Answer:
The vector equation of the line is [tex]\overrightarrow {r}=(11j-8k)+t(4i-4j+6k)[/tex] and parametric equations for the line are [tex]x=4t[/tex], [tex]y=11-4t[/tex], [tex]z=-8+6t[/tex].
Step-by-step explanation:
It is given that the line passes through the point (0,11,-8) and parallel to the line
[tex]x=-1+4t[/tex]
[tex]y=6-4t[/tex]
[tex]z=3+6t[/tex]
The parametric equation are defined as
[tex]x=x_1+at,y=y_1+bt,z=z_1+ct[/tex]
Where, (x₁,y₁,z₁) is point from which line passes through and <a,b,c> is cosine of parallel vector.
From the given parametric equation it is clear that the line passes through the point (-1,6,3) and parallel vector is <4,-4,6>.
The required line is passes through the point (0,11,-8) and parallel vector is <4,-4,6>. So, the parametric equations for the line are
[tex]x=4t[/tex]
[tex]y=11-4t[/tex]
[tex]z=-8+6t[/tex]
Vector equation of a line is
[tex]\overrightarrow {r}=\overrightarrow {r_0}+t\overrightarrow {v}[/tex]
where, [tex]\overrightarrow {r_0}[/tex] is a position vector and [tex]\overrightarrow {v}[/tex] is cosine of parallel vector.
[tex]\overrightarrow {r}=(11j-8k)+t(4i-4j+6k)[/tex]
Therefore the vector equation of the line is [tex]\overrightarrow {r}=(11j-8k)+t(4i-4j+6k)[/tex] and parametric equations for the line are [tex]x=4t[/tex], [tex]y=11-4t[/tex], [tex]z=-8+6t[/tex].
The vector equation and parametric equations for the line through the point (0, 11, −8) and parallel to the given line are established by using the point as the initial point and the direction ratios from the parallel line.
Explanation:The task is to find a vector equation and parametric equations for a line passing through the point (0, 11, −8) and parallel to given line equations x = −1 + 4t, y = 6 − 4t, z = 3 + 6t. To find these equations, we utilize the given point as the initial point and extract the direction ratios from the coefficients of t in the given parametric equations of the parallel line, which are (4, −4, 6).
The vector equation of the line can be written as ℓ = ℓ0 + t·d, where ℓ0 is the position vector of the initial point, and d is the direction vector. Since the given point is (0, 11, −8) and the parallel line's direction vector is (4, −4, 6), the vector equation is ℓ = (0, 11, −8) + t(4, −4, 6).
The parametric equations are derived directly from the vector equation. These are:
These equations represent the trajectory of the line through space, governed by the parameter t.
4.C.48 Calculate the current yield on the described bond. A $500 Treasury bond with a coupon rate of 2.8% that has a market value of $450 The current yield is %. (Round to two decimal places as needed.) tents hual uccess a Library Success le Resources
Answer:
The current yield is 3.11%.
Step-by-step explanation:
Given - A $500 Treasury bond with a coupon rate of 2.8% that has a market value of $450.
Now the face value of the bond is = $500
The rate of interest is = 2.8%
Then interest on $500 becomes:
[tex]0.028\times500=14[/tex] dollars
The current market value is $450
So, current yield is = [tex]\frac{14}{450}\times100= 3.11[/tex]%
The current yield is 3.11%.
Final answer:
The current yield on a $500 Treasury bond with a 2.8% coupon rate and a market value of $450 is 3.11%, calculated by dividing the annual coupon payment by the market value of the bond and then converting to a percentage.
Explanation:
To calculate the current yield on the described $500 Treasury bond with a coupon rate of 2.8% that has a market value of $450, follow these steps:
First, calculate the annual coupon payment by multiplying the face value of the bond by the coupon rate: $500 * 0.028 = $14.Next, divide the annual coupon payment by the market value of the bond: $14 / $450 = 0.0311.To express this as a percentage, multiply by 100: 0.0311 * 100 = 3.11%.Therefore, the current yield on the bond is 3.11% when rounded to two decimal places.
4. A study of 30 fathers was completed in which the fathers were asked the highest level of education they had completed. 10 completed only elementary school, 10 completed elementary school and high school; 7 completed elementary school, high school and college; 3 completed elementary school, high school, college and graduate school. What was the cumulative percentage of fathers who completed only elementary school?
Answer:
The cumulative percentage of fathers who completed only elementary school is nearle 33%.
Step-by-step explanation:
Among 30 fathers:
10 completed only elementary school;10 completed elementary school and high school;7 completed elementary school, high school and college;3 completed elementary school, high school, college and graduate school.You can fill these numbers into the table:
[tex]\begin{array}{cccc}&\text{Frequency}&\text{Cumulative frequency}&\text{Cumulative percentage}\\\text{Elementary school}&10&10&\approx 33\%\\\text{El. and high school}&10&20&\approx 67\%\\\text{El., high schools and college}&7&27&90\%\\\text{El., high, college and grad. sch.}&3&30&100\%\end{array}[/tex]
The cumulative percentage of fathers who completed only elementary school is nearle 33%.
Answer: 33%
Step-by-step explanation:
A vehicle factory manufactures cars. The unit cost C (the cost in dollars to make each car) depends on the number of cars made. If x cars are made, then the unit cost is given by the function C(x) = x^2 - 400x + 45,377 . What is the minimum unit cost?
Do not round your answer.
Answer:
5377
Step-by-step explanation:
C(x) = x^2 - 400x + 45,377
To find the location of the minimum, we take the derivative of the function
We know that is a minimum since the parabola opens upward
dC/dx = 2x - 400
We set that equal to zero
2x-400 =0
Solving for x
2x-400+400=400
2x=400
Dividing by2
2x/2=400/2
x=200
The location of the minimum is at x=200
The value is found by substituting x back into the equation
C(200) = (200)^2 - 400(200) + 45,377
=40000 - 80000+45377
=5377
Answer:
The minimum unit cost is 5377
Step-by-step explanation:
Note that we have a cudratic function of negative principal coefficient.
The minimum value reached by this function is found in its vertex.
For a quadratic function of the form
[tex]ax ^ 2 + bx + c[/tex]
the x coordinate of the vertex is given by the following expression
[tex]x=-\frac{b}{2a}[/tex]
In this case the function is:
[tex]C(x) = x^2 - 400x + 45,377[/tex]
So:
[tex]a=1\\b=-400\\c=45,377[/tex]
Then the x coordinate of the vertex is:
[tex]x=-\frac{-400}{2(1)}[/tex]
[tex]x=200\ cars[/tex]
So the minimum unit cost is:
[tex]C(200) = (200)^2 - 400(200) + 45,377[/tex]
[tex]C(200) = 5377[/tex]
Divide
(x^2 - 13x +40) divided by (x- 6)
Answer:
The quotient is: x-7
The remainder is: -2
Step-by-step explanation:
We need to divide (x^2 - 13x +40) divided by (x- 6)
The quotient is: x-7
The remainder is: -2
The division is shown in the figure attached
Color blindness is an inherited characteristic that is more common in males than in females. If M represents male and C represents red-green color blindness, using the relative frequencies of the incidences of males and red-green color blindness as probabilities results in the values below. P(C)=0.042, P ( M intersect C)=0.022, P ( M union C)=0.412What is the probability that a randomly selected person is not color blind?
Answer:
The probability that a randomly selected person is not color blind is 0.958
Step-by-step explanation:
Given,
C represents red-green color blindness,
Also, the probability that a randomly selected person is color blind,
P(C) = 0.042,
Thus, probability that a randomly selected person is not color blind,
P(C') = 1 - P(C) = 1 - 0.042 = 0.958
The mean length of six-year-old rainbow trout in the Arolik River in Alaska is 481 millimeters with a standard deviation of 41 millimeters. Assume these lengths are normally distributed. What proportion of six-year-old rainbow trout are less than 516 millimeters long?
Answer: 0.8023
Step-by-step explanation:
Given : [tex]\text{Mean}=\mu=481 \text{ millimeters}[/tex]
[tex]\text{Standard deviation}=41 \text{ millimeters}[/tex]
Assuming these lengths are normally distributed.
The formula to calculate the z-score is given by :-
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x= [tex]516 \text{ millimeters}[/tex]
[tex]z=\dfrac{516-481}{41}=0.853658536585\approx0.85[/tex]
The p-value = [tex]P(z\leq0.85)=0.8023374\approx0.8023[/tex]
Hence, the required probability : 0.8023
Use Gauss's approach to find the following sum
4+10+16+22+...+70
The sum of the sequence is
Each consecutive term in the sum is separated by a difference of 6, so the [tex]n[/tex]-th term is [tex]4+6(n-1)=6n-2[/tex] for [tex]n\ge1[/tex]. The last term is 70, so there are [tex]6n-2=70\implies n=12[/tex] terms in the sum.
Now,
[tex]S=4+10+\cdots+64+70[/tex]
but also
[tex]S=70+64+\cdots+10+4[/tex]
Doubling the sum and grouping terms in the same position gives
[tex]2S=(4+70)+(10+64)+\cdots+(64+10)+(70+4)=12\cdot74[/tex]
[tex]\implies\boxed{S=444}[/tex]
could someone give me an answer and explain how you got it ?
Answer:
A (9, 3)
Step-by-step explanation:
First the point is rotated 90° counterclockwise about the origin. To do that transformation: (x, y) → (-y, x).
So S(-3, -5) becomes S'(5, -3).
Next, the point is translated +4 units in the x direction and +6 units in the y direction.
So S'(5, -3) becomes S"(9, 3).
The distribution of the amount of money spent on book purchases for a semester by college students has a mean of $280 and a standard deviation of $40. If the distribution is bell-shaped and symmetric, what proportion of students will spend between $200 and $280 this semester? Round your answer to four decimal places.
Answer: 0.4772
Step-by-step explanation:
Given : The distribution is bell shaped , then the distribution must be normal distribution.
Mean : [tex]\mu=\$280[/tex]
Standard deviation :[tex]\sigma= \$40[/tex]
The formula to calculate the z-score :-
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x = $200
[tex]z=\dfrac{200-280}{40}=-2[/tex]
For x = $280
[tex]z=\dfrac{280-280}{40}=0[/tex]
The p-value = [tex]P(-2<z<0)=P(z<0)-P(z<-2)[/tex]
[tex]0.5-0.0227501=0.4772499\approx0.4772[/tex]
Hence, the proportion of students will spend between $200 and $280 this semester = 0.4772
Approximately 34% of the college students will spend between $200 and $280 on book purchases for a semester according to the empirical rule for a bell-shaped and symmetric distribution.
Explanation:The distribution of the amount of money spent on book purchases for a semester by college students is described as bell-shaped and symmetric with a mean of $280 and a standard deviation of $40. To find the proportion of students that will spend between $200 and $280, we can use the empirical rule (68-95-99.7 rule) which indicates that approximately 68% of the data fall within one standard deviation from the mean. Since we are concerned with the range from $200 (which is 2 standard deviations below the mean) to $280 (the mean), we are effectively looking at half of this 68% range. Therefore, approximately 34% of the students are expected to spend between $200 and $280 on books for a semester.
To calculate the proportion, we take 68% of the range (which covers -1 to +1 standard deviation from the mean) and divide it by 2:
68% / 2 = 34%Thus, the proportion of students spending between $200 and $280 is 0.34, which when rounded to four decimal places, gives us 0.3400.
Who directed Dr. Strangelove? a. Warren Beatty b. Peter Fonda c. Dustin Hoffman d. Stanley Kubrick
Answer:
D. Stanley Kubrick
Step-by-step explanation:
Answer:
[tex]\Large\textnormal{(D.) Stanley Kubrick}[/tex]
Step-by-step explanation:
Stanley Kubrick directed to Dr. Strangelove. I hope this helps, and have a wonderful day!
Exercise 1.4.4: Solve y' + xy = x.
Answer:
Step-by-step explanation:
Let's rewrite this with dy/dx in place of y', since they mean the same thing. But to solve a differential we will need to take the antiderivative by separation to find the general solution.
[tex]\frac{dy}{dx}+xy=x[/tex]
The goal is to get the x stuff on one side and the y stuff on the other side by separation. But we have an xy term there that we need to be able to break apart. So let's get everything on one side separate from the dy/dx and take it from there.
[tex]\frac{dy}{dx}=x-xy[/tex]
Now we can factor out the x:
[tex]\frac{dy}{dx}=x(1-y)[/tex]
And now we can separate:
[tex]\frac{dy}{(1-y)}=x dx[/tex]
Now we solve by taking the antiderivative of both sides:
[tex]\int\ {\frac{1}{1-y}dy }=\int\ {x} \, dx[/tex]
On the left side, the antiderivative of the derivative of y cancels out, and the other part takes on the form of the natural log, while we follow the power rule backwards on the right to integrate x:
[tex]ln(1-y)=\frac{1}{2}x^2+C[/tex]
That's the general solution. Not sure what your book has you solving for. Some books solve for the constant, C. Some solve for y when applicable. I'm leaving it like it is.
Find an equation of the line that contains the following pair of points. (3,2) and (1,3) The equation of the line is (Simnlif vour ancwor Llco.intogore or fractione.for anu.numborc.in.the aquatio
Answer: [tex]x+2y-7=0[/tex]
Step-by-step explanation:
We know that the equation of a line passing through points (a,b) and (c,d) is given by :-
[tex](y-b)=\dfrac{d-b}{c-a}(x-a)[/tex]
Then , the equation of a line passing through points (3,2) and (1,3) is given by :-
[tex](y-2)=\dfrac{3-2}{1-3}(x-3)\\\\\Rightarrow\ (y-2)=\dfrac{1}{-2}(x-3)\\\\\Rightarrow\ -2(y-2)=(x-3)\\\\\Rightarrow\ -2y+4=x-3\\\\\Rightarrow\ x+2y-7=0[/tex]
Hence, the equation of a line passing through points (3,2) and (1,3) is : [tex]x+2y-7=0[/tex]
Find x.
A. 124
B.56
C.62
D.28
Answer:
C. 62 degrees
Step-by-step explanation:
Alright I see a circle and a half a rotation where the diameter is at. A half of 360 degrees is 180 degrees.
So the arc measure in degrees for EG is 180 degrees (both the left piece and right piece have this measure).
Since EG is 180 then FG=EG-EF=180-56=124.
To find x we have to half 124 since it is the arc measure where x is but x is the inscribed angle.
x=124/2=62
C.
62 degrees ! All the rest wouldn’t make sense (:
write a 6th order homogeneous linear equation whose general solution is
y=C1+C2e-t+C3Cos(3t)+C4sin(3t)+C5tcos(3t)+C6tsin(3t)
Answer:
[tex] y''''''+y'''''+18y''''+18y'''+81y''+81y'=0[/tex]
Step-by-step explanation:
We are given that a general solution of 6th order homogeneous linear equation
[tex] y=C_1+C_2e^{-t}+C_3 Cos(3t)+C_4 Sin(3t)+C_5 tCos(3t) +C_6 sin t (3t)[/tex]
We have to find the 6th order homogeneous linear differential equation whose general solution is given above.
We know that imaginary roots are in pair
There two values of imaginary roots and the values of imaginary roots are repeat.
From first term of general solution we get D=0
From second term of general solution we get D=-1
Last four terms are the values of imaginary roots and roots are repeated.
Therefore, D=[tex]\pm 3i[/tex] and D=[tex]\pm 3i[/tex]
Substitute all values then we get
[tex]D(D+1)(D^2+9)^2=0[/tex]
[tex]D(D+1)(D^4+18D^2+81)=0[/tex]
[tex]D^6+D^5+18D^4+18D^3+81 D^2+81 D=0[/tex]
[tex](D^6+D^5+18D^4+18D^3+81 D^2+81 D)y=0[/tex]
[tex] y''''''+y'''''+18y''''+18y'''+81y''+81y'=0[/tex]
Therefore, the 6th order homogeneous linear differential equation is
[tex] y''''''+y'''''+18y''''+18y'''+81y''+81y'=0[/tex]
One card is selected at random from.a standard deck of 52 cards. Determine the probability that the card selected is a club or a picture cardst sd 10.
Answer:
The answer is [tex]\frac{11}{26}[/tex]
Step-by-step explanation:
Total number of cards in the deck = 52
Number of clubs = 13
Number of picture cards = 12
Number of picture cards that are clubs = 3
So, number of picture cards or clubs = [tex]13+12-3=22[/tex]
P(club or picture) = [tex]\frac{22}{52} =\frac{11}{26}[/tex]
The answer is [tex]\frac{11}{26}[/tex].
To determine the probability of selecting a club or a picture card from a standard deck of 52 cards, calculate the number of favorable outcomes and divide it by the total number of possible outcomes.
Explanation:To determine the probability that a card selected from a standard deck of 52 cards is a club or a picture card, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes.
There are 13 clubs and 12 picture cards (Jacks, Queens, and Kings) in a deck. However, we need to subtract the Queen of Clubs, as it has already been selected. So, the number of favorable outcomes is 13 + 12 - 1 = 24.
The total number of possible outcomes is 52, as there are 52 cards in a standard deck.
Therefore, the probability of selecting a club or a picture card is 24/52, which simplifies to 6/13 or approximately 0.4615.
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True or False: If p1, p2, . . . , pn are prime, then A = p1p2 . . . pn1 + 1 is also prime.
Answer:
The given statement is FALSE.
Step-by-step explanation:
If p1, p2, . . . , pn are prime, then A = p1p2 . . . pn1 + 1 is also prime.
No, this statement is false.
Let's take an example:
We take two prime numbers.
p1 = 3
p2 = 5
Now p1p2+1 becomes :
[tex](3\times5)+1=16[/tex]
And we know that 16 is not a prime number.
Note : A prime number is a number that is only divisible by 1 and itself like 3,5,7,11 etc.
Your Co. collects 50% of its sales in the month of the sales, 30% of the follow month, and 20% the second month after the sale.
Calculate budgeted cash receipts for March and April.
January=50
February=40
March=60
April=30
May=70
June=50
Answer:
March 52April 41Step-by-step explanation:
In March, Your Co. will collect 20% of January's sales, 30% of February's sales, and 50% of March's sales:
.20×50 +.30×40 +.50×60 = 10 +12 +30 = 52
Similarly, in April, collections will be ...
.20×40 + .30×60 + .50×30 = 8 +18 +15 = 41
Answer:Am nevoie de Puncte
Step-by-step explanation:
Solve the Differential equation (2x^3 - xy^2 - 2y + 3)dx - (x^2y + 2x)dy = 0
Notice that
[tex](2x^3-xy^2-2y+3)_y=-2xy-2[/tex]
[tex](-x^2y-2x)_x=-2xy-2[/tex]
so the ODE is exact, and we can find a solution [tex]F(x,y)=C[/tex] such that
[tex]F_x=2x^3-xy^2-2y+3[/tex]
[tex]F_y=-x^2y-2x[/tex]
Integrating both sides of the first equation wrt [tex]x[/tex] gives
[tex]F(x,y)=\dfrac{x^4}2-\dfrac{x^2y^2}2-2xy+3x+g(y)[/tex]
Differentiating wrt [tex]y[/tex] gives
[tex]F_y=-x^2y-2x=-x^2y-2x+g'(y)\implies g'(y)=0\implies g(y)=C[/tex]
So we have
[tex]\boxed{F(x,y)=\dfrac{x^4}2-\dfrac{x^2y^2}2-2xy+3x=C}[/tex]
The solution to this complex differential equation requires knowledge in calculus and differential equations. Without additional context, it's impossible to provide a specific solution. However, exploring technique utilization such as exact differential equations, integrating factors and substitution would be beneficial.
Explanation:To solve the given differential equation, which is (2x^3 - xy^2 - 2y + 3)dx - (x^2y + 2x)dy = 0, we will use an approach of factorization or grouping like terms to simplify the equation. In some cases, you might need to rearrange terms and identify if it's a special type of differential equation, like exact, separable, or homogeneous, and then apply the relevant techniques accordingly.
This difficult task requires excellent knowledge in calculus and differential equations. Unfortunately, due to the complexity of this particular equation, without additional context or information, it is impossible to provide a specific solution. I would recommend you to go through topics such as exact differential equations, as well as methods involving integrating factors and substitution. These may help you to analyze and solve this complex equation.
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1) Let A = {1, 2, 3, 4} and R be a relation on the set A defined by: R = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (4, 1), (4, 4)} Determine whether R is reflexive, irreflexive, symmetric, asymmetric, antisymmetric, or transitive. For each property, either explain why R has that property or give an example showing why it does not. (30 points)
Answer:
R is not reflexive,not irreflexive ,symmetric,not asymmetric,not antisymmetric and transitive.
Step-by-step explanation:
1.Reflexive : Relation R is not reflexive because it does not contain identity relation on A.
Identity relation on A:{(1,1),(2,2),(3,3),(4,4)}
If we take a relation R on A :{(1,1),(2,2),(3,3),(4,4),(1,2),(2,1)}
The relation R is reflexive on set A because it contain identity relation.
2.Irreflexive: Given relation R is not reflexive because it contain some elements of identity relation .If a relation is irreflexive then it does not contain any element of identity relation. We can say the intersection of R with identity relation is empty.
[tex]R\cap I=\left\{(1,1),(2,2),(4,4)\right\} \neq \phi[/tex]
If we take a relation on A R:{(1,2),(2,1)}
Then relation is irreflexive because it does not contain any element of identity relation
[tex]R\cap I=\phi[/tex]
3.Symmetric : The given relation R is symmetric because it satisfied the property of symmetric relation.
(1,2)belongs to relation (2,1) also belongs to relation ,(1,4) belongs to relation and (4,1) also belongs to given relation Hence we can say it is symmetric relation.
4.Asymmetric: The given relation is not asymmetric relation.Because it does not satisfied the property of asymmetric relation
Asymmetric relation: If (a,b) belongs to relation then (b,a) does not belongs to given relation.
Here (1,2) belongs to given relation and (2,1) also belongs to given relation Therefore, it is not asymmetric.
If we take a relation R on A
R:{(1,2),(1,3)}
It is asymmetric relation because it does not contain (2,1) and (3,1).
5.Antisymmetric: The given relation is not antisymmetric because it does not satisfied the property of antisymmetric relation.
Antisymmetric relation: If (a,b) and (b,a) belongs to relation then a=b
If we take a relation R
R;{(1,2) (1,1)}
It is antisymmetric because it contain (1,1) where 1=1 .Hence , it is antisymmetric.
6.Transitive: The given relation is transitive because it satisfied the property of transitive relation
Transitive relation: If (a,b) and (b,c) both belongs to relation R then (a,c) belongs to R.
Here, we have (1,2) and (2,1)belongs to relation then (1,1) and (2,2) are also belongs to relation.
(1,4) and (4,1) then (1,1) and (4,4) are also belongs to the relation.
A Social Security number consists of nine digits in a particular order, and repetition of digits is allowed. After seeing the last four digits printed on a receipt, if you randomly select the other digits, what is the probability of getting the correct Social Security number of the person who was given the receipt?
Answer: [tex]\dfrac{1}{100,000}[/tex]
Step-by-step explanation:
Given : The total number of digits in number system (0 to 9) = 10
The number of digits in a social security number = 9
After last four digits are printed, the number of digits remaining to print = 9-4=5
Since , repetition of digits is allowed, then the total number of ways to print 5 remaining digits is given by :-
[tex]10\times10\times10\times10\times10=100,000[/tex]
Now, the probability of getting the correct Social Security number of the person who was given the receipt is given by:-
[tex]\dfrac{\text{Number of correct code}}{\text{Total number of codes}}\\\\=\dfrac{1}{100,000}[/tex]
Final answer:
The probability of correctly guessing an entire Social Security number with the last four digits known is 0.001%, calculated by multiplying the probability of guessing each of the five unknown digits correctly, which is 1/10, resulting in (1/10)^5 or 1/100,000.
Explanation:
The question asks about the probability of correctly guessing an entire Social Security number given the last four digits. A Social Security number has nine digits and the digits can be repeated. If you know the last four digits, you need to guess the first five correctly.
Since each digit can be any number from 0 to 9, there are 10 possibilities for each digit. The probability of guessing one digit correctly is 1 out of 10 (1/10). To find the probability of guessing all five correctly, you need to multiply the probability for each digit, so the probability for all five is (1/10) x (1/10) x (1/10) x (1/10) x (1/10), which equals 1/100,000 or 0.00001. Therefore, the probability of getting the correct Social Security number is 0.00001 or 0.001%.
Analyze the diagram below and complete the instructions that follow.
Find the value of x.
A.√3
B. 3√2/2
C. 3√2
D. 3√3
Answer:
B. 3√2/2
Step-by-step explanation:
The value of x can be found with the tan rule.
Step 1: Identify the sides as opposite, adjacent and hypotenuse to apply the formula
Opposite = (opposite from angle 45 degrees)
Adjacent = x (between angle 45 degrees and 90 degrees)
Hypotenuse = 3 (opposite from 90 degrees)
Step 2: Apply the tan formula
Cos (angle) = adjacent/hypotenuse
Cos (45) = x/3
√2/2 = x/3
x = √2/2 x 3
x = 3√2/2
Therefore, the correct answer is B; x = 3√2/2
!!
Kyd and North are playing a game. Kyd selects one card from a standard 52-card deck. If Kyd selects a face card (Jack, Queen, or King), North pays him $5. If Kyd selects any other type of card, he pays North $2. a) What is Kyd's expected value for this game? b) What is North's expected value for this game? c) Who has the advantage in this game?
Step-by-step explanation:
In a 52 deck, there are 12 face cards (4 Jacks, 4 Queens, and 4 Kings). The remaining 40 cards are non-face cards.
The probability that Kyd draws a face card is 12/52, and the probability that he draws a non-face card is 40/52.
a) Kyd's expected value is:
K = (12/52)(5) + (40/52)(-2)
K = -5/13
K ≈ -$0.38
b) North's expected value is:
N = (12/52)(-5) + (40/52)(2)
N = 5/13
N ≈ $0.38
c) Kyd is expected to lose money, and North is expected to gain money. North has the advantage.
Kyd's expected value for this game is -$0.38.
North's expected value for this game is $0.38.Kyd is expected to lose money, and North is expected to gain money.
North has the advantage.
What is probability?The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a statement is true. An event's probability is a number between 0 and 1, where, roughly speaking, 0 denotes the event's impossibility and 1 denotes certainty.
Given
In a 52 deck, there are 12 face cards (4 Jacks, 4 Queens, and 4 Kings). The remaining 40 cards are non-face cards.
The probability that Kyd draws a face card is 12/52, and the probability that he draws a non-face card is 40/52.
a) Kyd's expected value is:
K = (12/52)(5) + (40/52)(-2)
K = -5/13
K ≈ -$0.38
b) North's expected value is:
N = (12/52)(-5) + (40/52)(2)
N = 5/13
N ≈ $0.38
c) Kyd is expected to lose money, and North is expected to gain money. North has the advantage.
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solve this system of linear equations. Separate the X- and Y- values with a comma. -9x+2y=-16 19x+3y=41
Answer:
(2, 1)
Step-by-step explanation:
The best way to do this to avoid tedious fractions is to use the addition method (sometimes called the elimination method). We will work to eliminate one of the variables. Since the y values are smaller, let's work to get rid of those. That means we have to have a positive and a negative of the same number so they cancel each other out. We have a 2y and a 3y. The LCM of those numbers is 6, so we will multiply the first equation by a 3 and the second one by a 2. BUT they have to cancel out, so one of those multipliers will have to be negative. I made the 2 negative. Multiplying in the 3 and the -2:
3(-9x + 2y = -16)--> -27x + 6y = -48
-2(19x + 3y = 41)--> -38x - 6y = -82
Now you can see that the 6y and the -6y cancel each other out, leaving us to do the addition of what's left:
-65x = -130 so
x = 2
Now we will go back to either one of the original equations and sub in a 2 for x to solve for y:
19(2) + 3y = 41 so
38 + 3y = 41 and
3y = 3. Therefore,
y = 1
The solution set then is (2, 1)
x=2 and y=1
proof:
-9x+2y=-16
-9(2)+2(1)=-16
which is a true statement
A simple random sample of size nequals15 is drawn from a population that is normally distributed. The sample mean is found to be x overbarequals31.1 and the sample standard deviation is found to be sequals6.3. Determine if the population mean is different from 25 at the alpha equals 0.01 level of significance.
Answer with explanation:
To test the Significance of the population which is Normally Distributed we will use the following Formula Called Z test
[tex]z=\frac{\Bar X - \mu}{\frac{\sigma}{n}}[/tex]
[tex]\Bar X =31.1\\\\ \sigma=6.3\\\\ \mu=25\\\\n=15\\\\z=\frac{31.1-25}{\frac{6.3}{\sqrt{15}}}\\\\z=\frac{6.1\times\sqrt{15}}{6.3}\\\\z=\frac{6.1 \times 3.88}{6.3}\\\\z=3.756[/tex]
→p(Probability) Value when ,z=3.756 is equal to= 0.99992=0.9999
⇒Significance Level (α)=0.01
We will do Hypothesis testing to check whether population mean is different from 25 at the alpha equals 0.01 level of significance.
→0.9999 > 0.01
→p value > α
With a z value of 3.75, it is only 3.75% chance that ,mean will be different from 25.
So,we conclude that results are not significant.So,at 0.01 level of significance population mean will not be different from 25.
An individual is planning a trip to a baseball game for 16 people. Of the people planning to go to the baseball game, 8 can go on saturday and 12 can go on sunday, some of them can go on both days. How many people can only go to the game on saturday?
Answer: There are 4 people who only go to the game on Saturday.
Step-by-step explanation:
Let the number of people go on Saturday be n(A).
Let the number of people go on Sunday be n(B).
Since we have given that
n(A) = 8
n(B) = 12
n(A∪B) = 16
According to rules, we get that
[tex]n(A)+n(B)-n(A\cap B)=n(A\cup B)\\\\8+12-n(A\cap B)=16\\\\20-n(A\cap B)=16\\\\n(A\cap B)=20-16=4[/tex]
So, n(only go on Saturday) = n(only A) = n(A) - n(A∩B) = 8-4 = 4
Hence, there are 4 people who only go to the game on Saturday.
The amount of sales tax on a new car is directly proportional to the purchase price of the car. Victor bought a new car for $30,000 and paid $1,500 in sales tax. Rita bought a new car from the same dealer and paid $2,375 sales tax. How much did Rita pay for her car?
Answer:
Rita paid 47,500 dollars for the purchase price.
Step-by-step explanation:
We are given the sales tax on a new car is directly proportional to the purchase price of the car which means there is is something k such that
when you multiply it to the sales tax you get the purchase price.
Let's set this equation:
y=kx
Let y represent the purchase price and x the sales tax.
The second sentence tells us that (x,y)=(1500,30000).
We can plug this into y=kx to find the constant k. (Constant means it stays the same no matter what the input and output is).
So we have:
30000=k(1500)
300 =k(15) I went ahead and divided previous equation by 100.
Now divide both sides by 15:
300/15=k
Simplify:
20=k
So the equation to use the answer the question is
y=20x
where y is purchase price and x is sales tax.
So we want to know the purchase price on a car if the sales tax is 2375.
So replace x with 2375:
y=20(2375)
y=47500
Answer:
$47500
Step-by-step explanation:
If the amount of sales tax on a new car is directly proportional to the purchase price of the car and Victor bought a new car for $30,000 and paid $1,500 in sales tax and Rita bought a new car from the same dealer and paid $2,375 sales tax, Rita payed $47,500 for her car.
y=20(2375)
y=47500
8. Show all computations for the following. Do not use a calculator.
(a) How many bit strings are there of length six which are palindromes? Explain
(b) How many bit strings are there of length 6 which contain exactly three 1’s? Explain
Answer:
a= 8 ways
b. 20
Step-by-step explanation:
Palindrome of length 6 means first three digits must be same as the last three in reverse. For example 123321 is palindrome of six digits.
There 2 bits 0 and 1
a.So each of first three digits can be filled in 2 ways
therefore, 2*2*2= 8 ways
number of different palindromes of 6 digits will be 8
b. In a 6 digit a palindrome there Are 6 spaces in which 3 spaces are to be filled with 1's
this cab be done in
[tex]_{6}^{3}\textrm{C}= \frac{6!}{3!\times3!}[/tex]
= 20
Suppose that 9 female and 6 male applicants have been successfully screened for 5 positions. If the 5 positions are filled at random from the 15 ?finalists, what is the probability of selecting no? females?
Answer: [tex]\dfrac{2}{1001}[/tex]
Step-by-step explanation:
Given : The number of female applicants = 9
The number of male applicants = 6
Total applicants = 15
The number of ways to select 5 applicants from 15 applicants :-
[tex]^{15}C_5=\dfrac{15!}{5!(15-5)!}=3003[/tex]
The number to select 5 applicants from 15 applicants such that no female applicant is selected:-
[tex]^{9}C_0\times^6C_5=1\times\dfrac{6!}{5!(6-5)!}=6[/tex]
Now, the required probability :-
[tex]\dfrac{6}{3003}=\dfrac{2}{1001}[/tex]
7) State the prime factorization of 30.
Answer:
[tex]30=2\: *3\:*5[/tex]
Step-by-step explanation:
We analyze between which prime numbers we can divide the number 30. The smallest prime number by which we divide is 2. Then:
[tex]\frac{30}{2}=15[/tex]
We now look for the smallest prime number that divides the 15. Since 15 is not a multiple of 2, we make the division with the number 3 that is divisor of 15.
[tex]\frac{15}{3}=5[/tex]
We now look for a number that divides to 5, but since 5 is a prime number, the only divisor other than 1 is 5. Then:
[tex]\frac{5}{5}=1[/tex]
This ends the decomposition of 30 and we find 3 prime factors:
2,3 and 5.
Analyze the diagram below and complete the instructions that follow.
Find the value of x and the value of y.
A. x=9, y=18√2
B.x=18, y=18
C.x=9√2, y=18√2
D. x=9√3, y=18
Answer:
D. x = 9√3 and y = 18
Step-by-step explanation:
This is an isosceles triangle divided into two equal parts.
Step 1: 18 can be divided into 2 parts which makes the base of both triangles 9.
Step 2: Find the value of x
The value of x can be found through the tan rule.
tan (angle) = opposite/adjacent
tan (60) = x/9
√3 = x/9
x = √3 x 9
x = 9√3
Step 3: Find the value of y
The value of y can be found through the cos rule.
Cos (angle) = adjacent/hypotenuse
Cos (60) = 9/y
1/2 = 9/y
y = 18
Therefore, the correct answer is D; x = 9√3 and y = 18
!!