Let assume that [tex]\sqrt3[/tex] is rational. Therefore we can express it as [tex]\dfrac{a}{b}[/tex] where [tex]a,b\in \mathbb{Z}[/tex] and [tex]\text{gcd}(a,b)=1[/tex].
[tex]\dfrac{a}{b}=\sqrt3\\\dfrac{a^2}{b^2}=3\\a^2=3b^2[/tex]
It means that [tex]3|a^2[/tex] and so also [tex]3|a[/tex].
Therefore [tex]a=3k[/tex] where [tex]k\in\mathbb{Z}[/tex].
[tex](3k)^2=3b^2\\9k^2=3b^2\\b^2=3k^2[/tex]
It means that [tex]3|b^2[/tex] and so also [tex]3|b[/tex].
If both [tex]a[/tex] and [tex]b[/tex] are divisible by 3, then it contradicts our initial assumption that [tex]\text{gcd}(a,b)=1[/tex]. Therefore [tex]\sqrt3[/tex] must be an irrational number.
If San Francisco accounts for 1.24 percent of total U.S. population, and has 1.43 percent of total U.S. laundry detergent sales, what is the CDI for this market? Also, what does this index mean? Remember the convention for CDIs and BDIs—they are expressed as whole numbers.
Answer:
CDI: 1.43/1.24x100= 115 What does this index mean? Good market potential.
Step-by-step explanation:
Answer: CDI: 1.43/1.24x100= 115 What does this index mean? Good market potential.
Step-by-step explanation:
Which of the following is the graph of y=-4 sqrt x
Answer: The answer should be D on edg
Please see the attachment for graph.
We are a given a equation and need to graph it.It is square root function. We make table of x and y and then plot the points on graph and join the points.We will take some random values of x and then find the value of y corresponding to the value of x.For x=1, Put x=1 into equation:
[tex]y = -4\sqrt{1} = - 4[/tex]
For x=4, Put x=4 into equation:
[tex]y = -4\sqrt{4} = - 8[/tex]
For x=9, Put x=9 into equation:
[tex]y = -4\sqrt{9} = - 12[/tex]
Table of x and y: x y1 -4
4 -8
9 -12
Now we plot the points on graph and join the points.
Please see the attachment for graph.
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A family has five children. The probability of having a girl is 2 What is the probability of having no girls? Round the answer to the fourth decimal place
Answer: Hence, the probability of having no girls is 0.0313.
Step-by-step explanation:
Since we have given that
Number of children a family has = 5
Number of outcomes would be [tex]2^5=32[/tex]
Probability of having a girl = [tex]\dfrac{1}{2}=0.5[/tex]
We need to find the probability of having no girls.
P(no girls ) = P( all boys )
So, it becomes,
[tex]P(all\ boys)=(0.5)^5=0.03125\approx 0.0313[/tex]
Hence, the probability of having no girls is 0.0313.
A student gently drops a ball from different heights and measures the time it takes to fall to the ground. Which statement BEST describes why the investigation is an experimental study? A) The student does not use a control group. B) There is only one independent variable involved. C) The student sets the values of the independent variable. D) It is possible to establish a cause-effect relation between the variables.
Answer:
c
Step-by-step explanation:
i think not 100 percent sure
Determine whether f(x)=-5x^2-10x+6 has a maximum or a minimum value. Find that value and explain how you know.
Answer:
(-1, 11) is a max value; parabola is upside down
Step-by-step explanation:
We can answer this question backwards, just from what we know about parabolas. This is a negative x^2 parabola, so that means it opens upside down. Because of this, that means that there is a max value.
The vertex of a parabola reflects either the max or the min value. In order to find the vertex, we put the equation into vertex form, which has the standard form:
[tex]y=a(x-h)^2+k[/tex]
where h and k are the coordinates of the vertex.
To put a quadratic into vertex form, you need to complete the square. That process is as follows. First, set the quadratic equal to 0. Then make sure that the leading coefficient is a positive 1. Ours is a -5 so we will have to factor it out. Then, move the constant to the other side of the equals sign. Finally, take half the linear term, square it, and add it to both sides. We will get that far, and then pick up with the rest of the process as we come to it.
[tex]-5x^2-10x+6=y[/tex]
Set it to equal zero:
[tex]-5x^2-10x+6=0[/tex]
Now move the 6 to the other side:
[tex]-5x^2-10x=-6[/tex]
Factor out the -5:
[tex]-5(x^2+2x)=-6[/tex]
Take half the linear term, square it, and add it to both sides. Our linear term is 2x. Half of 2 is 1, and 1 squared is 1, so add it to both sides. Keep it mind that we have the =5 out front of those parenthesis that will not be forgotten. So we are not adding in a +1, we are adding in a (+1)(-5) which is -5:
[tex]-5(x^2+2x+1)=-6-5[/tex]
In completing the square, we have created a perfect square binomial on the left. Stating that binomial along with simplifying on the right gives us:
[tex]-5(x+1)^2=-11[/tex]
Now, bring the -11 over to the other side and set it back to equal y and you're ready to state the vertex:
[tex]-5(x+1)^2+11=y[/tex]
The vertex is at (-1, 11)
Help need on 2 algebra problems !!!!!!! please
Evaluate the root without using a calculator, or note that the root isn't a real number.
1. square root 8√256
A. Not a real number
B. 16
C. 2
D. 4
2.square root 4√16
A. 2
B. –2
C. 3
D. Not a real number
Answer:
1) C. 2
2) A. 2
Step-by-step explanation:
1. We need to descompose 256 into its prime factors:
[tex]256=2*2*2*2*2*2*2*2=2^8[/tex]
We must rewrite the expression [tex]\sqrt[8]{256}[/tex]:
[tex]=\sqrt[8]{2^8}[/tex]
We need to remember that:
[tex]\sqrt[n]{a^n}=a[/tex]
Then:
[tex]=2[/tex]
2. Let's descompose 16 into its prime factors:
[tex]16=2*2*2*2=2^4[/tex]
We must rewrite the expression [tex]\sqrt[4]{16}[/tex]:
[tex]=\sqrt[4]{2^4}[/tex]
Then we get:
[tex]=2[/tex]
For [tex]\(\sqrt[8]{256}\)[/tex] , the answer is 2 (C), and for [tex]\(\sqrt[4]{16}\)[/tex], the answer is also 2(A), obtained through prime factorization and simplifying using the property [tex]\(\sqrt[n]{a^n} = a\)[/tex].
Let's go into more detail for both questions:
1. [tex]\(\sqrt[8]{256}\)[/tex]:
- First, find the prime factorization of 256: [tex]\(256 = 2^8\)[/tex]
- Rewrite the expression as [tex]\(\sqrt[8]{2^8}\)[/tex].
- Using the property [tex]\(\sqrt[n]{a^n} = a\)[/tex], simplify to 2.
- Therefore, [tex]\(\sqrt[8]{256} = 2\)[/tex]
- Correct answer: C. 2
2. [tex]\(\sqrt[4]{16}\)[/tex]:
- Start with the prime factorization of 16: [tex]\(16 = 2^4\)[/tex]
- Express the expression as [tex]\(\sqrt[4]{2^4}\)[/tex]
- Apply the property [tex]\(\sqrt[n]{a^n} = a\) to get \(2\).[/tex]
- Thus, [tex]\(\sqrt[4]{16} = 2\)[/tex]
- Correct answer: A. 2
In both cases, understanding the prime factorization and utilizing the property of radicals [tex](\(\sqrt[n]{a^n} = a\))[/tex] helps simplify the expressions and find the correct values.
Place the indicated product in the proper location on the grid. -4x3y2(7xy4)
Answer:
The product is:
[tex]-28x^4y^6[/tex]
Step-by-step explanation:
We need to find product of the terms:
-4x3y2(7xy4)
For multiplication we multiply constants with constants and power of same variables are added
[tex]-4x^3y^2(7xy^4)\\=(-4*7)(x^3*x)(y^2*y^4)\\=(-28)(x^{3+1})(y^{2+4})\\=(-28)(x^4)(y^6)\\=-28x^4y^6[/tex]
So, the product is:
[tex]-28x^4y^6[/tex]
Manuel and Ruben both have bank accounts. The system of equations models their balances after x weeks. y = 11.5x + 22 y = –13x + 218 Their balances will be the same after weeks. Their balances will be $
Answer:
The equal balances will be $114 after 8 weeks
Step-by-step explanation:
* Lets study the information in the problem
- Manuel and Ruben both have bank accounts
- The system of equations models their balances y after x weeks
- Manuel balance is y = 11.5x + 22
- Ruben balance is y = -13x + 218
- After x weeks they will have same balances, means the values of y
will be equal at the same values of x
- The solve the problem we will equate the two equations to find x
and then substitute this x in on of the equation s to find the
balance y
- Lets do that
∵ Manuel balance is y = 11.5x + 22
∵ Ruben balance is y = -13x + 218
∵ After x weeks their balances will be equal
- Equate the equations
∴ 11.5x + 22 = -13x + 218
- add 13 x for both sides
∴ 11.5x + 13x + 22 = 218
∴ 24.5x + 22 = 218
- subtract 22 from both sides
∴ 24.5x = 218 - 22
∴ 24.5x = 196
- Divide both sides by 24.5
∴ x = 8
- Their balances will be equals after 8 weeks
- To find the balance substitute x by 8 in any equation
∵ y = 11.5x + 22
∵ x = 8
∴ y = 11.5(8) + 22
∴ y = 92 + 22 = 114
∴ The equal balances will be $114
* The equal balances will be $114 after 8 weeks
Answer:
The equal balances will be $114 after 8 weeks
Step-by-step explanation:
David estimated he had about 20 fish in his pond. A year later, there were about 1.5 times as many fish. The year after that, the number of fish increased by a factor of 1.5 again. The number of fish is modeled by f(x)=20(1.5)^x.
Create a question you could ask that could be answered only by graphing or using a logarithm.
Answer:
After how many years is the fish population 60?
x=2.71 years
Step-by-step explanation:
The fish population increases by a factor of 1.5 each year. We have the equation that represents this situation
[tex]f (x) = 20 (1.5) ^ x[/tex]
Where x represents the number of years elapsed f(x) represents the amount of fish.
Given this situation, the following question could be posed
After how many years is the fish population 60?
So we do [tex]f (x) = 60[/tex] and solve for the variable x
[tex]60 = 20 (1.5) ^ x\\\\\frac{60}{20} = (1.5)^x\\\\3 = (1.5)^x\\\\log_{1.5}(3) = log_{1.5}(1.5)^x\\\\log_{1.5}(3) = x\\\\x =log_{1.5}(3)\\\\x=2.71\ years[/tex]
Observe the solution in the attached graph
Raise the quality in parentheses to the indicated exponent, and slim lift 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]\frac {-y ^ {- 2 * -2}} {4x ^ {- 5 * -2} * y ^ {- 4 * -2}} =\\\frac {-y ^ {4}} {4x ^ {10} * y ^ {8}} =[/tex]
SImplifying:
[tex]\frac {-y ^ {4-8}} {4x ^ {10}} =\\\frac {-y ^ {- 4}} {4x ^ {10}} =\\- \frac {1} {4x ^ {10} y^ {4}}[/tex]
Answer:
[tex]- \frac {1} {4x ^ {10} y ^ {4}}[/tex]
Suppose that the number of calls coming per minute into an airline reservation center follows a Poisson distribution. Assume that the mean is 3 calls per minute. The probability that at least two calls are received in a given two-minute period is _______.
Answer: 0.9826
Step-by-step explanation:
Given : Mean : [tex]\lambda =3\text{ calls per minute}[/tex]
For two minutes period the new mean would be :
[tex]\lambda_1=2\times3=6\text{ calls per two minutes}[/tex]
The formula to calculate the Poisson distribution is given by :_
[tex]P(X=x)=\dfrac{e^{-\lambda_1}\lambda_1^x}{x!}[/tex]
Then ,the required probability is given by :-
[tex]P(X\geq2)=1-(P(X\leq1))\\\\=1-(P(0)+P(1))\\\\=1-(\dfrac{e^{-6}6^0}{0!}+\dfrac{e^{-6}6^1}{1!})\\\\=1-0.0173512652367\\\\=0.982648734763\approx0.9826[/tex]
Hence, the probability that at least two calls are received in a given two-minute period is 0.9826.
Suppose θ is an angle in the standard position whose terminal side is in Quadrant IV and cotθ = -6/7 . Find the exact values of the five remaining trigonometric functions of θ. Find the exact values of the five remaining trigonometric functions of θ.
let's recall that on the IV Quadrant the sine/y is negative and the cosine/x is positive, whilst the hypotenuse is never negative since it's just a distance unit.
[tex]\bf \stackrel{\textit{on the IV Quadrant}}{cot(\theta )=\cfrac{\stackrel{adjacent}{6}}{\stackrel{opposite}{-7}}}\qquad \impliedby \textit{let's find the \underline{hypotenuse}} \\\\\\ \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies c=\sqrt{a^2+b^2} \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases} \\\\\\ c=\sqrt{6^2+(-7)^2}\implies c=\sqrt{36+49}\implies c=\sqrt{85} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\bf tan(\theta )=\cfrac{\stackrel{opposite}{-7}}{\stackrel{adjacent}{6}}\qquad \qquad sec(\theta )=\cfrac{\stackrel{hypotenuse}{\sqrt{85}}}{\stackrel{adjacent}{6}}\qquad \qquad csc(\theta )=\cfrac{\stackrel{hypotenuse}{\sqrt{85}}}{\stackrel{opposite}{-7}}[/tex]
[tex]\bf sin(\theta )=\cfrac{\stackrel{opposite}{-7}}{\stackrel{hypotenuse}{\sqrt{85}}}\implies \stackrel{\textit{and rationalizing the denominator}}{\cfrac{-7}{\sqrt{85}}\cdot \cfrac{\sqrt{85}}{\sqrt{85}}\implies -\cfrac{7\sqrt{85}}{85}} \\\\\\ cos(\theta )=\cfrac{\stackrel{adjacent}{6}}{\stackrel{hypotenuse}{\sqrt{85}}}\implies \stackrel{\textit{and rationalizing the denominator}}{\cfrac{6}{\sqrt{85}}\cdot \cfrac{\sqrt{85}}{\sqrt{85}}\implies \cfrac{6\sqrt{85}}{85}}[/tex]
Answer:
These are the five remaining trigonometric functions:
tanθ = - 7/6secθ = (√85) / 6cosθ = 6(√85) / 85sinθ = - 7(√85) / 85cscθ = - (√85)/7Explanation:
Quadrant IV corresponds to angle interval 270° < θ < 360.
In this quadrant the signs of the six trigonometric functions are:
sine and cosecant: negativecosine and secant: positivetangent and cotangent: negativeThe expected values of the five remaining trigonometric functions of θ are:
1) Tangent:
tan θ = 1 / cot (θ) = 1 / [ -6/7] = - 7/62) Secant
sec²θ = 1 + tan²θ = 1 + (-7/6)² = 1 + 49/36 = 85/36sec θ = ± (√85)/ 6
Choose positive, because secant is positive in Quadrant IV.
sec θ = (√85) / 6
3) Cosine
cosθ = 1 / secθ = 6 / (√85) = 6 (√85) / 854) Sine
sin²θ + cos²θ = 1 ⇒ sin²θ = 1 - cos²θ = 1 - [6(√85) / 85] ² =sin²θ = 1 - 36×85/(85)² = 1- 36/85 = 49/85
sinθ = ± 7 / (√85) = ± 7(√85)/85
Choose negative sign, because it is Quadrant IV.
sinθ = - 7 (√85) / 85
5) Cosecant
cscθ = 1 / sinθ = - 85 / (7√85) = - (√85) / 7The graph shows a distribution of data. What is the standard deviation of the data? A)0.5 B)1.5 C)2.0 D)2.5
Answer:
A)0.5
Step-by-step explanation:
We can see in the graph , that it is bell-shaped along x =2. A bell-shaped graph along one value is called symmetric graph and it represents a normal distribution.
Since, the give graph is symmetric around x=2, so the mean of the data is 2.
The point immediate left to the mean represents x-σ
so,
2 - σ = 1.5
So,
σ = 0.5
The sigma represents standard deviation.
Hence, Option A is correct ..
Answer:
its A
Step-by-step explanation:
What method would you choose to solve the equation 2x2 – 7 = 9? Explain why you chose this method.
The simplification method would be the best to solve the given equation.
What is simplification?simplify means making it in a simple form by reducing variables in an equation. we can achieve simplification easily by using PEMDAS.
Given equation 2x² - 7 = 9;
By simplify
2x² = 16
x² = 8
x = √8, -√8
Hence, for given equation simplification using PEMDAS is the best way of solving because it can be easily broken into parts to find the value of x.
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You have two exponential functions. One function has the formula g(x) = 3(2 x ). The other function has the formula h(x) = 2 x+1. Which option below gives formula for k(x) = (g – h)(x)? k(x) = 2x k(x) = 5(2x) k(x) = 5(2x+1) k(x) = 2
Answer:
[tex]k(x)=2^{x}[/tex] ⇒ 1st answer
Step-by-step explanation:
* Lets explain how to solve the problem
∵ [tex]g(x)=3(2^{x})[/tex]
∵ [tex]h(x)=2^{x+1}[/tex]
- Lets revise this rule to use it
# If [tex]a^{n}*a^{m}=a^{n+m}====then==== a^{n+m}=a^{n}*a^{m}[/tex]
- We will use this rule in h(x)
∵ [tex]h(x)=2^{x+1}[/tex]
- Let a = 2 , n = x , m = 1
∴ [tex]h(x)=2^{x}*2^{1}[/tex]
- Now lets find k(x)
∵ k(x) = (g - h)(x)
∵ [tex]g(x)=3(2^{x})[/tex]
∵ [tex]h(x)=2^{x}*2^{1}[/tex]
∴ [tex]k(x)=3(2^{x})-(2^{x}*2^{1})[/tex]
- We have two terms with a common factor [tex]2^{x}[/tex]
∵ [tex]2^{x}[/tex] is a common factor
∵ [tex]\frac{3(2^{x})}{2^{x}}=3[/tex]
∵ [tex]\frac{2^{x}*2^{1}}{2^{x}}=2^{1}=2[/tex]
∴ [tex]k(x) = 2^{x}[3 - 2]=2^{x}(1)=2^{x}[/tex]
* [tex]k(x)=2^{x}[/tex]
Use partial fraction expansion to evaluate: LaTeX: \int\frac{x-1}{x^2+3x+2}dx ∫ x − 1 x 2 + 3 x + 2 d x a. LaTeX: -2\ln\left|x+1\right|+3\ln\left|x+2\right|+C − 2 ln | x + 1 | + 3 ln | x + 2 | + C b. LaTeX: \frac{-2}{x+1}+\frac{3}{x+2}+C − 2 x + 1 + 3 x + 2 + C c. LaTeX: \frac{2}{\left(x+1\right)^2}+\frac{-3}{\left(x+2\right)^2}+C 2 ( x + 1 ) 2 + − 3 ( x + 2 ) 2 + C d. LaTeX: \frac{1}{\left(x+3+\frac{2}{x}\right)^2}+C 1 ( x + 3 + 2 x ) 2 + C
The correct answer is -2 ln|x + 1| + 3 ln|x + 2| + C which corresponds to option (a).
We are asked to evaluate the integral:
∫ (x - 1) ÷ (x² + 3x + 2) dx
First, factor the denominator:
x² + 3x + 2 = (x + 1)(x + 2)
This allows us to use partial fraction decomposition to rewrite the integral :
(x - 1) ÷ [(x + 1)(x + 2)] = A ÷ (x + 1) + B ÷ (x + 2)
Next, solve for A and B:
Multiply both sides by the denominator (x + 1)(x + 2):x - 1 = A(x + 2) + B(x + 1)Set up equations by plugging in values for x to solve for A and B:When x = -1 : -1 - 1 = A(-1 + 2) + B(-1 + 1)-2 = A(1) + B(0) , so A = -2When x = -2 : -2 - 1 = A(-2 + 2) + B(-2 + 1)-3 = -B, so B = 3So, we can write :
(x - 1) ÷ [(x + 1)(x + 2)] = -2 ÷ (x + 1) + 3 ÷ (x + 2)
Integrate both terms separately :
∫ (-2 ÷ (x + 1)) dx + ∫ (3 ÷ (x + 2)) dx
This gives us :
-2 ln|x + 1| + 3 ln|x + 2| + C
Hence, the solution is :
-2 ln|x + 1| + 3 ln|x + 2| + C
The correct answer is option (a).
Complete Question :
Use Partial fraction expansion to evaluate : ∫ (x - 1) ÷ (x² + 3x + 2) dx
a. -2 ln|x + 1| + 3 ln|x + 2| + C b. [tex]\frac{-2}{x+1}+\frac{3}{x+2}+C - 2 x + 1 + 3 x + 2 + C[/tex]
c. [tex]\frac{2}{\left(x+1\right)^2}+\frac{-3}{\left(x+2\right)^2}+C 2 ( x + 1 ) 2 + - 3 ( x + 2 ) 2 + C[/tex]
d. [tex]\frac{1}{\left(x+3+\frac{2}{x}\right)^2}+C 1 ( x + 3 + 2 x ) 2 + C[/tex]
An Access Ramp to a freeway extends horizontally a distance of 80 feet while it rises a total of 15 feet . Find the Slope of the Access Ramp. A) 16/3 B) 19/16 C) 65 D) 3/16 E) 95
Answer:
D.
Step-by-step explanation:
Slope is rise over run by definition, and we are given the values for each in the problem. The run is 80 and the rise is 15 so
[tex]m=\frac{15}{80}=\frac{3}{16}[/tex]
Find the two geometric means between 20 and 5. 7. Solve: 44-32-3 8. Develop the identity for sin 2.4 using the identity for sin(A+ B).
Answer with explanation:
1.
Let a, and b be two numbers between 20 and 5 , which is in geometric progression.
So,the series is as Follows =20 , a, b, 5
Common ratio
[tex]=\frac{\text{Second term}}{\text{First term}}[/tex]
[tex]\frac{20}{a}=\frac{a}{b}=\frac{b}{5}\\\\b^2=5 a---(1)\\\\a^2=20 b\\\\\frac{b^4}{25}=20 b-----\text{Using 1}\\\\b^3=500\\\\b=(500)^{\frac{1}{3}}\\\\b=5\times (4)^{\frac{1}{3}}\\\\5a=25\times (4)^{\frac{2}{3}}\\\\a=5\times (4)^{\frac{2}{3}}[/tex]
2.
44 -32-3
=12-3
=9
3.
⇒Sin (2.4)=Sin(2+0.4)
⇒Sin 2 ×Cos (0.4)+Cos 2 × Sin (0.4)
⇒Sin (A+B)=Sin A×Cos B+Cos A×Sin B
b. Two events are dependent if the occurrence of one event changes to occurrence of the second event. True or False
Answer:
true
Step-by-step explanation:
Answer:
True
Step-by-step explanation:
If 2 events are independent, then one event will not affect the other
what is the length of pr?
help me, thank tou so much :)
Answer:9
Step-by-step explanation:
1st triangle is similar to the second one as the angles of both of the triangles are the same..
So we know the ratio of the similar lines will be constant.it means,
XY/PQ=XZ/PR=YZ/QR
So,Xy/PQ=XZ/PR
21/7=27/x
X=(27×7)/21
X=9
Thats the value of pr..
How do I calculate this? Is there a formula?
A suspension bridge with weight uniformly distributed along its length has twin towers that extend 95 meters above the road surface and are 1200 meters apart. The cables are parabolic in shape and are suspended from the tops of the towers. The cables touch the road surface at the center of the bridge. Find the height of the cables at a point 300 meters from the center. (Assume that the road is level.)
Answer:
Height of cables = 23.75 meters
Step-by-step explanation:
We are given that the road is suspended from twin towers whose cables are parabolic in shape.
For this situation, imagine a graph where the x-axis represent the road surface and the point (0,0) represents the point that is on the road surface midway between the two towers.
Then draw a parabola having vertex at (0,0) and curving upwards on either side of the vertex at a distance of [tex]x = 600[/tex] or [tex]x = -600[/tex], and y at 95.
We know that the equation of a parabola is in the form [tex]y=ax^2[/tex] and here it passes through the point [tex](600, 95)[/tex].
[tex]y=ax^2[/tex]
[tex]95=a \times 600^2[/tex]
[tex]a=\frac{95}{360000}[/tex]
[tex]a=\frac{19}{72000}[/tex]
So new equation for parabola would be [tex]y=\frac{19x^2}{72000}[/tex].
Now we have to find the height [tex](y)[/tex]of the cable when [tex]x= 300[/tex].
[tex]y=\frac{19 (300)^2}{72000}[/tex]
y = 23.75 meters
Answer: 23.75 meters
Step-by-step explanation:
If we assume that the origin of the coordinate axis is in the vertex of the parabola. Then the function will have the following form:
[tex]y = a (x-0) ^ 2 + 0\\\\y = ax ^ 2[/tex]
We know that when the height of the cables is equal to 95 then the horizontal distance is 600 or -600.
Thus:
[tex]95 = a (600) ^ 2[/tex]
[tex]a = \frac{95} {600 ^ 2}\\\\a = \frac {19} {72000}[/tex]
Then the equation is:
[tex]y = \frac{19}{72000} x ^ 2[/tex]
Finally the height of the cables at a point 300 meters from the center is:
[tex]y = \frac{19}{72000}(300) ^ 2[/tex]
[tex]y =23.75\ meters[/tex]
Complete the table for the function and find the indicated limit.
limx→0 (x^3−6x+8/x−2)
(EQUATION AND ANSWER CHOICES BELOW)
Answer:
The last choice is the one you want
Step-by-step explanation:
If you plug in the values of x to our rational function, the y values you get back match those in the last choice. The limit is -4; we see that as our x value approach 0 (but cannot equal 0!!), the y values get closer and closer to -4. So that's the limit!
Five Card Draw is one of most basic forms of poker, and it's the kind of poker you're used to seeing in movies and on TV. This game has been around for a long time, and has been played in countless home games and card rooms across the nation. Play begins with each player being dealt five cards, one at a time, all face down. The remaining deck is placed aside, often protected by placing a chip or other marker on it. Players pick up the cards and hold them in their hands, being careful to keep them concealed from the other players, then a round of betting occurs. Some combinations of five-card hand have special names such as full house, royal flush, four of a kind, etc. Let`s find some 5-card combinations. Order of the drawn card does not matter. a) A flush is a poker hand, where all five cards are of the same suit, but not in sequence. Compute the number of a 5-card poker hands containing all diamonds.
Answer:
1287
Step-by-step explanation:
The number of combinations of 13 diamonds taken 5 at a time is ...
13C5 = 13·12·11·10·9/(5·4·3·2·1) = 13·11·9 = 1287
CNNBC recently reported that the mean annual cost of auto insurance is 954 dollars. Assume the standard deviation is 234 dollars. You take a simple random sample of 61 auto insurance policies. Find the probability that a single randomly selected value is at least 960 dollars.
Answer: 0.42
Step-by-step explanation:
Given: Mean : [tex]\mu=954\text{ dollars}[/tex]
Standard deviation : [tex]234\text{ dollars}[/tex]
Sample size : [tex]n=61[/tex]
The formula to calculate z score is given by :-
[tex]z=\dfrac{X-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
For X=960
[tex]z=\dfrac{960 -954}{\dfrac{234}{\sqrt{61}}}=0.200262812203\approx0.2[/tex]
The p-value =[tex]P(X\geq960)=1-P(X<960)=1-P(z<0.2)=1-0.5792597=0.4207403\approx0.42[/tex]
Hence, the probability that a single randomly selected value is at least 960 dollars = 0.42
A study claims that the mean age of online dating service users is 40 years. Some researchers think this is not accurate and want to show that the mean age is not 40 years. Identify the null hypothesis, H0, and the alternative hypothesis, Ha, in terms of the parameter μ. Select the correct answer below: H0: μ≥40; Ha: μ<40 H0: μ≤40; Ha: μ>40 H0: μ≠40; Ha: μ=40 H0: μ=40; Ha: μ≠40
Answer: [tex]H_0:\mu=40[/tex]
[tex]H_0:\neq40[/tex]
Step-by-step explanation:
A null hypothesis is a hypothesis where a researcher generally try to disprove, it says that there is no statistically significant relationship between the two variables . An alternative hypothesis says that there is a statistical significance between two variables.Claim 1. : Mean age of online dating service users is 40 years.
i.e. [tex]\mu=40[/tex], since it has equals sign so we take this as null hypothesis.
Claim 2. : Mean age of online dating service users is not 40 years.
[tex]\mu\neq40[/tex]
⇒ Null Hypothesis : [tex]H_0:\mu=40[/tex]
Alternative hypothesis : [tex]H_0:\neq40[/tex]
Lockheed Martin, the defense contractor designs and build communication satellite systems to be used by the U.S. military. Because of the very high cost the company performs numerous test on every component. These test tend to extend the component assembly time. Suppose the time required to construct and test (called build time) a particular component is thought to be normally distributed, with a mean equal to 45 hours and a standard deviation equal to 6.75 hours. To keep the assembly flow moving on schedule, this component needs to have a build time between 37.5 and 54 hours. Find the propability that the bulid time will be such that assembly will stay on schedule.
Answer:
p(on schedule) ≈ 0.7755
Step-by-step explanation:
A suitable probability calculator can show you this answer.
_____
The z-values corresponding to the build time limits are ...
z = (37.5 -45)/6.75 ≈ -1.1111
z = (54 -45)/6.75 ≈ 1.3333
You can look these up in a suitable CDF table and find the difference between the values you find. That will be about ...
0.90879 -0.13326 = 0.77553
The probability assembly will stay on schedule is about 78%.
Using the normal distribution, it is found that there is a 0.7747 = 77.47% probability that the build time will be such that assembly will stay on schedule.
Normal Probability Distribution
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
It measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score , which is the percentile of measure X.In this problem:
Mean of 45 hours, thus [tex]\mu = 45[/tex].Standard deviation of 6.75 hours, thus [tex]\sigma = 6.75[/tex].The probability of the time being between 37.5 and 54 hours is the p-value of Z when X = 54 subtracted by the p-value of Z when X = 37.5, then:X = 54
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{54 - 45}{6.75}[/tex]
[tex]Z = 1.33[/tex]
[tex]Z = 1.33[/tex] has a p-value of 0.9082.
X = 37.5
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{37.5 - 45}{6.75}[/tex]
[tex]Z = -1.11[/tex]
[tex]Z = -1.11[/tex] has a p-value of 0.1335.
0.9082 - 0.1335 = 0.7747.
0.7747 = 77.47% probability that the build time will be such that assembly will stay on schedule.
A similar problem is given at https://brainly.com/question/24663213
Determine whether the random variable is discrete or continuous. In each case, state the possible values of the random variable. (a) The number of fish caught during a fishing tournament . (b) The time it takes for a light bulb to burn out .
Answer: a.- discrete b.- continous
Step-by-step explanation: Discrete Variable. Variables that can only take on a finite number of values are called "discrete variables." All qualitative variables are discrete. Some quantitative variables are discrete, such as performance rated as 1,2,3,4, or 5, or temperature rounded to the nearest degree.
Continuous Variable. If a variable can take on any value between its minimum value and its maximum value, it is called a continuous variable; otherwise, it is called a discrete variable.
The number of fish caught during a fishing tournament is a discrete variable, and the time it takes for a light bulb to burn out is a continuous variable
Further explanationLet's define what variables are. Variables are any representation of a phenomenon or property that changes over time. In simple terms, variables are "things" that change, meaning they don't have a constant value. Variables can be either discrete or continuous.
To understand these concepts it's better to understand first what continuous means, continuous variables are those which can take any value whatsoever over time. This last statement is the main idea but it's not self-explanatory, a test to check whether a variable is continuous or not is to take any 2 possible outcomes of that variable, and check if that variable can take any value between those 2 possible outcomes. If the test gives positive then our variable is continuous, if not then it's discrete.
Let's test the first question. During a fishing tournament, each person can fish only one fish at a time, therefor possible outcomes are 1 fish, or 2 fish, or 3, or 4, and so on. This means that we will never be able to get, for example, 1.5 fish (which is a value between 2 possible outcomes, 1 fish and 2 fish), therefor our variable is discrete.
Let's test the second question. The time it takes for a light bulb to burn out has many possible outcomes, examples are 1 second, 2.5 seconds, 10 minutes, etc. If we check between any of those possible outcomes, we will always be able to find a time, doesn't matter how precise, in which the light bulb could burn. This means that the time for a light bulb to burn is continuous.
Learn moreComparison of other variables: https://brainly.com/question/12967959Analysis of type of variable: https://brainly.com/question/10697348KeywordsVariable, Continuous, Discrete, Interval
Which of the following vectors can be written as a linear combination of the vectors (1, 1, 2), (1, 2, 1) and (2, 1, 5)? (0.4,3.7,-1.5) (0.2,0) None of the selections is correct. All the selections are correct
Answer with explanation:
Let, A=[1,1,2]
B=[1,2,1]
C=[2,1,5]
⇒Now, Writing vector , A in terms of Linear combination of C and B
A=x B +y C
⇒[1,1,2]=x× [1,2,1] + y×[2,1,5]
1.→1 = x +2 y
2.→ 1=2 x +y
3.→ 2= x+ 5 y
Equation 3 - Equation 1
→3 y=1
[tex]y=\frac{1}{3}[/tex]
[tex]1=x+\frac{2}{3}\\\\x=1 -\frac{2}{3}\\\\x=\frac{1}{3}[/tex]
So, Vector A , can be written as Linear Combination of B and C.
⇒Now, Writing vector , B in terms of Linear combination of A and C
Now, let, B = p A+q C
→[1,2,1]=p× [1,1,2] +q ×[2,1,5]
4.→1= p +2 q
5.→2=p +q
6.→1=2 p +5 q
Equation 5 - Equation 4
-q =1
q= -1
→2= p -1
→p=2+1
→p=3
So, Vector B , can be written as Linear Combination of A and C.
⇒Now, Writing vector , C in terms of Linear combination of A and B
C=m A + n B
[2,1,5] = m×[1,1,2] + n× [1,2,1]
7.→2= m+n
8.→1=m +2 n
9.→5=2 m + n
Equation 8 - Equation 7
n= -1
→m+ (-1)=2
→m=2+1
→m=3
So, Vector C , can be written as Linear Combination of A and B.
So, All the three vectors , A=[1,1,2],B=[1,2,1],C=[2,1,5] can be written as Linear combination of each other.
⇒≡But , the two vectors, (0.4,3.7,-1.5) (0.2,0),can't be written as Linear combination of each other as first vector is of order, 1×3, and second is of order, 1×2.
None of the selections is correct.
How to apply linear combinations and linear independence to determine the existence of a relationship with a given vector
In this case, we must check the existence of a set of real coefficients such that the following two linear combinations exist:
[tex]\alpha_{1}\cdot (1, 1, 2)+\alpha_{2}\cdot (1, 2,1)+\alpha_{3}\cdot (2, 1, 5) = (0.4, 3.7, -1.5)[/tex] (1)
[tex]\alpha_{4}\cdot (1,1,2)+\alpha_{5}\cdot (1,2,1) + \alpha_{6}\cdot (2,1,5) = (0, 2, 0)[/tex] (2)
Now we proceed to solve each linear combination:
First system[tex]\alpha_{1}+\alpha_{2}+2\cdot \alpha_{3} = 0.4[/tex]
[tex]\alpha_{1}+2\cdot \alpha_{2}+\alpha_{3} = 3.7[/tex]
[tex]2\cdot \alpha_{1}+\alpha_{2}+5\cdot \alpha_{3} = -1.5[/tex]
The system has no solution, since the third equation is a linear combination of the first and second ones.
Second system[tex]\alpha_{4}+\alpha_{5}+2\cdot \alpha_{6} = 0[/tex]
[tex]\alpha_{4}+2\cdot \alpha_{5}+\alpha_{6} = 2[/tex]
[tex]2\cdot \alpha_{4}+\alpha_{5}+5\cdot \alpha_{6} = 0[/tex]
The system has no solution, since the third equation is a linear combination of the first and second ones.
None of the selections is correct. [tex]\blacksquare[/tex]
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Chords and arcs. Can someone please help me with this and explain???20 points
Answer:
89
Step-by-step explanation:
So the line segment CD is 12.7 and half that is 6.35. I wanted this 6.35 so I can look at the right triangle there and find the angle there near the center. This will only be half the answer. So I will need to double that to find the measure of arc CD.
Anyways looking at angle near center in the right triangle we have the opposite measurement, 6.35, given and the hypotenuse measurement, 9.06, given. So we will use sine.
sin(u)=6.35/9.06
u=arcsin(6.35/9.06)
u=44.5 degrees
u represented the angle inside that right triangle near the center.
So to get angle COD we have to double that which is 89 degrees.
So the arc measure of CD is 89.
A university knows from historical data that 25% of students in an introductory statistics class withdraw before completing the class. Assume that 16 students have registered for the course. What is the probability that exactly 2 will withdraw?
Answer:
13.4%
Step-by-step explanation:
Use binomial probability:
P = nCr p^r q^(n-r)
where n is the number of trials,
r is the number of successes,
p is the probability of success,
and q is the probability of failure (1-p).
Here, n = 16, r = 2, p = 0.25, and q = 0.75.
P = ₁₆C₂ (0.25)² (0.75)¹⁶⁻²
P = 120 (0.25)² (0.75)¹⁴
P = 0.134
There is a 13.4% probability that exactly 2 students will withdraw.
The probability that exactly 2 out of 16 students will withdraw from an introductory statistics class, given a historical withdrawal rate of 25%, can be calculated using the binomial probability formula.
Explanation:This problem falls into the category of binomial probability. We define 'success' as a student withdrawing from the course. The number of experiments is 16 (as there are 16 students), the number of successful experiments we are interested in is 2 (we want to know the probability of exactly 2 student withdrawing), and the probability of success on a single experiment is 0.25 (as per the given 25% withdrawal rate).
To calculate binomial probability, we can use the binomial formula P(X=k) = C(n, k)*(p^k)*((1-p)^(n-k)), where:
P(X=k) = probability of k successes
C(n, k) = combination of n elements taken k at a time
p = probability of success
n, k = number of experiments, desired number of successes respectively.
Substituting our values into this formula, we get:
P(X=2) = C(16, 2) * (0.25^2) * ((1-0.25)^(16-2)).
You will have to calculate the combination and simplify the expression to get your final probability.
Learn more about Binomial Probability here:https://brainly.com/question/33993983
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