f : X → Y and g: Y → Z
Now we have to show:
If gof is one-to-one then f must be one-to-one.
Given:
gof is one-to-one
To prove:
f is one-to-one.
Proof:
Let us assume that f(x) is not one-to-one .
This means that there exist x and y such that x≠y but f(x)=f(y)
On applying both side of the function by the function g we get:
g(f(x))=g(f(y))
i.e. gof(x)=gof(y)
This shows that gof is not one-to-one which is a contradiction to the given statement.
Hence, f(x) must be one-to-one.
Now, example to show that gof is one-to-one but g is not one-to-one.Let A={1,2,3,4} B={1,2,3,4,5} C={1,2,3,4,5,6}
Let f: A → B
be defined by f(x)=x
and g: B → C be defined by:
g(1)=1,g(2)=2,g(3)=3,g(4)=g(5)=4
is not a one-to-one function.
since 4≠5 but g(4)=g(5)
Also, gof : A → C
is a one-to-one function.
You are going to buy a new car worth $25,800. The dealer computes your monthly payment to be $509.55 for 60 months of financing. What is the dealer's effective rate of return on this loan transaction? The dealer's effective rate of return is 1 1%. (Round to one decimal place.)
Answer:
6.9%
Step-by-step explanation:
Interest rate is the one variable in an amortization formula that cannot be determined explicitly. An iterative solution is required, which means the computation must be done by a calculator, spreadsheet, or web site.
My TI-84 TVM Solver tells me that for the given loan amount and payment schedule, the APR is about 6.9%.
Police response time to an emergency call is the difference between the time the call is first received by the dispatcher and the time a patrol car radios that it has arrived at the scene. Over a long period of time, it has been determined that the police response time has a normal distribution with a mean of 8.1 minutes and a standard deviation of 1.9 minutes. For a randomly received emergency call, find the following probabilities. (Round your answers to four decimal places.)(a) the response time is between 5 and 10 minutes(b) the response time is less than 5 minutes(c) the response time is more than 10 minutes
Answer:
a) 0.7898
b) 0.0516
c) 0.1587
Step-by-step explanation:
Given : Mean : [tex]\mu=8.1\text{ minutes}[/tex]
Standard deviation : [tex]\sigma =1.9\text{ minutes}[/tex]
Since , the police response time has a normal distribution.
The formula to calculate the z-score :-
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x=5 minutes.
[tex]z=\dfrac{5-8.1}{1.9}=-1.63[/tex]
For x=10 minutes.
[tex]z=\dfrac{10-8.1}{1.9}=1[/tex]
a) The p-value =[tex]P(-1.63<z<1)=P(z<1)-P(z<-1.63)[/tex]
[tex]=0.8413447-0.0515507=0.789794\approx0.7898[/tex]
b) The p-value =[tex]P(z<-1.63)=0.0515507\approx0.0516[/tex]
c) The p-value =[tex]P(z>1)=1-P(z<1)[/tex]
[tex]=1-0.8413447=0.1586553\approx0.1587[/tex]
We calculated the probability of different police response times using the z-score method. The probability of a response time between 5 and 10 minutes is 0.7897, the probability for less than 5 minutes is 0.0516, and more than 10 minutes is 0.1587.
Explanation:To answer this question, we need to first standardize the response times using the z-score formula: z = (X - μ) / σ, where X is the value we're interested in, μ is the mean, and σ is the standard deviation.
(a) To find the probability that the response time is between 5 and 10 minutes, we first calculate the z-scores for 5 and 10 minutes:
Z(5) = (5 - 8.1) / 1.9 = -1.632 Z(10) = (10 - 8.1) / 1.9 = 1
Next, we find these values in the z-table which yields: P(Z<1) = 0.8413, P(Z<-1.632) = 0.0516. The probability that the response time is between 5 and 10 minutes is the difference between these values, so P(5 < X < 10) = 0.8413 - 0.0516 = 0.7897.
(b) For the response time less than 5 minutes, we calculate the probability using the z-score for 5 minutes. Z(5) = -1.632, looking in the z-table, we find this value equals to 0.0516. Therefore, the response time is less than 5 minutes is 0.0516.
(c) Lastly, the probability for a response time more than 10 minutes is P(Z > 1) which is equal to 1 - P(Z < 1). From the z-table, we find P(Z<1) = 0.8413. Then, P(Z > 1) = 1 - 0.8413 = 0.1587. So, the probability that the response time is more than 10 minutes is 0.1587.
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Please help me with this
Answer:
Definition of mid-point
Step-by-step explanation:
Midpoint is the center point of a line segment
At midpoints the line segment is divided into two equal lengths
Point K is the midpoint of segment MJ and length of segment MK = length of segment KJ
Point K is the midpoint of segment OL and length of segment OK = length of segment KL
This extreme value problem has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. f(x, y) = x2 − y2; x2 + y2 = 81 maximum value minimum value
The Lagrangian is
[tex]L(x,y,\lambda)=x^2-y^2+\lambda(x^2+y^2-81)[/tex]
with critical points where the partial derivatives are identically zero:
[tex]L_x=2x+2\lambda x=0\implies 2x(1+\lambda)=0\implies x=0\text{ or }\lambda=-1[/tex]
[tex]L_y=-2y+2\lambda y=0\implies-2y(1-\lambda)=0\implies y=0\text{ or }\lambda=1[/tex]
[tex]L_\lambda=x^2+y^2-81=0\implies x^2+y^2=81[/tex]
If [tex]x=0[/tex], then [tex]L_\lambda=0\implies y=\pm9[/tex]If [tex]\lambda=-1[/tex], then [tex]L_y=0\implies y=0[/tex] and [tex]L_\lambda=0\implies x=\pm9[/tex]If [tex]y=0[/tex], we get the previous conclusion of [tex]x=\pm9[/tex]If [tex]\lambda=1[/tex], then [tex]L_x=0\implies x=0[/tex] and we again get [tex]y=\pm9[/tex]So we have four critical points to consider: (0, -9), (0, 9), (-9,0), and (9, 0). We have
[tex]f(0,-9)=-81[/tex]
[tex]f(0,9)=-81[/tex]
[tex]f(-9,0)=81[/tex]
[tex]f(9,0)=81[/tex]
So the maximum value is 81 and the minimum value is -81.
The extreme values of a function are the minimum and the maximum values of the function.
The extreme values are -81 and 81
The function is given as:
[tex]\mathbf{f(x,y) = x^2 - y^2}[/tex]
[tex]\mathbf{x^2 + y^2 = 81}[/tex]
Subtract 81 from both sides of [tex]\mathbf{x^2 + y^2 = 81}[/tex]
[tex]\mathbf{x^2 + y^2 - 81 = 0}[/tex]
Using Lagrange multiplies, we have:
[tex]\mathbf{L(x,y,\lambda) = f(x,y) + \lambda(0)}[/tex]
Substitute [tex]\mathbf{f(x,y) = x^2 - y^2}[/tex] and [tex]\mathbf{x^2 + y^2 - 81 = 0}[/tex]
[tex]\mathbf{L(x,y,\lambda) = x^2 - y^2 + \lambda(x^2 + y^2 - 81)}[/tex]
Differentiate
[tex]\mathbf{L_x = 2x + 2\lambda x}[/tex]
[tex]\mathbf{L_y = -2y + 2\lambda y}[/tex]
[tex]\mathbf{L_{\lambda} = x^2 + y^2 -81}[/tex]
Equate to 0
[tex]\mathbf{2x + 2\lambda x = 0}[/tex]
[tex]\mathbf{-2y + 2\lambda y = 0}[/tex]
[tex]\mathbf{x^2 + y^2 -81 = 0}[/tex]
So, we have:
[tex]\mathbf{2\lambda x = -2x}[/tex]
[tex]\mathbf{2\lambda y = 2y}[/tex]
Divide both sides of [tex]\mathbf{2\lambda x = -2x}[/tex] by -2x
[tex]\mathbf{\lambda = -1}[/tex]
Divide both sides of [tex]\mathbf{2\lambda y = 2y}[/tex] by 2y
[tex]\mathbf{\lambda = 1}[/tex]
The above means that:
[tex]\mathbf{\lambda = -1\ or\ x = 0}[/tex]
[tex]\mathbf{\lambda = 1\ or\ y = 0}[/tex]
Recall that: [tex]\mathbf{x^2 + y^2 = 81}[/tex]
When x = 0, we have:
[tex]\mathbf{0^2 + y^2 = 81}[/tex]
Take square roots of both sides
[tex]\mathbf{y = \±9}[/tex]
When y = 0, we have:
[tex]\mathbf{x^2 + 0^2 = 81}[/tex]
Take square roots of both sides
[tex]\mathbf{x = \±9}[/tex]
To determine the critical points, we consider:
[tex]\mathbf{\lambda = -1\ or\ x = 0}[/tex] or [tex]\mathbf{y = \±9}[/tex]
[tex]\mathbf{\lambda = 1\ or\ y = 0}[/tex] or [tex]\mathbf{x = \±9}[/tex]
So, the critical points are:
[tex]\mathbf{(x,y) = \{ (0, -9), (0, 9), (-9,0), (9, 0)\}}[/tex]
Substitute the above values in [tex]\mathbf{f(x,y) = x^2 - y^2}[/tex]
[tex]\mathbf{f(0,-9) = 0^2 - (-9)^2 = -81}[/tex]
[tex]\mathbf{f(0,-9) = 0^2 - (9)^2 = -81}[/tex]
[tex]\mathbf{f(-9,0) = (-9)^2 - 0^2 = 81}[/tex]
[tex]\mathbf{f(9,0) = (9)^2 - 0^2 = 81}[/tex]
Considering the above values, we have:
[tex]\mathbf{Minimum= -81}[/tex]
[tex]\mathbf{Maximum= 81}[/tex]
Hence, the extreme values are -81 and 81, respectively.
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Use an element argument to prove each statement. Assume that all sets are subsets of a universal set U.
For all sets A, B, andC, if A ⊆ B then A∪C ⊆ B∪C
We are asked to prove the statement,
For all sets A, B, and C, if A ⊆ B then A∪C ⊆ B∪C
Let us consider set A as:
A={1,3,4,5}
and B={1,2,3,4,5,6,7}
Clearly we may observe that A is a subset of B.
( Since, all the elements of set A are contained in set B .
Hence, A is a subset of B)
Now let us consider set C as:
C={1,2}
Hence,
A∪C={1,2,3,4,5}
and
B∪C={1,2,3,4,5,6,7}
Still we observe that:
A∪C ⊆ B∪C
Since all the elements of the set A∪C are contained in the set B∪C.
The time it takes to ring up a customer at the grocery store follows an exponential distribution with a mean of 3.5 minutes. What is the probability density function for the time it takes to ring up a customer?
The probability density function for the time it takes to ring up a customer at the grocery store, following an exponential distribution with a mean of 3.5 minutes, is (1/3.5) × e[tex]^{(-x/3.5)[/tex].
Explanation:The probability density function for the time it takes to ring up a customer at the grocery store, following an exponential distribution with a mean of 3.5 minutes, is given by:
f(x) = (1/3.5) × e[tex]^{(-x/3.5)[/tex]
Where x is the time it takes to ring up a customer.
In this case, the exponential distribution models the time between events, which in this context is the time between customer arrivals at the grocery store.
The exponential distribution is a continuous probability distribution that is often used to model random events that occur independently and exponentially over time.
A plumbing supply company has fixed costs of $9,000 per month and average variable costs of $9.30 per unit manufactured. The company has $90,000 available to cover the monthly costs. How many units can the company manufacture? (Fixed costs are those that occur regardless of the level of production. Variable costs depend on the level of production. Your answer should be in terms of whole units produced.)
Answer:
8710 units
Step-by-step explanation:
Step 1: Write all the data
Fixed cost: $9000
Average variable cost: 9.3 per unit
Total cost: 90,000
Total units: x
Step 2: Find the total variable cost
Average variable cost is per unit so it has to be multiplied by the number of units to find the total variable cost.
Total variable cost = average variable cost per unit x number of units
Total variable cost = 9.3x
Step 3: Make the formula for finding x
Total cost = total fixed cost + total variable cost
90,000 = 9000 + 9.3x
81000 = 9.3x
x = 8709.67
Rounded off to 8710 units
!!
A meteorologist is studying the speed at which thunderstorms travel. A sample of 10 storms are observed. The mean of the sample was 12.2 MPH and the standard deviation of the sample was 2.4. What is the 95% confidence interval for the true mean speed of thunderstorms?
Answer:
The 95% confidence interval for the true mean speed of thunderstorms is [10.712, 13.688].
Step-by-step explanation:
Given information:
Sample size = 10
Sample mean = 12.2 mph
Standard deviation = 2.4
Confidence interval = 95%
At confidence interval 95% then z-score is 1.96.
The 95% confidence interval for the true mean speed of thunderstorms is
[tex]CI=\overline{x}\pm z*\frac{s}{\sqrt{n}}[/tex]
Where, [tex]\overline{x}[/tex] is sample mean, z* is z score at 95% confidence interval, s is standard deviation of sample and n is sample size.
[tex]CI=12.2\pm 1.96\frac{2.4}{\sqrt{10}}[/tex]
[tex]CI=12.2\pm 1.487535[/tex]
[tex]CI=12.2\pm 1.488[/tex]
[tex]CI=[12.2-1.488, 12.2+1.488][/tex]
[tex]CI=[10.712, 13.688][/tex]
Therefore the 95% confidence interval for the true mean speed of thunderstorms is [10.712, 13.688].
6.Sarah is planning to fence in her backyard garden. One side of the garden is 34 feet long, another side is 30 feet long, and the third side is 67 feet long.If the fencing material costs $3.00 per foot, how much will Sarah’s fence cost?
A.$154.00
B.$938.00
C.$750.00
D.$393.00
Answer:
D. $393.00
Step-by-step explanation:
We have to sum the three distances to get the perimeter.
P= 34 ft + 67 ft+ 30 ft
P = 131 ft
And each foot of material cost $3.00
we get the cost with the distance and de cost of the material:
Cost = 131 ft * 3.00 $ per ft = $393.00
Sarah needs $393.00 to fence her backyard garden
9. An RSA cryptosystem has modulus n 391, which is a product of the primes 23 and 17. Which of the following is suitable as an encoding key e? (a) 163 (b) 353 (c) 351 (d) 277 (e) none of these. 10. Which of the following polynomials p(x) is complete over Zalr? (a) z4+1 (e) none of these
Answer:
163
Step-by-step explanation:
So n=391.
This means p=23 and q=17 where p*q=n.
[tex] \lambda (391)=lcm(23-1,17-1)=lcm(22,16)=2*8*11=16*11=176. [/tex]
We want to choose e so that e is between 1 and 176 and the gcd(e,176)=1.
There is only one number in your list that is between 1 and 176... Hopefully the gcd(163,176)=1.
It does. See notes below for checking it:
176=2(88)=2(4*22)=2(2)(2)(2)(11)
None of the prime factors of 176 divide 163 so we are good.
The answer is 163.
Show that the equation is exact and find an implicit solution. y cos(xy) + 3x^2 + [x cos(xy) + 2y]y' = 0
We have
[tex]\dfrac{\partial(y\cos(xy)+3x^2)}{\partial y}=\cos(xy)-xy\sin(xy)[/tex]
[tex]\dfrac{\partial(x\cos(xy)+2y)}{\partial x}=\cos(xy)-xy\sin(xy)[/tex]
so the ODE is indeed exact. Then there's a solution of the form [tex]f(x,y)=C[/tex] such that
[tex]\dfrac{\partial f}{\partial x}=y\cos(xy)+3x^2[/tex]
[tex]\implies f(x,y)=\sin(xy)+x^3+g(y)[/tex]
Differentiating wrt [tex]y[/tex] gives
[tex]\dfrac{\partial f}{\partial y}=x\cos(xy)+2y=x\cos(xy)+g'(y)[/tex]
[tex]\implies g'(y)=2y\implies g(y)=y^2+C[/tex]
Then the solution to the ODE is
[tex]f(x,y)=\boxed{\sin(xy)+x^3+y^2=C}[/tex]
Which of the following is a solution to the second order differential equation LaTeX: y''=-4y y ″ = − 4 y ? To answer this question, attempt to verify each of the following proposed solutions. a. LaTeX: y=\sin2t y = sin 2 t b. LaTeX: y=-\frac{2}{3}t^3 y = − 2 3 t 3 c. LaTeX: y=\cos2t y = cos 2 t d. LaTeX: y=e^{2t} y = e 2 t e. LaTeX: y=\frac{y''}{-4}
Answer:
y = sin(2t)y = cos(2t)Step-by-step explanation:
In the case of each of the answers listed above, the second derivative is equal to -4 times the function, as required by the differential equation.
For y = 2/3t^3, the second derivative is 4t, not -4y.
For y = e^(2t), the second derivative is 4y, not -4y.
__
The graph shows the sum of the second derivative and 4y is zero for the answers indicated above, and not zero for the other two proposed answers.
Find the first 2 terms of each of two power series solutions: y" + x^2 y'+ xy = 0
Apparently this solution is too long for posting, so I've written it elsewhere and am attaching screenshots of it.
The first four terms of each solution are
[tex]1-\dfrac23x^3+\dfrac5{36}x^6-\dfrac{10}{567}x^9[/tex]
and
[tex]x-\dfrac1{24}x^4+\dfrac1{315}x^7-\dfrac7{32,400}x^{10}[/tex]
A vacuum cleaner dealership sold 370 units in 2011 and 411 units in 2012. Find the percent increase or decrease in the number of units sold.
The number of units sold increased or decreased? by about what percent?
Answer:
The percent of Increase is of 11.08% (0.1108)
Step-by-step explanation:
Hello, great question. These types are questions are the beginning steps for learning more advanced Algebraic Equations.
Since we have two different values for two different years, we can use the following Algebraic Expression to calculate the percent difference of sales between both years. The Expression would be the following,
[tex]370 * (x+1) = 411[/tex]
Where x is the percent difference. Now we solve for x,
[tex]370 * (x+1) = 411[/tex]
[tex]x+1 = 1.1108[/tex]
[tex]x = 0.1108[/tex]
so now we see that the percent of Increase is of 11.08% (0.1108)
I hope this answered your question. If you have any more questions feel free to ask away at Brainly.
The number of units sold increased by about 11.08%.
Explanation:To find the percent increase or decrease in the number of units sold, we need to calculate the difference between the number of units sold in 2012 and 2011, and then divide that difference by the number of units sold in 2011.
The amount of increase or decrease is calculated as: (Number of units sold in 2012 - Number of units sold in 2011)/Number of units sold in 2011 x 100
In this case, the calculation is: (411 - 370)/370 x 100 = (41/370) x 100 = 11.08%
Therefore, the number of units sold increased by about 11.08%.
A bag of 29 tulip bulbs contains 10 red tulip bulbs, 10 yellow tulip bulbs, and 9 purple tulip bulbs. (a) What is the probability that two randomly selected tulip bulbs are both red? (b) What is the probability that the first bulb selected is red and the second yellow? (c) What is the probability that the first bulb selected is yellow and the second red? (d) What is the probability that one bulb is red and the other yellow?
Answer:
a.The probability that two randomly selected tulip bulbs both are red=[tex]\frac{45}{406}[/tex].
b.The probability that the first bulb selected is red and second yellow=[tex]\frac{50}{406}[/tex].
c.The probability that the first bulb selected is yellow and the second red=[tex]\frac{50}{406}[/tex].
d.The probability that one bulb is red and other yellow=[tex]\frac{50}{203}[/tex].
Step-by-step explanation:
Given
Total number of bulbs= 29
Number of bulbs of red=10
Number of yellow bulbs=10
Number of purple bulbs=9
Formula of probability, P(E)=[tex]\frac{favourable \; cases}{total\;number\; of \; cases}[/tex]
a.The probability that two randomly selected tulip bulbs are both red=[tex]\frac{10}{29}\times\frac{9}{28}=\frac{45}{406}[/tex].
b.The probability of getting first bulb is red=[tex]\frac{10}{29}[/tex].
The probability of getting second bulb is yellow=[tex]\frac{10}{28}[/tex]
Hence,the probability that the firs bulb selected is red and the second bulb yellow=[tex]\frac{10}{29}\times\frac{9}{28}=\frac{45}{406}[/tex]
c. The probability of getting firs bulb is yellow =[tex]\frac{10}{29}[/tex]
The probability of getting second bulb is red=[tex]\frac{10}{28}[/tex]
Hence,the probability that the firs bulb selected is yellow and the second bulb red=[tex]\frac{10}{29}\times\frac{10}{28}=\frac{50}{406}[/tex].
d.The probability of getting first bulb is red and second is yellow=[tex]\frac{50}{406}[/tex]
The probability of getting first bulb is yellow and second is red=[tex]\frac{50}{406}[/tex]
The probability that one bulbe is red and other is yellow= probability of getting first bulb is red and other yellow+ probability of getting first bulb is yellow and other is red
Hence, the probability of getting one bulb is red and other is yellow=[tex]\frac{50}{406}+\frac{50}{406}=\frac{50}{203}[/tex]
John took all his money from his savings account. He spent $110 on a radio and 4/11 of what was left on presents for his friends. John then put 2/5 of his remaining money into a checking account and donated the $420 that was left to charity. How much money did John originally have in his savings account?
Answer:
$1210
Step-by-step explanation:
Let x be total amount
First John spent $110 on a radio and 4/11 of what was left on presents for his friends so he was left with
[tex]\frac{7}{11}(x-110)=\frac{7}{11}x-70[/tex]
Then he put 2/5 of his remaining money into a checking account
[tex]\frac{3}{5}\left(\frac{7}{11}x-70\right)[/tex]
Rest he donated to charity
[tex]420=\frac{3}{5}\left(\frac{7}{11}x-70\right)\\\Rightarrow \left(\frac{7}{11}x-70\right)=\frac{2100}{3}\\\Rightarrow \frac{7}{11}x-70=700\\\Rightarrow \frac{7}{11}x=770\\\Rightarrow x=1210[/tex]
Hence total amount of money John originally had was $1210
The arithmetic mean of any two nonnegative real numbers a and b is greater than or equal to their geometric mean vab. [Hint: consider (Va - vb) 0.]
Answer with explanation:
Here, a and b are two real numbers.
Arithmetic Mean of a and b
[tex]A=\frac{a+b}{2}[/tex]
[tex]\rightarrow a< \frac{a+b}{2}<b[/tex]
Geometric Mean of a and b
[tex]G=\sqrt{ab}[/tex]
[tex]\rightarrow a< \sqrt{ab}<b[/tex]
[tex]A-G\\\\=\frac{a+b}{2}-\sqrt{ab}\\\\=\frac{a+b-2\sqrt{ab}}{2}\\\\=[\frac{\sqrt{a}-\sqrt{b}}{\sqrt{2}}]^2>0\\\\A-G>0\\\\A>G[/tex]
Square of difference of any two numbers is greater than or equal to 0.
⇒A.M of two Numbers > G.M of two Numbers
49. Prejudice operates mainly through the use of stereotyping. A stereotype is/are______________.
a) a reward system for the dominant group in order to continue the social stratification of minority groups
b) relatively enduring social arrangements that distribute and exercise power.
c) an ethnic or racial slur intended to display the “less than” characteristics of a minority group.
d) the disenfranchisement of individuals or groups based on the media construed images of members of those groups.
e) oversimplified ideas about a group or a social category; generalization or assumptions about the characteristics of a group or an individual
Answer:
d) the disenfranchisement of individuals or groups based on the media construed images of members of those groups.
Step-by-step explanation:
Prejudice operates mainly through the use of stereotyping. A stereotype is/are the disenfranchisement of individuals or groups based on the media construed images of members of those groups.
The price of gasoline purchased varies directly with the number of gallons of gas purchased. If 19 gallons are purchased for $22.50, what is the price of purchasing 14 gallons? Let x represent the number of gallons purchased, and let y represent the total price. Round your answer to the nearest hundredth.
Answer:
[tex]\text{16.57 dollars}[/tex]
Step-by-step explanation:
A proportion is needed to find the value of x
[tex]$\frac{22.50}{19} =\frac{x}{14}\Longrightarrow 19x = 315 \Longrightarrow x = 16.57$ \\ \\ \text{It would cost 16.57 dollars to purchase 14 gallons.}[/tex]
The cost of 14 gallons of gasoline is $16.58.
What is proportion?Proportions are of two types one is the direct proportion in which if one quantity is increased by a constant k the other quantity will also be increased by the same constant k and vice versa.
In the case of inverse proportion if one quantity is increased by a constant k the quantity will decrease by the same constant k and vice versa.
Given, The price of gasoline purchased varies directly with the number of gallons of gas purchased and 19 gallons are purchased for $22.50.
Let k be the proportionality constant.
∴ y = kx.
22.50 = 19k.
k = 22.50/19.
k = 1.184.
So, the cost of 14 gallons of gasoline is (14×1.184) = $16.58.
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Consider the random variables X and Y with joint density function ???? f(x,y)= x+y, 0≤x≤1;0≤y≤1 0, elsewhere. (a) Find the marginal distributions of X and Y . (b) Find P(X > 0.25,Y > 0.5).
a. The marginal densities
[tex]f_X(x)=\displaystyle\int_0^1(x+y)\,\mathrm dy=x+\frac12[/tex]
and
[tex]f_Y(y)=\displaystyle\int_0^1(x+y)\,\mathrm dx=y+\frac12[/tex]
b. This can be obtained by integrating the joint density over [0.25, 1] x [0.5, 1]:
[tex]P(X>0.25,Y>0.5)=\displaystyle\int_{1/4}^1\int_{1/2}^1(x+y)\,\mathrm dx\,\mathrm dy=\frac{33}{64}[/tex]
Final answer:
To find the marginal distributions of X and Y, we integrate the joint density function over the range of the other variable. The marginal distribution of X is f(x) = x+1/2, for 0≤x≤1. The marginal distribution of Y is f(y) = y+1/2, for 0≤y≤1.
Explanation:
To find the marginal distributions of X and Y, we need to integrate the joint density function over the range of the other variable. For the marginal distribution of X, we integrate f(x,y) with respect to y from 0 to 1:
(∫⁰₁(x+y) dy) = (x+y/2)∣⁰₁ = x+1/2
So, the marginal distribution of X is given by f(x) = x+1/2, for 0≤x≤1.
Similarly, for the marginal distribution of Y, we integrate f(x,y) with respect to x from 0 to 1:
(∫⁰₁(x+y) dx) = (x2/2+xy)∣⁰₁ = y+1/2
Therefore, the marginal distribution of Y is given by f(y) = y+1/2, for 0≤y≤1.
A loan of $1000 is to be paid back, with interest, at the end of 1 year. Aft er 3 months, a partial payment of $300 is made. Use the US Rule to determine the balance due at the end of one year, considering the partial payment. Assume a simple interest rate of 9%.
Answer:
total balance due at the end of 1 year is $769.75
Step-by-step explanation:
Given data
loan amount = $1000
time period = 1 year
return = $300
rate = 9%
to find out
balance due at the end of one year
solution
we know in question $300 return after 3 month so we first calculate interest of $1000 for 3 month and than we after 3 month remaining 9 month we calculate interest for $700
interest for first 3 month = ( principal × rate × time ) / 100 .............1
here time is 3 month so = 3/12 will take and rate 9 % and principal $1000
put all these value in equation 1 we get interest for first 3 month
interest for first 3 month = ( principal × rate × time ) / 100
interest for first 3 month = ( 1000 × 9 × 3/12 ) / 100
interest for first 3 month = $22.5
now we calculate interest for remaining 9 months i.e.
interest for next 9 months = ( principal × rate × time ) / 100
here principal will be $700 because we pay $300 already
interest for next 9 months = ( 700 × 9 × 9/12 ) / 100
interest for next 9 months = $47.25
now we combine both interest that will be
interest for first 3 months +interest for next 9 months = interest of 1 year
interest of 1 year = $22.5 + $47.25
interest of 1 year = $69.75
so amount will be paid after 1 year will be loan amount + interest
amount will be paid after 1 year = 1000 + 69.75
amount will be paid after 1 year is $1069.75
so total balance due at the end of 1 year = amount will be paid after 1 year - amount paid already
total balance due at the end of 1 year = $1069.75 - $300
total balance due at the end of 1 year is $769.75
Eliminate the parameter.
x = 3 cos t, y = 3 sin t
Answer:
x^2+y^2 = 3^2
Step-by-step explanation:
We need to eliminate the parameter t
Given:
x = 3 cos t
y = 3 sin t
Squaring the above both equations
(x)^2=(3 cos t)^2
(y)^2 =(3 sin t)^2
x^2 = 3^2 cos^2t
y^2=3^2 sin^2t
Now adding both equations
x^2+y^2=3^2 cos^2t+3^2 sin^2t
Taking 3^2 common
x^2+y^2=3^2 (cos^2t+sin^2t)
We know that cos^2t+sin^2t = 1
so, putting the value
x^2+y^2=3^2(1)
x^2+y^2 = 3^2
Hence the parameter t is eliminated.
To eliminate the parameter in the given equations x = 3 cos t and y = 3 sin t, we can substitute cos(t) and sin(t) in terms of x and y to eliminate the parameter. The resulting equations represent the line y = x.
Explanation:To eliminate the parameter in the given equations x = 3 cos t and y = 3 sin t, we need to express x and y in terms of each other without the parameter 't'. Using the identity cos^2(t) + sin^2(t) = 1, we can solve for cos(t) and sin(t), and substitute them into the equations to eliminate the parameter.
Using the fact that cos(t) = x/3 and sin(t) = y/3, we can rewrite the equations as x = 3 cos(t) = 3(x/3) = x and y = 3 sin(t) = 3(y/3) = y. Therefore, eliminating the parameter results in x = x and y = y, which simply means that the equations represent the line y = x.
Learn more about Eliminating parameters from equations here:https://brainly.com/question/32043775
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8. (8 marks) Prove that for all integers m and n, m + n and m-n are either both even or both odd
Answer with explanation:
Let m and n are integers
To prove that m+n and m-n are either both even or both odd.
1. Let m and n are both even
We know that sum of even number is even and difference of even number is even.
Suppose m=4 and n=2
m+n=4+2=6 =Even number
m-n=4-2=2=Even number
Hence, we can say m+n and m-n are both even .
2. Let m and n are odd numbers .
We know that sum of odd numbers is even and difference of odd numbers is even.
Suppose m=7 and n=5
m+n=7+5=12=Even number
m-n=7-5=2=Even number
Hence, m+n and m-n are both even .
3. Let m is odd and n is even.
We know that sum of an odd number and an even number is odd and difference of an odd and an even number is an odd number.
Suppose m=7 , n=4
m+n=7+4=11=Odd number
m-n=7-4=3=Odd number
Hence, m+n and m-n are both odd numbers.
4.Let m is even number and n is odd number .
Suppose m=6, n=3
m+n=6+3=9=Odd number
m-n=6-3=3=Odd number
Hence, m+n and m-n are both odd numbers.
Therefore, we can say for all inetegers m and n , m+n and m-n are either both even or both odd.Hence proved.
Final answer:
The problem is solved by expressing the conditions under which m and n are both even or odd, and their sum and difference in terms of 2k (for even) and 2k+1 (for odd), demonstrating that m+n and m-n are both even or both odd.
Explanation:
To prove that for all integers m and n, m + n and m - n are either both even or both odd, we start by recalling the definition of even and odd numbers. An even number can be expressed as 2k, where k is an integer, and an odd number can be expressed as 2k + 1, where k is an integer.
If m and n are both even, then m = 2a and n = 2b for some integers a and b. Thus, m + n = 2a + 2b = 2(a + b) and m - n = 2a - 2b = 2(a - b), proving that m + n and m - n are both even.If m and n are both odd, then m = 2a + 1 and n = 2b + 1 for some integers a and b. Consequently, m + n = (2a + 1) + (2b + 1) = 2(a + b + 1) and m - n = (2a + 1) - (2b + 1) = 2(a - b), showing that m + n and m - n are both even.If one is even and the other is odd, for example, m = 2a and n = 2b + 1, then m + n = 2a + (2b + 1) = 2(a + b) + 1 and m - n = 2a - (2b + 1) = 2(a - b) - 1, indicating that m + n and m - n are both odd.This argument shows that m + n and m - n must either be both even or both odd for any integers m and n.
Find i (the rate per period) and n (the number of periods) for the following loan at the given annual rate.
Annual payments of $3,600 are made for 12 years to repay a loan at 5.7% compounded annually.
i=
n=
Answer:
i = 5.7%
n = 12
Step-by-step explanation:
Compounded annually means once per year. So the rate per period is 5.7%, and the number of periods is 12.
Answer:
i = 5.7%
n = 12
Step-by-step explanation:
i (the rate per period) and n (the number of periods) for the following loan at the given annual rate.
Annual payments of $3,600 are made for 12 years to repay a loan at 5.7% compounded annually.
Therefore,
i = 5.7%
n = 12
2. A multiple-choice test contains 10 questions. There are four possible answers for each question lo how many ways can a student answer the questions on the test if the student answers every question? the test if the studest
Answer:
The number of ways are 1,048,576.
Step-by-step explanation:
Consider the provided information.
Product rule: If one event occurs in n contexts and the second event occurs in m contexts, then the number of ways in which the two events happen is n×m.
There are 10 questions and each question has 4 choices.
Therefore, for first question we have 4 choices, for second question we have 4 choices similarly for 10th question we have 4 choices which can be represented as:
4×4×4×4×4×4×4×4×4×4 = [tex]4^{10}[/tex]
4×4×4×4×4×4×4×4×4×4 = 1048576
Thus, the number of ways are 1,048,576.
Use induction to prove that 2? ?? for any integer n>0 . Indicate type of induction used.
I proved the base case using n = 1, and for my induction hypothesis, I said that we assume n = k for 2^k > k, but I am stuck trying to get to n = k + 1.
So far I have:
2^k > k
2*2^k > 2*k
2^{k+1} > 2k
Answer with explanation:
The given statement is which we have to prove by the principal of Mathematical Induction
[tex]2^{n}>n[/tex]
1.→For, n=1
L H S =2
R H S=1
2>1
L H S> R H S
So,the Statement is true for , n=1.
2.⇒Let the statement is true for, n=k.
[tex]2^{k}>k[/tex]
---------------------------------------(1)
3⇒Now, we will prove that the mathematical statement is true for, n=k+1.
[tex]\rightarrow 2^{k+1}>k+1\\\\L H S=\rightarrow 2^{k+1}=2^{k}\times 2\\\\\text{Using 1}\\\\2^{k}>k\\\\\text{Multiplying both sides by 2}\\\\2^{k+1}>2k\\\\As, 2 k=k+k,\text{Which will be always greater than }k+1.\\\\\rightarrow 2 k>k+1\\\\\rightarrow2^{k+1}>k+1[/tex]
Hence it is true for, n=k+1.
So,we have proved the statement with the help of mathematical Induction, which is
[tex]2^{k}>k[/tex]
You borrow $680 from your brother and agree to pay back $750 in 3 months. What simple interest rate will you pay?
Answer:
hence rate interest r = 41.176%
Step-by-step explanation:
The amount borrowed= $680
amount payed back = $750
therefore, interest incurred = 750-680= $70
time, t= 3 months = 3/12= 0.25 years
rate%, r
we know that SI= [tex]\frac{PRT}{100}[/tex]
70= [tex]\frac{680\timesr\0.25}{100}[/tex]
r=[tex]\frac{7000}{0.25\times680} = 41.176[/tex]
hence rate interest r = 41.176%
10. Determine whether or not, vectors ui(1,-2, 0, 3), u2 = (2, 3,0,-1), u3 = (3,9,-4,-2) e R is a linear combination of the (2,-1,2,1) 2
If (2, -1, 2, 1) is a linear combination of the three given vectors, then there should exist [tex]c_1,c_2,c_3[/tex] such that
[tex](2,-1,2,1)=c_1(1,-2,0,3)+c_2(2,3,0,-1)+c_3(3,9,-4,-2)[/tex]
or equivalently, there should exist a solution to the system
[tex]\begin{cases}c_1+2c_2+3c_3=2\\-2c_1+3c_2+9c_3=-1\\-4c_3=2\\3c_1-c_2-2c_3=1\end{cases}[/tex]
Right away we get [tex]c_3=-\dfrac12[/tex], so the system reduces to
[tex]\begin{cases}c_1+2c_2=\dfrac72\\\\-2c_1+3c_2=\dfrac72\\\\3c_1-c_2=0\end{cases}[/tex]
Notice that the first equation is the sum of the latter two. The third equation gives us
[tex]3c_1-c_2=0\implies 3c_1=c_2[/tex]
so that in the second equation,
[tex]-2c_1+3c_2=\dfrac72\implies7c_1=\dfrac72\implies c_1=\dfrac12[/tex]
which in turn gives
[tex]3c_1=c_2\implies c_2=\dfrac32[/tex]
and hence the (2, -1, 2, 1) is a linear combination of the given vectors, with
[tex]\boxed{(2,-1,2,1)=\dfrac12(1,-2,0,3)+\dfrac32(2,3,0,-1)-\dfrac12(3,9,-4,-2)}[/tex]
Customer account "numbers" for a certain company consist of 3 letters followed by 2 numbers.Step 1 of 2 : How many different account numbers are possible if repetitions of letters and digits are allowed?
Final answer:
If 3 letters are followed by 2 numbers and repetitions are allowed, there are a total of 1757600 possible different account numbers. Each letter position has 26 choices and each number position has 10 choices.
Explanation:
The student has asked to determine the number of different customer account "numbers" a company can have if the accounts consist of 3 letters followed by 2 numbers, with repetitions allowed for both letters and numbers.
To calculate the total number of possible account numbers, we can use the multiplication principle of counting. The number of options for each position of the account number is multiplied together to get the total number of combinations.
For the 3 letters, each position can contain any letter from A-Z, which gives us 26 choices per position. Since repeats are allowed, each of the 3 positions has 26 possible choices.
For the 2 numbers, each position can contain any digit from 0-9, which gives us 10 choices per position.
Therefore, to find the total number of possible account numbers, we calculate:
26 × 26 × 26 × 10 × 10 = 1757600 possible account numbers.
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) = 0.9x^2 -234x + 23,194 . How many cars must be made to minimize the unit cost?
Do not round your answer.
Answer:
130 cars.
Step-by-step explanation:
The cost function is given by:
C(x) = 0.9x^2 -234x + 23,194; where x is the input and C is the total cost of production.
To find the minimum the unit cost, there must be a certain number of cars which have to be produced. To find that, take the first derivative of C(x) with respect to x:
C'(x) = 2(0.9x) - 234 = 1.8x - 234.
To minimize the cost, put C'(x) = 0. Therefore:
1.8x - 234 = 0.
Solving for x gives:
1.8x = 234.
x = 234/1.8.
x = 130 units of cars.
To check whether the number of cars are minimum, the second derivative of C(x) with respect to x:
C''(x) = 1.8. Since 1.8 > 0, this shows that x = 130 is the minimum value.
Therefore, the cars to be made to minimize the unit cost = 130 cars!!!