Answer:
C. P69,256.82
Step-by-step explanation:
We know that,
The amount formula in compound interest is,
[tex]A=P(1+\frac{r_1}{n_1})^{n_1t_1} (1+\frac{r_2}{n_2})^{n_2t_2}.......[/tex]
Where, P is the principal amount,
[tex]r_1, r_2....[/tex] are the annual rate for the different periods,
[tex]t_1, t_2,.....[/tex] are the number of year for different periods,
[tex]n_1, n_2, n_3...[/tex] are the number of periods,
Given,
A = P 200,000,
[tex]r_1=10%=0.1[/tex], [tex]n_1=4[/tex], [tex]t_1=5[/tex],[tex]r_2=12%=0.12[/tex], [tex]n_2=1[/tex], [tex]t_2=5[/tex]
Thus, by the above formula the final amount would be,
[tex]200000=P(1+\frac{0.1}{4})^{4\times 5}(1+\frac{0.12}{1})^{1\times 5}[/tex]
[tex]200000=P(1+0.025)^{20}(1+0.12)^5[/tex]
[tex]200000=P(1.025)^{20}(1.12)^5[/tex]
[tex]\implies P=69,256.824\approx 69,256.82[/tex]
Option C is correct.
The correct option is C. P69,256.82. The father should invest approximately P69,256.82 today to ensure his son receives P200,000 in ten years.
To determine the amount the father needs to invest today to give his son P200,000 ten years from now, we will break the problem into two phases due to different interest rates and compounding periods.
Phase 1: First Five Years (10% compounded quarterly)
→ Future Value (FV) needed after 10 years: P200,000
→ Future Value (FV) after first five years at 12% annual interest for the next five years:
Let's use the formula for compound interest to calculate amount required after the first five years.
Here,
→ n is the number of times the interest is compounded per year
→ t is time in years.
→ [tex]FV = PV*(1 + r/n)^{(nt)[/tex]
After the first five years, the amount needs to grow at 12% compounded annually for 5 years to reach P200,000.
We can calculate the present value (PV) at the end of the first five years needed to achieve P200,000 after next 5 years.
→ P200,000 = [tex]PV * (1 + 0.12/1)^{(1*5)[/tex]
→ P200,000 = [tex]PV * (1.12)^5[/tex]
Calculating PV:
→ PV = P200,000 / (1.7623)
≈ P113,477.57
Phase 2: First Five Years Investment Calculation
Now, we need to find out the amount the father should invest today to reach P113,477.57 after five years with 10% interest compounded quarterly.
Here,
→ r is the quarterly rate,
→ nt is the total number of quarters.
We use the same compound interest formula:
→ [tex]FV = PV * (1 + r/n)^{(nt)[/tex]
→ P113,477.57 = [tex]PV * (1 + 0.10/4)^{(4*5)[/tex]
→ P113,477.57 = [tex]PV * (1.025)^{20[/tex]
→ [tex](1.025)^{20} \approx 1.6386[/tex]
Calculating PV:
→ PV = P113,477.57 / 1.6386
≈ P69,256.82
So, the father needs to invest approximately P69,256.82 today.
For the equation x^2y" - xy' = 0, find two solutions, show that they are linearly independent and find the general solution. Equations of the form ax^2y" + bxy' + cy = 0 are called Euler's equations or Cauchy-Euler equations. They are solved by trying y = x^r and solving for r (assume that x greaterthanorequalto 0 for simplicity).
Answer:
[tex]y=A+Bx^{2}[/tex]
Step-by-step explanation:
The given Cauchy-Euler equation is: [tex]x^2y''-xy'=0[/tex]
Comparing to the general form: [tex]ax^2y''+bxy'+cy=0[/tex], we have a=1,b=-1 and c=0
The auxiliary solution is given by: [tex]am(m-1)+bm+c=0[/tex]
[tex]\implies m(m-1)-m=0[/tex]
[tex]\implies m(m-1-1)=0[/tex]
[tex]\implies m(m-2)=0[/tex]
[tex]\implies m=0\:\:or\:\:m=2[/tex]
The general solution to this is of the form [tex]y=Ax^{m_1}+Bx^{m_2}[/tex], where A and B are constants.
[tex]y=Ax^{0}+Bx^{2}[/tex]
Therefore the general solution is;
[tex]y=A+Bx^{2}[/tex]
Let [tex]y_1=A[/tex] and [tex]y_2=Bx^2[/tex]
Since we CANNOT express the two solutions as constant multiple of each other, we say the two solutions are linearly independent.
[tex]y_1\neCy_2[/tex], where C is a constant.
The solutions to the differential equation x^2y" - xy' = 0 are y = 1 and y = x, which are linearly independent. The general solution is y = C1 + C2x, where C1 and C2 are constants.
Explanation:For the Cauchy-Euler equation x^2y" - xy' = 0, we find solutions by substituting y = x^r. Differentiating yields y' = rx^{r-1} and y" = r(r-1)x^{r-2}. Substituting these into the equation and simplifying gives us r(r-1)x^r - rx^r = 0, which simplifies to the characteristic equation r^2 - r = 0. Solving this equation, we get the roots r1 = 0 and r2 = 1. Therefore, the two solutions are y1 = x^0 = 1 and y2 = x^1 = x. To show that they are linearly independent, we evaluate the Wronskian determinant W(y1,y2) = |1 x| = x which is nonzero for x > 0. The general solution is a combination of the two: y = C1y1 + C2y2, where C1 and C2 are arbitrary constants.
According to a study done by De Anza students, the height for Asian adult males is normally distributed with an average of 66 inches and a standard deviation of 2.5 inches. Suppose one Asian adult male is randomly chosen. Let X = height of the individual. A) Give the distribution of X.
X ~ _____ (____,____)
B) Find the probability that the person is between 64 and 69 inches.
Write the probability statement.
P(______
What is the probability?
The distribution of X is a normal distribution with a mean of 66 inches and a standard deviation of 2.5 inches. To find the probability that a person's height is between 64 and 69 inches, we need to determine the z-scores for each height value and use a z-table or calculator to find the probabilities.
Explanation:A) Give the distribution of X.
X ~ Normal(66, 2.5)
B) Find the probability that the person is between 64 and 69 inches.
P(64 ≤ X ≤ 69)
To find this probability, we need to calculate the z-score for each height value using the formula z = (X - μ) / σ. Then, we can use a z-table or a calculator to find the corresponding probabilities.
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The height X of Asian adult males follows a normal distribution with parameters X ~ N(66, 2.5). The probability that one randomly chosen male is between 64 and 69 inches tall can be found using Z-scores and the area under the normal curve.
The distribution of the height X for Asian adult males according to the given study is X ~ N(66, 2.5), where 66 is the mean height in inches and 2.5 is the standard deviation.
Probability that the height is between 64 and 69 inches
To find the probability that the height of a randomly chosen Asian adult male is between 64 and 69 inches, we can use the properties of the normal distribution. We first convert the raw scores to Z-scores using the formula Z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.
The probability statement is P(64 < X < 69). To calculate this probability, we find the Z-scores for 64 and 69 inches and then find the area under the normal curve between these two Z-scores using a standard normal distribution table or a calculator with normal distribution functions.
Assume that on a standardized test of 100 questions, a person has a probability of 80% of answering any particular question correctly. Find the probability of answering between 74 and 84 questions
Answer: 0.7264
Explanation:
The number of independent questions (n) = 100
Probability of answering a question (p) = 0.80
Let X be the no. of questions that need to be answered.
[tex]\therefore[/tex] random variable X follows binomial distribution
The probability function of a binomial distribution is given as
[tex]P(X=x) = \binom{n}{x}\times p^{x}(1-p)^{n-x}[/tex]
Now , we nee to find P(74 ≤ X ≤ 84)
[tex]\therefore P(74\leq X\leq 84) = P(X=74) + P(X=75).........+ P(X=84)[/tex]
P(74 ≤ X ≤ 84) = [tex]\sum_{74}^{84}\binom{100}{x}\times (0.80)^{x}(0.20)^{100-x}[/tex]
P(74 ≤ X ≤ 84) = 0.7264
Timelimit: 15 minutes. 11:08 remaining, x A radio tower is located 250 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is 38 and that the angle of depression to the bottom of the tower is 27. How tall is the tower? Preview feet Points possible: 1 This is attempt 1 of 1. Submit
Answer:
The height of tower is 322.7 feet.
Step-by-step explanation:
Given
Distance between a building and tower= 250 feet
BCDE is a rectangle .Therefore, we have BC=ED and CD=BE=250 feet
In triangle ABE
[tex]tan\theta=\frac{perpendicula \; side }{hypotenuse}[/tex]
[tex]\theta=38^{\circ}[/tex]
[tex] tan38^{\circ}=\frac{AB}{BE}[/tex]
[tex]\frac{AB}{250}=0.781[/tex]
[tex] AB=0.781\times250[/tex]
AB=195.25 feet
In triangle EDC
[tex]\theta=27^{\circ}[/tex]
[tex]tan27^{\circ}=\frac{ED}{CD}[/tex]
[tex]\frac{ED}{250}=0.509[/tex]
[tex]ED=250\times0.509[/tex]
ED=127.25 feet
ED=BC=127.25 feet
The height of tower=AB+BC
The height of tower=195.25+127.25=322.5 feet
How many solutions can a nonhomogeneous system of linear (algebraic) equations have? What if the system is homogeneous?
Answer:
Step-by-step explanation:
Infinitely many.
Infinitely many.
A non-homogeneous system of linear equations can have a unique solution, no solution, or infinitely many solutions and a homogeneous system always has at least the trivial solution, and may have infinitely many non-trivial solutions depending on the rank of A.
A non-homogeneous system of linear equations (where the vector b on the RHS is non-zero) can have three types of solutions:
Unique solution: If the matrix A is square (m=n) and its determinant is not zero, the system has exactly one unique solution.No solution: If the system is inconsistent (meaning the rows of matrix A are linearly dependent and it does not satisfy the Rouche-Capelli theorem), there is no solution.Infinitely many solutions: If the matrix A has more variables than equations (m<n) and the columns are linearly dependent, the system has infinitely many solutions.In contrast, a homogeneous system of linear equations (where the vector b on the RHS is zero) always has at least the trivial solution, where all variables equal zero. Depending on the rank of matrix A:
Trivial solution: If the rank of A is equal to the number of variables (n), the only solution is the trivial one.Non-trivial solutions: If the rank of A is less than the number of variables (m<n), there are infinitely many non-trivial solutions.These rules are concisely summarized by the Rouche-Capelli theorem.
Engineers want to design seats in commercial aircraft so that they are wide enough to fit 90% of all males. (Accommodating 100% of males would require very wide seats that would be much too expensive.) Men have hip breadths that are normally distributed with a mean of 14.9 in. and a standard deviation of 0.9 in. Find Upper P 90. That is, find the hip breadth for men that separates the smallest 90% from the largest 10%.
Answer:
16.053 inches
Step-by-step explanation:
Many such probability questions are easily answered by a suitable calculator or spreadsheet.
Let R be the relation on N x N defined by (a, b) R(c, d) if and only if ad bc. Show that R Equivalence Relations. is an equivalence relation on N x N.
If 40% of a school consists of boys, 40 % of the students have blonde hair, and three times as many girls as boys have blond hair, what percentage of the school are blonde haired boys?
Answer:
10% of the school are blonde haired boys.
Step-by-step explanation:
Let x be the total number of students,
Number of students who have blonde hair = 40% of the total students
= 40% of x
= 0.4x
Also, the ratio of three times as many girls as boys have blond hair,
So, the ratio of blond hair boys and blond hair girls = 1 : 3
Let the number of boys who have blonde hair = y
And, the number of girls who have blonde hair = 3y
So, y + 3y = 0.4x
4y = 0.4x ⇒ y = 0.1x
Thus, the number of boys who have blonde hair = 0.1x
Hence, the percentage of the school are blonde haired boys
= [tex]\frac{0.1x}{x}\times 100=10\%[/tex]
john bates invested $5,000.00 in an account that paid 5% interest annually and is compounded anually.what is the amount after 2years
Answer:$500
Step-by-step explanation:
The hemisphere of radius r is made from a stack of very thin plates such that the density varies with height, r = kz, where k is a constant. Determine its mass and the distance z to the center of mass G.
Answer:
M = ¼ k π R⁴
zG = 8/15 R
Step-by-step explanation:
Note: I'm using lower case r as the radius of each plate and upper case R as the radius of the hemisphere.
The mass of each plate is density times volume:
dm = ρ dV
Each plate has a radius r and a thickness dz. So the volume of each plate is:
dV = π r² dz
Substituting:
dm = ρ π r² dz
We're told that ρ = kz. Substituting:
dm = kz π r² dz
Next, we need to write the radius r in terms of the height z. To do that, we need to look at the cross section (see image below).
The height z and the radius r form a right triangle, where the hypotenuse is the radius of the hemisphere R.
Using Pythagorean theorem:
z² + r² = R²
r² = R² − z²
Substituting:
dm = kπ z (R² − z²) dz
We now have the mass of each plate as a function of its height. To find the total mass, we integrate between z=0 and z=R.
M = ∫ dm
M = ∫₀ᴿ kπ z (R² − z²) dz
M = kπ ∫₀ᴿ (R² z − z³) dz
M = kπ (½ R² z² − ¼ z⁴) |₀ᴿ
M = kπ (½ R⁴ − ¼ R⁴)
M = ¼ k π R⁴
Next, to find the center of gravity, we use the weighted average:
zG = (∫ z dm) / (∫ dm)
zG = (∫ z dm) / M
We already found M, we just have to evaluate the other integral:
∫ z dm
∫₀ᴿ kπ z² (R² − z²) dz
kπ ∫₀ᴿ (R² z² − z⁴) dz
kπ (⅓ R² z³ − ⅕ z⁵) |₀ᴿ
kπ (⅓ R⁵ − ⅕ R⁵)
²/₁₅ k π R⁵
Plugging in:
zG = (²/₁₅ k π R⁵) / (¼ k π R⁴)
zG = ⁸/₁₅ R
Find all the roots of the given function. Use preliminary analysis and graphing to find good initial approximations. f(x)equals=cosine left parenthesis 3 x right parenthesis minus 7 x squared plus 4 xcos(3x)−7x2+4x
Answer:
The given function is
f(x)=cos 3x-7 x²+ 4x
f'(x)=-3 sin 3 x-14 x+4
When you will draw the graph of the function , you will find that root of the function lie between (-1,0).
Consider initial root as,
[tex]x_{0}=0[/tex]
Using Newton method to find the roots of the equation
[tex]x_{n+1}=x_{n}-\frac{f{x_n}}{f'{x_{n}}}\\\\x_{1}=x_{0} - \frac{cos 3x_{0}-7 x_{0}^2+ 4x_{0}}{-3 sin 3 x_{0}-14 x_{0}+4}\\\\x_{1}=-\frac{\cos 0^{\circ}-0+0}{-3 \times 0-0+4}\\\\x_{1}=\frac{-1}{4}\\\\x_{1}= -0.25\\\\x_{2}=x_{1} - \frac{cos 3x_{1}-7 x_{1}^2+ 4x_{1}}{-3 sin 3 x_{1}-14 x_{1}+4}\\\\x_{2}=-0.25 -\frac{cos (-0.75)-7\times (0.0625)- 1}{-3 sin (-0.75)+3.50+4}\\\\x_{2}= -0.176054[/tex]
[tex]x_{3}=x_{2} - \frac{cos 3x_{2}-7 x_{2}^2+ 4x_{2}}{-3 sin 3 x_{2}-14 x_{2}+4}\\\\x_{3}=-0.176054 -\frac{cos (3\times -0.176054)-7\times (-0.176054)^2+4 \times -0.176054}{-3 sin (-0.176054)-14 \times (-0.176054)+4}\\\\x_{3}= -0.1689[/tex]
So, root of the equation is
=0.1688878
=0.1689(approx)
a bank charges a 1% fee to process a credit card cash advance whis is taken out of the cash advance amount. A customer wants a cash advance of $4,000. how much money will he receive from the bank after the cash advance has been processed.
Answer: [tex]\$\ 3600[/tex]
Step-by-step explanation:
Given : A bank charges a 1% fee to process a credit card cash advance which is taken out of the cash advance amount.
The rate of fee can be also written as 0.01.
The amount of money a customer wants as a cash advance = $4,000
The amount of bank fee = [tex]0.01\times4000=40[/tex]
The amount which customer receive after cash advance has been processed = [tex]4000-40=\$3600[/tex]
Hence, the amount of money will he receive from the bank after the cash advance has been processed = [tex]\$\ 3600[/tex]
A given binomial distribution has 10 trials and probability of success p=1/3. Compute the standard deviation and explain your solution
Answer: 1.49
Step-by-step explanation:
We know that the standard deviation in a binomial distribution is given by :-
[tex]\sigma=\sqrt{n\cdot p\cdot (1-p)}[/tex], where n is the number of trials and p is the probability of success.
Given : The number of trials : [tex]n=10[/tex]
The probability of success = [tex]p=\dfrac{1}{3}[/tex]
Then , the standard deviation will be :-
[tex]\sigma=\sqrt{10\cdot \dfrac{1}{3}\cdot (1-\dfrac{1}{3})}\\\\\Rightarrow\ \sigma=1.490711985\approx1.49[/tex]
The standard deviation of a given binomial distribution with 10 trials and a probability of success of 1/3 is calculated using the formula σ = √npq, resulting in approximately 1.49. This value helps us understand the spread of successes in the experiment.
Explanation:To compute the standard deviation of a given binomial distribution with 10 trials and a probability of success p=1/3, we first identify the probability of failure, which is q=1-p=2/3. The formula for the standard deviation (σ) of a binomial distribution is σ = √npq, where n is the number of trials, p is the probability of success, and q is the probability of failure.
By substituting the given values into the formula:
n = 10p = 1/3q = 2/3We get σ = √(10)(1/3)(2/3) = √(20/9) ≈ 1.49. Therefore, the standard deviation of the binomial distribution is approximately 1.49.
This calculation helps us understand the dispersion or spread of the distribution of successes in our binomial experiment, indicating how far the number of successes can deviate from the expected value on average.
When situations arise in which your organization cannot meet one or more standards immediately, it is vitally important to recognize an exception to standards to determine where problems may exist. Answer: A Reference: p 159 True False
Answer:
open fb and search ravi verma
Step-by-step explanation:
send friend request
8) What does the mathematical symbol TT represent? 9) What does the mathematical symbol E represent?
Answer:
TT means pi and e means Euler
Step-by-step explanation:
The Office of Student Services at a large western state university maintains information on the study habits of its full-time students. Their studies indicate that the mean amount of time undergraduate students study per week is 20 hours. The hours studied follows the normal distribution with a standard deviation of six hours. Suppose we select a random sample of 144 current students. What is the probability that the mean of this sample is between 19.25 hours and 21.0 hours?
(A) 0.4332
(B) 0.8664
(C) 0.9104
(D) 0.0181
Answer: (C) 0.9104
Step-by-step explanation:
Given : The hours studied follows the normal distribution
Mean : [tex]\mu=\text{20 hours}[/tex]
Standard deviation : [tex]\sigma=6\text{ hours}[/tex]
Sample size : [tex]n=144[/tex]
The formula to calculate the z-score is given by :-
[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
Let x be the number hours taken by randomly selected undergraduated student.
Then for x = 19.25 , we have
[tex]z=\dfrac{19.25-20}{\dfrac{6}{\sqrt{144}}}=-1.5[/tex]
for x = 21.0 , we have
[tex]z=\dfrac{20-21}{\dfrac{6}{\sqrt{144}}}=2[/tex]
The p-value : [tex]P(19.25<x<21)=P(-1.5<z<2)[/tex]
[tex]P(2)-P(-1.5)= 0.9772498- 0.0668072=0.9104426\approx0.9104[/tex]
Thus, the probability that the mean of this sample is between 19.25 hours and 21.0 hours = 0.9104.
How many triangles can be made from the following three lengths: 3.1 centimeters, 9.8 centimeters, and 5.2 centimeters?
one
none
more than one
Answer:
none
Step-by-step explanation:
We have three sides
Adding the two smallest sides together must be bigger than the third side
3.1+5.2 > 9.8
8.3> 9.8
This is false, so we cannot make a triangle
A tank initially holds 100 gallons of a brine solution containing 1 lb. of salt. At t = 0 another brine solution containing 1 pound of salt per gallon is poured into the tank at the rate of 4 gal/min., while the well-stirred mixture leaves the tank at the same rate. Find (a) the amount of salt in the tank at any time t and (b) the time at which the mixture in the tank contains 2 lb of salt.
Answer:
[tex]\boxed{\text{(a)}A = 100 - 99e^{-t/25}; \,\text{(b) 15 s}}[/tex]
Step-by-step explanation:
(a) Expression for mass of salt as function of time
[tex]\text{Let A = mass of salt after t min}\\\text{and }r_{i} = \text{rate of salt coming into talk}\\\text{and }r_{o}$ =\text{rate of salt going out of tank}[/tex]
i. Set up an expression for the rate of change of salt concentration.
[tex]\dfrac{\text{d}A}{\text{d}t} = r_{i} - r_{o}\\\\r_{i} = \dfrac{\text{4 gal}}{\text{1 min}} \times \dfrac{\text{1 lb}}{\text{1 gal}} = \text{4 lb/min}\\\\r_{o} = \dfrac{\text{4 gal}}{\text{1 min}} \times \dfrac {A\text{ lb}}{\text{100 gal}} =\dfrac{A}{25}\text{ lb/min}\\\\\dfrac{\text{d}A}{\text{d}t} = 4 - \dfrac{x}{25}[/tex]
ii. Integrate the expression
[tex]\dfrac{\text{d}A}{\text{d}t} = \dfrac{100 - A}{25}\\\\\dfrac{\text{d}A}{100 - A} = \dfrac{\text{d}t}{25}\\\\\int \frac{\text{d}A}{100 - A} = \int \frac{\text{d}t}{25}\\\\-\ln(100 - A) = \dfrac{t}{25} + C[/tex]
iii. Find the constant of integration
[tex]-\ln (100 - A) = \dfrac{t}{25} + C\\\\\text{At $t$ = 0, $A$ = 1, so}\\\\-\ln (100 - 1) = \dfrac{0}{25} + C\\\\C = -\ln 99[/tex]
iv. Solve for A as a function of time.
[tex]\text{The integrated rate expression is}\\\\-\ln (100-A) = \dfrac{t}{25} - \ln 99\\\\\text{Solve for } A\\\\\ln(100 - A) = \ln 99 - \dfrac{t}{25}\\\\100 - A = 99e^{-t/25}\\\\A = \boxed{\mathbf{100 - 99e^{-t/25}}}[/tex]
The diagram shows A as a function of time. The mass of salt in the tank starts at 1 lb and increases asymptotically to 100 lb.
(b) Time to 2 lb salt
[tex]A = 100 - 99e^{-t/25}\\\\2 = 100 - 99e^{-t/25}\\\\99e^{-t/25} = 98\\\\e^{-t/25} = 0.989899\\\\-t/25 = -0.01015\\\\t = 25\times 0.01015 =\text{0.25 min = 15 s}\\\\\text{The tank will contain 2 lb of salt after } \boxed{\textbf{15 s}}[/tex]
To find the amount of salt in the tank at any time t, we need to consider the rate at which the brine solution is being poured in and leaving the tank. Initially, there is 1 pound of salt in the tank. The rate at which salt is added is 1 pound per gallon, and the rate at which the mixture is being removed is 4 gallons per minute. Therefore, the amount of salt in the tank at time t can be calculated using the equation. To find the time at which the mixture in the tank contains 2 pounds of salt, we can set up the equation. Solving this equation for t will give us the time at which the mixture contains 2 pounds of salt.
Explanation:To find the amount of salt in the tank at any time t, we need to consider the rate at which the brine solution is being poured in and leaving the tank. Initially, there is 1 pound of salt in the tank. The rate at which salt is added is 1 pound per gallon, and the rate at which the mixture is being removed is 4 gallons per minute. Therefore, the amount of salt in the tank at time t can be calculated using the equation:
Amount of salt at time t = 1 + (1 pound/gallon)(4 gallons/minute)(t minutes)
To find the time at which the mixture in the tank contains 2 pounds of salt, we can set up the equation:
2 = 1 + (1 pound/gallon)(4 gallons/minute)(t minutes)
Solving this equation for t will give us the time at which the mixture contains 2 pounds of salt.
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Please someone help me with these equations
Answer:
[tex]f (-2) =-\frac{8}{3}[/tex]
[tex]f (4) =\frac{4}{3}[/tex]
[tex]f (1) = - 4[/tex]
Step-by-step explanation:
For this case it has a piecewise function composed of two functions.
To evaluate the piecewise function observe the condition.
[tex]f (x) = \frac{1}{3}x ^ 2 -4[/tex] when [tex]x \neq 1[/tex]
[tex]f (x) = -4[/tex] when [tex]x = 1[/tex]
We start by evaluating [tex]f(-2)[/tex], note that [tex]x = -2\neq 1[/tex]. Then we use the quadratic function:
[tex]f (-2) = \frac{1}{3}(-2) ^ 2 -4 = -\frac{8}{3}[/tex]
Now we evaluate [tex]f(4)[/tex] note that [tex]x = 4\neq 1[/tex]. Then we use the quadratic function:
[tex]f (4) = \frac{1}{3}(4) ^ 2 -4 = \frac{4}{3}[/tex]
Finally we evaluate [tex]f(1)[/tex] As [tex]x = 1[/tex] then
[tex]f (1) = - 4[/tex]
Express the following repeating decimal as a fraction in simplest form.
0.342 repeating (line over entire decimal)
Final answer:
The repeating decimal 0.342 can be expressed as the fraction 38/111 in simplest form, by using algebraic methods to eliminate the repeating pattern and then simplifying the resulting fraction.
Explanation:
Expressing the repeating decimal 0.342 as a fraction involves using algebra to convert the decimal into a form that can be simplified into a fraction. First, we set the repeating decimal equal to a variable:
Let x = 0.342
To remove the repeating part, we multiply x by a power of 10 that corresponds to the number of digits in the repeating sequence. Since there are three digits repeating (342), we multiply by 1000:
1000x = 342.342
We now subtract the original equation (1x) from this new equation (1000x) to get rid of the decimal:
1000x - x = 342.342 - 0.342
999x = 342
Now we simplify by dividing both sides by 999:
x = 342 / 999
This fraction can be further simplified by dividing both the numerator and denominator by their greatest common divisor:
x = 38 / 111. Therefore, the fractional form of 0.342 is 38/111 in its simplest form.
To convert the repeating decimal 0.342 (with 342 repeating indefinitely) to a fraction, let's follow these steps:
1. Let x equal the repeating decimal:
\( x = 0.342342342... \)
2. Recognize the pattern: the digits 342 repeat every three places. To isolate the repeating sequence, we need to multiply x by a power of 10 that has the same number of digits as the repeating sequence.
3. The repeating sequence has three digits, so we'll multiply x by \( 10^3 = 1000 \):
\( 1000x = 342.342342... \)
4. Now, subtract the original number x from this new number to remove the repeating part:
\( 1000x - x = 342.342342... - 0.342342342... \)
5. Notice that on the right-hand side, the repeating decimals cancel each other out, leaving whole numbers:
\( 999x = 342.000000... \)
6. Now solve for x:
\( x = \frac{342}{999} \)
7. To simplify the fraction, we find the greatest common divisor (GCD) of the numerator (342) and the denominator (999).
To find the GCD, we can list the factors of each:
Factors of 342 include: \( 1, 2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342 \).
Factors of 999 include: \( 1, 3, 9, 27, 37, 111, 333, 999 \).
The largest factor that appears in both lists is 9.
8. Divide both the numerator and the denominator by their GCD:
\( x = \frac{342 ÷ 9}{999 ÷ 9} \)
9. Doing the division:
\( x = \frac{38}{111} \)
Thus, the repeating decimal 0.342 (with a line over the entire decimal) can be expressed as the fraction \( \frac{38}{111} \) in its simplest form.
Last year, a person wrote 123 checks. Let the random variable x represent the number of checks he wrote in one day, and assume that it has a Poisson distribution. What is the mean number of checks written per day? What is the standard deviation? What is the variance?
Answer:The mean number of checks written per day = 0.3370
The standard deviation = 0.5805
The variance = 0.3370
Step-by-step explanation:
Let the random variable x represent the number of checks he wrote in one day.
Given : The number of checks written in last year = 123
Let the number of days in the year must be 365.
Now, the mean number of checks written per day will be :-
[tex]\lambda=\dfrac{123}{365}=0.33698630137\approx0.3370[/tex]
We know that in Poisson distribution , the variance is equals to the mean value .
[tex]\text{Thus , Variance }=\sigma^2= 0.3370[/tex]
[tex]\Rightarrow\ \sigma=\sqrt{0.3370}=0.580517010948\approx0.5805[/tex]
Thus, Standard deviation = 0.5805
Use Theorem 7.4.1. THEOREM 7.4.1 Derivatives of Transforms If F(s) = ℒ{f(t)} and n = 1, 2, 3, . . . , then ℒ{tnf(t)} = (−1)n dn dsn F(s). Evaluate the given Laplace transform. (Write your answer as a function of s.) ℒ{te2t sin "3t"}'
Answer:
[tex]L\left(te^{2t }sin3t\right)=\frac{6s-12}{(s^2-4s+13)^2}[/tex].
Step-by-step explanation:
If F(s)= L{f(t)}
Then [tex]L\left\{(t^nf(t)\right\}=(-1)^n\frac{\mathrm{d^n}F(s)}{\mathrm{d^n}s}[/tex]
[tex]L\left\{te^{2t}sin3t\right\}[/tex]
f(t)=[tex]e^{2t}sin3t[/tex]
[tex]L\left\{e^{at}sinbt\right\}=\frac{b}{(s-a)^2+b^2}[/tex]
Therefore,[tex] L\left\{e^{2t}sin3t\right\}=\frac{3}{(s-2)^2+(3)^2}[/tex]
[tex]L\left\{e^{2t}sin3t\right\}=\frac{3}{s^2-4s+13}[/tex]
[tex]L\left\{te^{2t}sin3t\right}=-\frac{\mathrm{d}F(s)}{\mathrm{d}s}[/tex]
=-[tex]\frac{\mathrm{d}e^{2t}sin3t}{\mathrm{d}s}[/tex]
[tex]L\left\{te^{2t}sin3t\right\}[/tex]
[tex]=\frac{3(2s-4)}{(s^2-4s+13)^2}[/tex]
[tex]L\left\{te^{2t}sin3t\right\}=\frac{6s-12}{(s^2-4s+13)^2}[/tex].
The Laplace transform of te2tsin(3t) is 3(-1)(1)/(s2+9)2.
Explanation:According to Theorem 7.4.1, the Laplace transform of a function tnf(t) can be calculated as (−1)n * dn/dsn * F(s), where F(s) is the Laplace transform of f(t). In this case, we need to evaluate the Laplace transform of te2tsin(3t).
First, we need to find the Laplace transform of te2t. Applying the formula F(s) = ℒ{f(t)} = ℒ{te2t}:
F(s) = -d/ds[2/(s-2)2]
Next, we need to find the Laplace transform of sin(3t). Applying the formula F(s) = ℒ{f(t)} = ℒ{sin(3t)}:
F(s) = 3/(s2+9)
Finally, we can evaluate the Laplace transform of te2tsin(3t) as the product of the Laplace transforms obtained:
F(s) = (-1)*(d1/d1s)*(-d1/d1s)[3/(s2+9)]
F(s) = 3(-1)(1)/(s2+9)2
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What are the coordinates of the x-intercept and y-intercept of the line -3y-x=-9
Answer:
Step-by-step explanation:
The x-intercept exists when y = 0; the coordinate takes on the form ( , 0)
The y-intercept exists when x = 0; the coordinate takes on the form (0, ).
We will fill in the blanks in a sec.
If the x-intercept exists when y = 0, then we fill in 0 for y and solve for x:
-3(0) - x = -9 so
-x = -9 and
x = 9
The coordinate is (9, 0) for the x-intercept.
If the y-intercept exists when x = 0, then we fill in 0 for x and solve for y:
-3y - 0 = -9 so
-3y = -9 and
y = 3
The coordinate is (0, 3) for the y-intercept.
A study of king penguins looked for a relationship between how deep the penguins dive to seek food and how long they stay under water. For all but the shallowest dives, there is a linear relationship between depth of dive and length of time under water. The study report gives a scatterplot for a random sample of penguins. The dive duration is measured in minutes and depth (x value) is in meters. The depths are all positive numbers. The dives varied from 40 meters to 300 meters in depth. The report then says, "The regression equation for this study is: y = 2.56 + 0.0135x."
(a) What is the intercept of the regression line? (Use 2 decimal places)
(b) What is the slope of the regression line? (Use 4 decimal places)
(c) What is the correct interpretation of the slope?
For every increase of 1 meter in depth, the mean dive duration increases by exactly 0.0135 minutes.
For every increase of 1 meter in depth, the mean dive duration decreases by exactly 0.0135 minutes.
For every increase of 1 meter in depth, the mean dive duration increases by approximately 0.0135 minutes.
For every increase of 1 meter in depth, the mean dive duration decreases by approximately 0.0135 minutes.
Answer:
given y=2.69+0.0138x the slope is the coefficient of x slope=0.0138 this means that the dive duration is expected to increase by about 0.0138 per minute
Step-by-step explanation:
Answer:
Hi!
A) The intercept of the regression line is 2.56.
B)The slope of the regression line is 0.0135.
C)For every increase of 1-meter depth, the mean dive duration increases by exactly 0.0135 minutes.
Explanation:
The definition of the regression line of a function is the value of Y(x) when x=0.
The slope of a function is the value that configures the ratio of change vertically on the regression.
In this case is represented by 0.0135 multiplied by the x.
If we are talking about depth, and the values of x are all positives values.
Y(50)=2.56+0.0135(50)=3.235
So, when the penguins go deeper, they remain more time diving.
A Halloween trick-or-treat group consists of 3 Trolls, 4 Nazguls, 4 Ents, and 5 Pokemon. A committee of 4 is to be picked to represent the group at the Monster Bash. (a) Find the probability that the committee will consist of 1 from each type of monster. (b) Suppose that the Pokemon refuse to be on the same committee as a the Trolls. Find the probability that this type of committee is formed.
Answer:
The probability that the committee will consist of 1 from each type of monster is 0.1318 and Suppose that the Pokemon refuse to be on the same committee as a the Trolls.So,the probability that this type of committee is formed is 0.5357
Step-by-step explanation:
No. of Trolls = 3
No. of Nazguls =4
No. of Ents = 4
No. of Pokemon = 5
Total Monsters = 16
We are given that A committee of 4 is to be picked to represent the group at the Monster Bash.
(a) Find the probability that the committee will consist of 1 from each type of monster.
So, the probability that the committee will consist of 1 from each type of monster:
= [tex]\frac{^3C_1 \times ^4C_1 \times ^4C_1 \times ^5C_1}{^{16}C_4}[/tex]
= [tex]\frac{\frac{3!}{1!(3-1)!} \times \frac{4!}{1!(4-1)!} \times \frac{4!}{1!(4-1)!}\times \frac{5!}{1!(5-1)!}}{\frac{16!}{4!(16-4)!}}[/tex]
= [tex]\frac{\frac{3!}{1!(2)!} \times \frac{4!}{1!(3)!} \times \frac{4!}{1!(3)!}\times \frac{5!}{1!(4)!}}{\frac{16!}{4!(12)!}}[/tex]
= [tex]0.1318[/tex]
b)Suppose that the Pokemon refuse to be on the same committee as a the Trolls.
So, the probability that this type of committee is formed
= [tex]\frac{(^3C_3 \times ^8C_1)+(^3C_2 \times ^8C_2)+(^3C_1 \times ^8C_3)+(^3C_0 \times ^8C_4)+(^5C_4 \times ^8C_0)+(^5C_3 \times ^8C_1)+(^5C_2 \times ^8C_2)+(^5C_1 \times ^8C_3)}{^{16}C_4}[/tex]
= [tex]\frac{(\frac{3!}{3!(3-3)!} \times \frac{8!}{1!(8-1)!})+(\frac{3!}{2!(3-2)!} \times\frac{8!}{2!(8-2)!})+(\frac{3!}{1!(3-1)!} \times\frac{8!}{3!(8-3)!})+(\frac{3!}{0!(3-0)!} \times \frac{8!}{4!(8-4)!})+(\frac{5!}{4!(5-4)!} \times \frac{8!}{0!(8-0)!})+(\frac{5!}{3!(5-3)!} \times \frac{8!}{1!(8-1)!})+(\frac{5!}{2!(5-2)!} \times\frac{8!}{2!(8-2)!})+(\frac{5!}{1!(5-1)!} \times \frac{8!}{3!(8-3)!})}{\frac{16!}{4!(16-4)!}}[/tex]
= [tex]0.5357[/tex]
Hence The probability that the committee will consist of 1 from each type of monster is 0.1318 and Suppose that the Pokemon refuse to be on the same committee as a the Trolls.So,the probability that this type of committee is formed is 0.5357
The probability that the committee will consist of 1 member from each type of monster is 1. If the Pokemon refuse to be on the same committee as a Troll, the probability that this type of committee is formed is 4/5 or 0.8.
Explanation:(a) To find the probability that the committee will consist of 1 member from each type of monster, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes. There are 3 choices for a Troll, 4 choices for a Nazgul, 4 choices for an Ent, and 5 choices for a Pokemon. So the total number of possible outcomes is 3 * 4 * 4 * 5 = 240. Now, we need to calculate the number of favorable outcomes. Since we want 1 member from each type of monster, we choose 1 Troll from 3, 1 Nazgul from 4, 1 Ent from 4, and 1 Pokemon from 5. So the number of favorable outcomes is 3 * 4 * 4 * 5 = 240. Therefore, the probability is 240/240, which simplifies to 1.
(b) If the Pokemon refuse to be on the same committee as a Troll, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes. To do this, we need to consider two cases: one where a Troll is chosen and one where a Troll is not chosen. Case 1: If a Troll is chosen, then there are 3 choices for a Troll, 4 choices for a Nazgul, 4 choices for an Ent, and 4 choices for a Pokemon (since one Pokemon is excluded). So the total number of outcomes in this case is 3 * 4 * 4 * 4 = 192. Case 2: If a Troll is not chosen, then there are 0 choices for a Troll, 4 choices for a Nazgul, 4 choices for an Ent, and 5 choices for a Pokemon. So the total number of outcomes in this case is 0 * 4 * 4 * 5 = 0. Therefore, the total number of favorable outcomes is 192 + 0 = 192. The total number of possible outcomes is still 240. So the probability is 192/240, which simplifies to 4/5 or 0.8.
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The size of a bicycle is determined by the diameter of the wheel. You want a 20-in bicycle for your birthday. What is the area of one of the wheels? 385.47 in2 314.16 in2 484.37 in2 345.54 in2
Answer:
314.16 in²
Step-by-step explanation:
If the diameter is 20", then the radius is 10" (1/2 the diameter).
Area of a circle is [tex]\pi r[/tex]²
So, π 10² = 100π = 314.16 in²
Guesstimate how much total time do Americans spend driving in one year? Express the answer in hours, years and lifetimes.
How would one go about guestimating a problem like this.
Answer:
1,460 hours per year.
113,880 hours in a lifetime.
Step-by-step explanation:
Great question, it is always good to ask away and get rid of any doubts that you may be having.
Let's assume that on an average day Americans drive 4 hours a day. This includes going and coming from work as well as any groceries they might need to do. Since there are 365 days in a year we can multiply this by the hours a day driven.
[tex]365*4 = 1,460[/tex]
We can "guesstimate" that on average an American drives 1,460 hours per year.
Based on my research the average lifetime of an American is 78 years old. We can now multiply this by the number of hours driven in a year to have an idea of the number of hours driven in a lifetime.
[tex]1,460*78 = 113,880[/tex]
On average we "guesstimate" that an American drives 113,880 hours in a lifetime.
I hope this answered your question. If you have any more questions feel free to ask away at Brainly.
Andrea has a jug containing 1.5 liters of chocolate milk. She fills one cup with 250 milliliters and another cup with 0.6 liters. How much chocolate milk is left in the jug in milliliters?
0.65 milliliters
249.1 milliliters
450 milliliters
650 milliliters
Answer:
650 milliliters.
Step-by-step explanation:
Amount in the Jug = 1.5 * 1000 = 1,500 mls.
She pours out 250 mls and 0.6 * 1000 = 600 mls
The amount left = 1500 - 250 - 600
= 650 milliliters.
The amount of chocolate milk left in the jug is; D: 650 milliliters
How do you convert units?
We are told;
Amount of chocolate milk in the jug = 1.5 litres
She fills one cup with 250 mL = 0.25 liters
She fills another cup with 0.6 liters
Thus;
Total amount used = 0.6 + 0.25 = 0.85 litres
Amount left = 1.5 - 0.85 = 0.65 Liters
Converting to mL gives; 650 milliliters
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1. Using the Euclidian algorithm, compute (91,39) and (73,21)
Answer:
HCF(91,39) = 13 and HCF(73,21) = 1
Step-by-step explanation:
As per euclidian algorithm, a = bq + r, where a is dividend, b is divisor, q is quotient and r is remainder.
We can use euclidian algorithm to find the HCF of numbers.
To find: HCF ( 91, 39 ):
On dividing 91 by 39, we get
91=39×2+13
Here, remainder = 13 [tex]\neq 0[/tex]
So, again applying division algorithm on 39 and 13, we get
[tex]39=13\times 3+0[/tex]
As remainder = 0 and divisor at this step is equal to 13, HCF = 13 .
To find: HCF ( 73, 21 )
On dividing 73 by 21, we get
[tex]73=21\times 3+10[/tex]
Here, remainder = 10 [tex]\neq 0[/tex]
On applying division algorithm on 21 and 10, we get
[tex]21=10\times 2+1[/tex]
Here, remainder = 1 [tex]\neq 0[/tex]
On applying division algorithm on 10 and 1, we get
[tex]10=1\times 10+0[/tex]
As remainder = 0 and divisor at this step is 1, HCF = 1
Compute the following quantities: a) i^21-i^32
b) 2-i/3-2i.
Please show work
Answer: a) i-1; b) 4+i/13
Step-by-step explanation:
The complex number [tex]i[/tex] is defined as the number such that [tex]i^{2}=-1[/tex];
We use the propperty to notice that [tex]i^1=i \quad i^2 = -1 \quad i^3=-i \quad i^4=1 \quad i^5=1 \quad i^6=-1 \quad i^7=-i \quad i^8=1 \quad i^9=i \quad i^10= -1 etc...[/tex].
a) We notice that [tex]i^{21}=i \quad \text{ and \text} \quad i^{32}=1[/tex]. Hence, [tex]i^{21}-i^{32}=i-1[/tex].
b) We multiply the expression by [tex]1=\frac{3+2\cdot i}{3 + 2 \cdot i}[/tex]. Then we get that
[tex]\frac{2-i}{3-2 \cdot i}=\frac{2-i}{3-2\cdot i}\cdot\frac{3+2 \cdot i}{3 + 2\cdot i } = \frac{(2-i)\cdot(3+2 \cdot i)}{3^2+2^2}= \frac{6+4i-3i+2i^2}{13}=\frac{6+i-2}{13} = \frac{4+i}{13}[/tex]