x}\right)^2\qquad\text{use}\ \left(\dfrac{a}{b}\right)^n=\dfrac{a^n}{b^n}\\\\=\dfrac{\sin^2x}{\cos^2x}\qquad\text{use}\ \sin^2x+\cos^2x=1\to\cos^2x=1-\sin^2x\\\\=\dfrac{\sin^2x}{1-\sin^2x}[/tex]
[tex]7.\\\\\cos x\cdot\cot x+\sin x\qquad\text{use}\ \cot x=\dfrac{\cos x}{\sin x}\\\\=\cos x\cdot\dfrac{\cos x}{\sin x}+\sin x=\dfrac{\cos^2x}{\sin x}+\dfrac{\sin^2x}{\sin x}=\dfrac{\cos^2x+\sin^2x}{\sin x}=\dfrac{1}{\sin x}[/tex]
From a point on a straight road, Pablo and Elena ride bicycles in the same direction. Pablo rides at 9 mph and Elena rides at 13 mph. In how many hours will they be 38 mi apart?
Pablo and Elena will be 38 miles apart in ____ hrs.
(Type an integer or a simplified fraction.)
Answer:
Pablo and Elena will be 38 miles apart in 19/2 hrs.
Step-by-step explanation:
Their separation speed is 13 mph - 9 mph = 4 mph. Using the relation ...
time = distance/speed
we can find the time from ...
time = (38 mi)/(4 mi/h) = 38/4 h = 19/2 h
Find the vector equation of the line through (00,00,00) and (11,55,44) where tequals=0 corresponds to the first given point and where tequals=1 corresponds to the second given point.
Answer:
(x, y, z) = (0, 0, 0) +t(1, 5, 4)
Step-by-step explanation:
Such a parametric equation can be written in the form ...
(first point) + t×(change in point values)
where the change in point values is the difference between the point coordinates.
Since the first point is (0, 0, 0), the change is easy to find. It is exactly equal to the second point's coordinates. Hence our equation is ...
(x, y, z) = (0, 0, 0) +t(1, 5, 4)
The sum can be "simplified" to ...
(x, y, z) = (t, 5t, 4t)
_____
Comment on answer forms
I like the first form for many uses, but the second form can preferred in some circumstances. Use the one consistent with your reference material (textbook or teacher preference).
Apples are on sale for \$3.12$3.12dollar sign, 3, point, 12 per kilogram. Is the total cost of the apples proportional to the total mass?
Answer:
Since every kilo of apples cost $3.12, it is directly proportional to the total weight (which is not exactly the same as mass).
Step-by-step explanation:
Answer:
:p
Step-by-step explanation:
Length of a rectangle is four times its width if the area of the rectangle is 196 yards find its perimeter
Answer:
22 yd
Step-by-step explanation:
Length is four times its width -----> L=4W
area of rectangle is 196 -----> 196=LW
Plug L=4W into 196=LW giving you 196=(4W)W
Simplify a bit 196=4W^2
Divide both sides by 4: 196/4=W^2
Simplify a bit: 49=W^2
Square root both sides: 7 or -7=W
The width is 7 yd
The length is 4(7)=28 yd
Now its final part is for you to find the perimeter of this rectangle. The rectangle is a 7 yd by 4 yd rectangle.
Double both then add.. 14+8=22 yd
Answer:
21 yd
Step-by-step explanation:
Length of a rectangle is four times its width =
L = 4 × w
The area of the rectangle is 196
L = 4W
A = L*W = 196
---
4W*W = 196
W*W = 196/4
W*W = 49
W = 7
L = 28
---
Answer:
P = 2(L + W)
P = 2(28 + 7)
P = 14 + 7
P = 21 yd
need help on this math question !!!
17. Which relation is not a function?
A. {(–7,2), (3,11), (0,11), (13,11)}
B. {(7, 11), (11, 13), (–7, 13), (13, 11)}
C. {(7,7), (11, 11), (13, 13), (0,0)}
D. {(7, 11), (0,5), (11, 7), (7,13)}
Answer:
D. {(7, 11), (0,5), (11, 7), (7,13)}
Step-by-step explanation:
Because the relations (7, 11), (7,13) are false.
f(7) = 11
f(7) = 13
=> 11 = 13 (F)
Answer:
The correct answer option is: D. {(7, 11), (0,5), (11, 7), (7,13)}.
Step-by-step explanation:
We are given some paired values of inputs and outputs (x and y) for a function and we are to determine whether which of them is not a function.
For a relation to be function, no value of x should be repeated. It means that for each value of y, there should be a unique value of x (input).
In option D, two of the pairs have same value of x. Therefore, it is not a function.
{(7, 11), (0,5), (11, 7), (7,13)}
Suppose that the commuting time on a particular train is uniformly distributed between 42 and 52 minutes. Bold a. What is the probability that the commuting time will be less than 44 minutes
Answer: 0.2
Step-by-step explanation:
Given: The commuting time on a particular train is uniformly distributed over the interval (42,52).
∴ The probability density function of X will be :-
[tex]f(x)=\dfrac{1}{b-a}\\\\=\dfrac{1}{52-42}=\dfrac{1}{10}, 42<x<52[/tex]
Thus, the required probability :-
[tex]P(X<44)=\int^{44}_{42}f(x)\ dx\\\\=\int^{44}_{42}\dfrac{1}{10}\ dx\\\\=\dfrac{1}{10}[x]^{44}_{42}=\dfrac{1}{10}(44-42)=\dfrac{1}{5}=0.2[/tex]
Hence, the probability that the commuting time will be less than 44 minutes= 0.2
the probability that the commuting time will be less than 44 minutes is 0.2 or 20%.
To calculate the probability that the commuting time will be less than 44 minutes, given that it is uniformly distributed between 42 and 52 minutes, we use the formula for the probability of a continuous uniform distribution:
P(a < X < b) = (b - a) / (max - min)
Here, min is 42, max is 52, a is 42 (since that's the minimum time), and b is 44 (the time we are interested in). So the probability that the commuting time is less than 44 minutes will be:
P(42 < X < 44) = (44 - 42) / (52 - 42) = 2 / 10
This simplifies to 1/5 or 0.2.
Therefore, the probability that the commuting time will be less than 44 minutes is 0.2 or 20%.
Determine where, if anywhere, the tangent line to f(x)=x^3−5x^2+x is parallel to the line y=4x+23
Answer:
(1/3(5-√34), 1/27(-205+32√34)) and (1/3(5+√34), 1/27(-205-32√34))
Step-by-step explanation:
The slope of the given line is the x-coefficient, 4. Then you're looking for points on the f(x) curve where f'(x) = 4.
f'(x) = 3x^2 -10x +1 = 4
3x^2 -10x -3 = 0
x = (5 ±√34)/3 . . . . . x-coordinates of tangent points
Substituting these values into f(x), we can find the y-coordinates of the tangent points. The desired tangent points are ...
(1/3(5-√34), 1/27(-205+32√34)) and (1/3(5+√34), 1/27(-205-32√34))
_____
The graph shows the tangent points and approximate tangent lines.
To find where the tangent line to the function f(x)=[tex]x^3-5x^2+x[/tex] is parallel to the line y=4x+23, we need to find where the derivative of the function is equal to the slope of the given line.
Explanation:In the field of mathematics, the problem asks us to find at which points the tangent line to function f(x)=[tex]x^3-5x^2+x[/tex] is parallel to the line y=4x+23. A line is tangent to a function at a point where the derivative of that function equals the slope of the line. The first step in solving this problem is to calculate the derivative of function f(x).
The derivative of the function f(x) is f'(x) =[tex]3x^2 - 10x +1[/tex]. Next, we set this equal to the slope of the given line, which is 4. Solving this quadratic equation will give us the x-values where the tangent line is parallel to y=4x+23.
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Identify the y-intercept of the function, f(x) = 3x2 -5x + 2.
O (0,-2)
O (0,2)
O(-2,0)
O (2,0)
f(x) = 3x^2 - 5x +2
Let me show you the picture below and the answer is (0 , 2)
Answer:
(0,2)
Step-by-step explanation:
the y-intercept of the function, f(x) = 3x² -5x + 2 when : x = 0
f(0) = 3(0)² - 5(0)+2 = 2
Consider the differential equation x^2 y''-xy'-3y=0. If y1=x3 is one solution use redution of order formula to find a second linearly independent solution
Suppose [tex]y_2(x)=y_1(x)v(x)[/tex] is another solution. Then
[tex]\begin{cases}y_2=vx^3\\{y_2}'=v'x^3+3vx^2//{y_2}''=v''x^3+6v'x^2+6vx\end{cases}[/tex]
Substituting these derivatives into the ODE gives
[tex]x^2(v''x^3+6v'x^2+6vx)-x(v'x^3+3vx^2)-3vx^3=0[/tex]
[tex]x^5v''+5x^4v'=0[/tex]
Let [tex]u(x)=v'(x)[/tex], so that
[tex]\begin{cases}u=v'\\u'=v''\end{cases}[/tex]
Then the ODE becomes
[tex]x^5u'+5x^4u=0[/tex]
and we can condense the left hand side as a derivative of a product,
[tex]\dfrac{\mathrm d}{\mathrm dx}[x^5u]=0[/tex]
Integrate both sides with respect to [tex]x[/tex]:
[tex]\displaystyle\int\frac{\mathrm d}{\mathrm dx}[x^5u]\,\mathrm dx=C[/tex]
[tex]x^5u=C\implies u=Cx^{-5}[/tex]
Solve for [tex]v[/tex]:
[tex]v'=Cx^{-5}\implies v=-\dfrac{C_1}4x^{-4}+C_2[/tex]
Solve for [tex]y_2[/tex]:
[tex]\dfrac{y_2}{x^3}=-\dfrac{C_1}4x^{-4}+C_2\implies y_2=C_2x^3-\dfrac{C_1}{4x}[/tex]
So another linearly independent solution is [tex]y_2=\dfrac1x[/tex].
The tread life of a particular brand of tire is a random variable best described by a normal distribution with a mean of 60,000 miles and a standard deviation of 1000 miles. What warranty should the company use if they want 96% of the tires to outlast the warranty? 59,000 miles 58,250 miles 61,000 miles 61,750 miles
Answer: 61,750 miles
Step-by-step explanation:
Given : The p-value of the tires to outlast the warranty = 0.96
The probability that corresponds to 0.96 from a Normal distribution table is 1.75.
Mean : [tex]\mu=60,000\text{ miles}[/tex]
Standard deviation : [tex]\sigma=1000\text{ miles}[/tex]
The formula for z-score is given by : -
[tex]z=\dfrac{x-\mu}{\sigma}\\\\\Rightarrow\ 1.75=\dfrac{x-60000}{1000}\\\\\Rightarrow\ x-60000=1750\\\\\Rightarrow\ x=61750[/tex]
Hence, the tread life of tire should be 61,750 miles if they want 96% of the tires to outlast the warranty.
The company looking to ensure 96% of the tires outlast the warranty should use a mileage warranty of 58,250. This ensures a failure rate of only 4%, as calculated using statistics and the concept of normal distribution.
Explanation:The question deals with the concept of normal distribution in statistics, specifically, with an application to a real-life situation - to decide the warranty for a product (in this case, a type of tire). The mean and standard deviation given represent the average life and variation in life of the tires respectively. If the company wants 96% of the tires to outlast the warranty, then they are looking to find the lifespan beyond which only 4% of the tires would fail.
The Z-score corresponding to the 96th percentile in a standard normal distribution table is roughly 1.75. Since standard deviation is 1,000 miles, that implies that 1.75 standard deviations below the average is acceptable for the warranty. Therefore, we calculate the warranty as Mean - 1.75*Standard Deviation, which results in 60,000 - 1.75*1000 = 58,250 miles. Thus, the company should use a warranty of 58,250 miles.
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