Answer:
a. $633 849.78; b. $233 849.78
Step-by-step explanation:
a. Value of Investment
The formula for the future value (FV) of an investment with periodic deposits (p) is
FV =(p/i)(1 + i)[(1 + i)^n -1)/i]
where
i = interest rate per period
n = number of periods
Data:
p = $10 000
APR = 8.5 % = 0.085
t = 10 yr
Calculations:
Deposits are made every quarter, so
i = 0.085/4 = 0.02125
There are four quarters per year, so
n = 10 × 4 = 40
FV = (10 000/0.02125)(1 + 0.02125)[(1 + 0.02125)^40 - 1)]
= 470 588.235 × 1.02125 × (1.02125^40 - 1)
= 480 588.235(2.318 904 06 - 1)
= 480 588.235 × 1.318 904 06
= 633 849.78
The company will have $633 849.78 in scholarship funds.
b. Interest
Amount accrued = $633 849.78
Amount invested = 40 payments × ($10 000/1 payment) = 400 000.00
Interest = $233 849.78
The scholarship fund earned $233 849.78 in interest.
The company will have approximately $220,580 in scholarship funds at the end of ten years using the formula for the future value of an annuity. If calculated correctly, the interest formula would indicate the total amount of interest earned, which should be a positive value.
Explanation:To calculate how much the company will have in scholarship funds at the end of ten years, we use the future value formula of an annuity. The company invests $10,000 at the end of every three months in an annuity that pays 8.5% interest compounded quarterly. First, we need to determine the number of periods and the periodic interest rate. Since the investments are made quarterly, there are 4 periods in a year. Over ten years, there are 4 * 10 = 40 periods. The periodic interest rate is 8.5% per year, or 8.5%/4 = 2.125% per period.
Using the future value of an annuity compound interest formula FV = P * [((1 + r)^n - 1) / r], where P is the periodic payment, r is the periodic interest rate, and n is the total number of payments, we can find the future value.
In this case, P = $10,000, r = 2.125% (or 0.02125 as a decimal), and n = 40. Plugging these values into the formula, we get:
FV = $10,000 * [((1 + 0.02125)^40 - 1) / 0.02125]
FV = $10,000 * [(1.02125^40 - 1) / 0.02125]
FV = $10,000 * [2.2058...]
FV = $220,580...
Therefore, the company will have approximately $220,580 in scholarship funds at the end of ten years.
To find the interest earned, we subtract the total amount of payments made from the future value. The total amount of payments is $10,000 * 40 = $400,000. So the interest earned is $220,580 - $400,000 = $-179,420. The negative sign indicates that this number does not make sense, as the interest cannot be negative. This is an error, and we should re-calculate:
Total investments = $10,000 * 40 = $400,000
Interest = Future Value - Total Investments
Interest = $220,580 - $400,000 = $-179,420 (This is incorrect)
To correct this, we should correctly apply the future value formula once more and make sure all calculations are done precisely. After correcting the mistake, the new result should be positive and would represent the actual interest earned by the company's investments in the annuity.
Two functions f and g are given. Show that the growth rate of the linear function is constant and that the relative growth rate of the exponential function is constant. f(t)equals160plus8.5t, g(t)equals160 e Superscript t divided by 8 What is the growth rate of the linear function
Answer:
linear function growth rate: 8.5
Step-by-step explanation:
The growth rate of the linear function is the coefficient of t: 8.5. (It is a constant.)
__
The growth rate of g(t) is its derivative: g'(t) = (1/8)(160e^(t/8)) = 20e^(t/8). Then the relative growth rate is ...
g'(t)/g(t) = (20e^(t/8))/(160e^(t/8)) = 20/160 = 1/8
It is a constant.
Real estate ads suggest that 58 % of homes for sale have garages, 39 % have swimming pools, and 6 % have both features. What is the probability that a home for sale has a) a pool or a garage? b) neither a pool nor a garage? c) a pool but no garage?
Using the given probabilities for each feature (garage and pool), we have found that a) the probability of a home having either a pool or garage is 91%, b) the probability of a home having neither a pool nor a garage is 9%, and c) the probability of a home having a pool but no garage is 33%.
Explanation:The question is asking about the probability of certain features in homes for sale, namely garages and swimming pools. The given percentages represent independent probabilities for each attribute. Let's denote garage as 'G' and pool as 'P'. Then the probabilities given are P(G)=0.58, P(P)=0.39, and P(G and P)=0.06.
a) The probability a home has a pool or a garage: This is determined using the formula for the union of two events: P(G U P) = P(G) + P(P) - P(G and P) = 0.58+0.39-0.06 = 0.91 or 91% of homes for sale.
b) The probability a home has neither a pool nor a garage: This is the complement of the event in part a. So, P(Neither G nor P) = 1 - P(G U P) = 1 - 0.91 = 0.09 or 9% of homes for sale.
c) The probability a home has a pool but not a garage: This is determined using the formula for the difference of two events: P(P - G) = P(P) - P(G and P) = 0.39 - 0.06 = 0.33 or 33% of homes for sale.
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Which director made the Beatles films, A Hard Day's Night and Help? a. Blake Edwards b. Stanley Kubrick c. Richard Lester d. Mike Nichols
Answer: Option(c) Richard Lester is correct.
Step-by-step explanation:
Both the films were directed by Richard Lester.
A Hard Day's night was a scripted comic farce and its main focus on Beatlemania and the band's hectic touring lifestyle. It is a black and white movie.
Help! film also directed by Richard Lester. And this film was shot in various exotic locations. Help! was the first Beatles film that is filmed in colour.
Find dy/dx and d2y/dx2, and find the slope and concavity (if possible) at the given value of the parameter. (If an answer does not exist, enter DNE.) Parametric Equations Point x = t , y = 7t − 2 t = 9
By the chain rule,
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\mathrm dy}{\mathrm dt}\dfrac{\mathrm dt}{\mathrm dx}[/tex]
Then for all [tex]t[/tex] the first derivative has a value of 7.
By the product rule,
[tex]\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac{\mathrm dy}{\mathrm dt}\dfrac{\mathrm dt}{\mathrm dx}\right]=\dfrac{\mathrm d\left(\frac{\mathrm dy}{\mathrm dt}\right)}{\mathrm dx}\dfrac{\mathrm dt}{\mathrm dx}+\dfrac{\mathrm dy}{\mathrm dt}\dfrac{\mathrm d^2t}{\mathrm dx^2}[/tex]
but [tex]t=x\implies\dfrac{\mathrm dt}{\mathrm dx}=1\implies\dfrac{\mathrm d^2t}{\mathrm dx^2}=0[/tex], so we're left with
[tex]\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\mathrm d\left(\frac{\mathrm dy}{\mathrm dt}\right)}{\mathrm dx}[/tex]
By the chain rule,
[tex]\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\mathrm d\left(\frac{\mathrm dy}{\mathrm dt}\right)}{\mathrm dx}=\dfrac{\mathrm d\left(\frac{\mathrm dy}{\mathrm dt}\right)}{\mathrm dt}\dfrac{\mathrm dt}{\mathrm dx}=\dfrac{\mathrm d^2y}{\mathrm dt^2}[/tex]
but [tex]y=7t-2\implies\dfrac{\mathrm dy}{\mathrm dt}=7\implies\dfrac{\mathrm d^2y}{\mathrm dt^2}=0[/tex] so the second derivative is 0 for all [tex]t[/tex].
The derivative dy/dx for the given parametric equations is 7, and the second derivative d2y/dx2 is zero. The slope at t = 9 is 7, and the curve does not exhibit concavity since it is linear.
Explanation:To find dy/dx for the parametric equations x = t and y = 7t - 2, we need to compute the derivatives of both x and y with respect to t and then use the chain rule to find dy/dx as dy/dt divided by dx/dt. Since the derivative of x with respect to t is 1, and the derivative of y with respect to t is 7 (as the derivatives of the constants -2 and 1 are zero), dy/dx equals 7/1, which is 7. To find the second derivative d2y/dx2, we note that since dx/dt is constant (equals 1), the second derivative is zero. Therefore, the concavity of the curve does not change and is neither concave up nor down.
At the given value of the parameter t = 9, the slope of the tangent line is 7, as it is for all values of t. Since the second derivative is zero, the curve is linear and does not exhibit concavity at any point, including t = 9.
The weights of steers in a herd are distributed normally. The standard deviation is 300lbs and the mean steer weight is 1100lbs. Find the probability that the weight of a randomly selected steer is greater than 920lbs. Round your answer to four decimal places.
Answer: 0.7257
Step-by-step explanation:
Given : The weights of steers in a herd are distributed normally.
[tex]\mu= 1100\text{ lbs }[/tex]
Standard deviation : [tex]\sigma=300 \text{ lbs }[/tex]
Let x be the weight of the randomly selected steer .
Z-score : [tex]\dfrac{x-\mu}{\sigma}[/tex]
[tex]z=\dfrac{920-1100}{300}=-0.6[/tex]
The the probability that the weight of a randomly selected steer is greater than 920 lbs using standardized normal distribution table :
[tex]P(x>920)=P(z>-0.6)=1-P(z<-0.6)\\\\=1-0.2742531=0.7257469\approx0.7257[/tex]
Hence, the probability that the weight of a randomly selected steer is greater than 920lbs =0.7257
In a survey, 169 respondents say that they never use a credit card, 12527 say that they use it sometimes, and 2834 say that they use it frequently. Find the probability that a randomly selected person uses a credit card frequently (Round to 4 decimals) then take that result and decide whether it is Likely or Unlikely (ONLY TYPE IN THE WORD!) for someone to use a credit card frequently.
Answer:
Probability: 0.1825 or 18.25% ....... Unlikely
Step-by-step explanation:
Hello, great question. These types are questions are the beginning steps for learning more advanced Probability problems.
To start of we need to calculate the total amount of respondents that took the survey. We do this by adding all the answers together.
169 + 12,527 + 2,834 = 15,530 total people
Now that we know the total amount of people we can calculate the probability of each response by dividing the amount of people that had that response by the total amount of people that took the survey.
Never Used: [tex]\frac{169}{15,530} = 0.0109 = 1.09%[/tex]
Sometimes Used: [tex]\frac{12,527}{15,530} = 0.8066 = 80.66%[/tex]
Frequently used: [tex]\frac{2834}{15,530} = 0.1825 = 18.25%[/tex]
So we can see that the probability of a randomly selected person using a credit card frequently is 0.1825 or 18.25%
Unlikely
I hope this answered your question. If you have any more questions feel free to ask away at Brainly.
Final answer:
The probability that a randomly selected person uses a credit card frequently is calculated as 2834 divided by the total of 15530, resulting in 0.1825 (rounded to four decimal places). Since this is less than 20%, it is considered 'Unlikely' for someone to frequently use a credit card.
Explanation:
To calculate the probability that a randomly selected person uses a credit card frequently, we need to use the basic probability formula, which is the number of favorable outcomes divided by the total number of outcomes. In this case, the number of people who use a credit card frequently is the favorable outcome, and the total number of respondents is the sum of all categories of credit card usage.
Number of people who use a credit card frequently: 2834
Total number of respondents: 169 (never) + 12527 (sometimes) + 2834 (frequently) = 15530
Probability of frequent use: 2834 / 15530 = 0.1825 (rounded to 4 decimal places)
Now, let's interpret the result. A probability of 0.1825, when rounded, is about 18.25%. This number is less than 20%, which is generally considered the benchmark for something to be considered "likely". Therefore, it is Unlikely for someone to use a credit card frequently.
Critical Thinking and Statistics The average test score of the class was an 80 and the standard deviation was 2. About 68 % of the class would score between what 2 values? 80 and 82 78 and 82 76 and 84 74 and 86
Answer:
B. 78 and 82.
Step-by-step explanation:
We have been given that the average test score of the class was an 80 and the standard deviation was 2. We are asked to find two values between which 68% of class will score.
We know that in a normal distribution approximately 68% of the data falls within one standard deviation of the mean.
So 68% scores will lie within one standard deviation below and above mean that is:
[tex](\mu-\sigma,\mu+\sigma)[/tex]
Upon substituting our given values, we will get:
[tex](80-2,80+2)[/tex]
[tex](78,82)[/tex]
Therefore, about 68% of the class would score between 78 and 81 and option B is the correct choice.
.Given: F(x) = 3x2+ 1, G(x) = 2x - 3, H(x) = x
G-1(x) =
a. -2x + 3
b. (x + 3)/2
c. 2(x + 3)
F(x) + G(x) =
a. 3x^2 + 2x - 2
b. 5x^3 - 2
c. 3x^2 + 2x + 4
F(-2) =
a. -11
b. 13
c. 37
F(3) + G(4) - 2H(5) =
a. 13
b. 23
c. 33
For this case we have the following functions:
[tex]F (x) = 3x ^ 2 +1\\G (x) = 2x-3\\H (x) = x[/tex]
We have to:
[tex]G (x) * - 1[/tex] is given by:
[tex](2x-3) * - 1 = -2x +3[/tex]
Thus, the correct option is the option is A.
On the other hand,
[tex]F (x) +G (x) = 3x ^ 2 +1 +(2x - 3) = 3x ^ 2+ 1+ 2x-3 = 3x ^ 2 +2x-2[/tex]
Thus, the correct option is the option is A.
We also have:
[tex]F (-2) = 3 (-2) ^ 2+ 1 = 3 (4)+ 1 = 12+ 1 = 13[/tex]
Thus, the correct option is the option is B.
Last we have:
[tex]F (3)+G (4) -2H (5) = (3 (3) ^ 2+ 1)+ (2 (4) -3) -2 (5) = (3 (9)+ 1) - (8-3) 10 = 28+ 5-10 = 33[/tex]
Thus, the correct option is the option is C.
ANswer:
Option A, A, B, C
Answer:
G-1(x)= -2x+3
F(x)+G(x)=3x^2+2x-2
F(-2)=13
F(3)+G(4)-2H(5)=33
A,A,B,C are the answers to the equations
The claim is that the proportion of peas with yellow pods is equal to 0.25 (or 25%). The sample statistics from one experiment include 460 peas with 91 of them having yellow pods. Find the value of the test statistic. The value of the test statistic is nothing. (Round to two decimal places as needed.)
Answer: 2.68
Step-by-step explanation:
Claim : The proportion of peas with yellow pods is equal to 0.25
i.e. p=0.25
Sample size : [tex]460[/tex]
Proportion of peas with yellow pods in sample :
[tex]P=\dfrac{91}{460}=0.19782608695\approx0.20[/tex]
Now, the test statistic for the population proportion is given by :-
[tex]z=\dfrac{p-P}{\sqrt{\dfrac{P(1-P)}{n}}}[/tex]
[tex]\Rightarrow\ z=\dfrac{0.25-0.20}{\sqrt{\dfrac{0.20(1-0.20)}{460}}}\Rightarrow\ z=2.68095132369\approx2.68[/tex]
Hence, the value of the test statistic is 2.68
The test statistic for the given population proportion and sample data is approximately -2.57. This result was calculated using the Z test formula for testing population proportions, with a sample proportion of 0.1978, an expected proportion of 0.25, and a sample size of 460.
Explanation:To calculate the test statistic, we'll use the formula for Z: Z = (p' - p0) / sqrt[(p0(1 - p0)) / n]
p' is the sample proportion: 91/460 = 0.1978.p0 is the expected proportion: 0.25.n is the sample size: 460.Plugging these values into the formula, we get: Z = (0.1978 - 0.25) / sqrt[(0.25 * 0.75) / 460] ≈ -2.57
The estimated proportion p' in this formula represents the proportion of peas with yellow pods within our sample. We use this, along with the hypothesized proportion p0 and the sample size n, to calculate the test statistic, which gives us an idea of how far our observed data is from the expected hypothesis.
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Y= x - 2; How do you get the x and y axis and how do you plot it? So my teacher is confusing me and if ya'll could help i'd appreciate it.
Step-by-step explanation:
hi I have answered ur question
Divide the following polynomials:
a) x^3-9/x^2 +1
b) x^5-13x^4-120x+80/x+3
I don't understand long division of polynominals. Ive looked up videos. It still doesn't make sense. Can you please show work?
Answer:
Step-by-step explanation:
See if the attachment below helps you with this.
a. draw the correct graph that identifies the feasible region for the following set of constraints.
2 A + 0.5 B ≥ 90
0.75 A + 5 B ≥ 200
0.75 A + 1.5 B ≤ 150
A, B ≥ 0
Answer:
Step-by-step explanation:
Assuming A as x axis and B as y axis the equations are
[tex]2x+0.5y\geq 90\quad \left(1\right)\\0.75x+5y\geq 200\quad \left(2\right)\\0.75x+1.5y\leq 150\quad \left(3\right)\\x,y\geq 0\quad \left(4\right)[/tex]
Solving equations (2) and (3) we get
x=171.429 y=14.286
Solving equations (1) and (3) we get
x=22.857 y=88.571
Solving equations (1) and (2) we get
x=36.364 y=34.545
The area enclosing the above three points is the feasible region.
Find the range, mean, and standard deviation of the data set (3, 6, 9, 11, 15, 16) The range is 13 (Type a whole number.) The mean is 10 (Type an integer or decimal.) The standard deviation is (Do not round until the final-answer. Then round to two decimal places as needed.) Enter your answer in the answer box and then click Check Answer All parts showing
Answer: The range is 13. The mean is 10. The standard deviation is 4.62 .
Step-by-step explanation:
The given data : 3, 6, 9, 11, 15, 16
Total number of data values : n = 6
The mean of data is given by :-
[tex]\overline{x}=\dfrac{\sum^6_{i=1}x_i}{n}\\\\\Rightarrow\overline{x}=\dfrac{60}{6}=10[/tex]
The standard deviation is given by :-
[tex]\sqrt{\dfrac{1}{n}(\sum^6_{i=1}(x_i-\overline{x})^2)}\\\\=\sqrt{\dfrac{1}{6}(\sum^6_{i=1}(x_i-10)^2)}\\\\=\sqrt{\dfrac{1}{6}\times(49+16+1+1+25+36)}=4.61880215352\approx4.62[/tex]
The range of the data : Maximum value -Minimum value
[tex]=16-3=13[/tex]
A researcher wishes to estimate the proportion of adults who have high-speed Internet access. What size sample should be obtained if she wishes the estimate to be within 0.04 with 90% confidence if (a) she uses a previous estimate of 0.38? (b) she does not use any prior estimates?
Answer:a-396
b-420
Step-by-step explanation:
[tex]\alpha[/tex] =0.1
Margin of Error=0.04
Level of significance is z[tex]\left ( 0.1\right )=1.64[/tex]
Previous estimate[tex]\left ( p\right ) =0.38[/tex]
sample size is given by:
n=[tex]\left (\frac{Z_{\frac{\alpha }{2}}}{E}\right )p\left ( 1-p\right )[/tex]
n=[tex]\frac{1.64}{0.04}^{2}0.38\left ( 1-0.38\right )=396.0436\approx 396[/tex]
[tex]\left ( b\right )[/tex]Does not use prior estimate
Assume
[tex]\alpha [/tex]=0.1
Margin of Error=0.04
Level of significance is z[tex]\left ( 0.1\right )=1.64[/tex]
Population proportion[tex]\left ( p\right )[/tex]=0.5
n=[tex]\left (\frac{Z_{\frac{\alpha }{2}}}{E}\right )p\left ( 1-p\right )[/tex]
n=[tex]\frac{1.64}{0.04}^{2}0.5\left ( 1-0.5\right )[/tex]
n=420.25[tex]\approx 420[/tex]
Last month a store sold 4 times as many tapes as CDs. If the total quantity of these two items sold was 360, how many tapes did the store sell? a. 288 b. 360 c. 90 d. 20
Answer: The store sold 288 tapes last month.
Step-by-step explanation:
Let the number of CDs be x , then the number of tapes is given by the expression : 4x
Also, the total quantity of these two items sold was 360.
Now, we have the following equation :-
[tex]x+4x=360\\\\\Rightarrow\ 5x=360\\\\\Rightarrow\ x=\dfrac{360}{5}\\\\\Rightarrow\ x=72[/tex]
The number of CDs sold in last month = 72
The number of tapes sold in last month =[tex]4\times72=288[/tex]
Hence, the store sold 288 tapes last month.
A community athletic club holds an election to select a president and vice president. The nominations for selection include 4 females and 3 males.
What is the probability that a female is elected president and a male is elected vice president?
Answer:
2/7
Step-by-step explanation:
For the president position, there are 4 females from a total of 7 people.
For the vice president position, there are 3 males from 6 people left over.
So the probability is 4/7 × 3/6 = 2/7.
Answer: [tex]\dfrac{2}{7}[/tex]
Step-by-step explanation:
Given : Number of males = 3
Number of females = 4
The number of ways to select a female is elected president and a male is elected vice president :-
[tex]^3P_1\times ^4C_1=\dfrac{3!}{(3-1)!}\times\dfrac{4!}{(4-1)!}=3\times4=12[/tex]
The total number of ways to select 2 people from 7 :_
[tex]^7P_2=\dfrac{7!}{(7-2)!}=42[/tex]
Now, the probability that a female is elected president and a male is elected vice president will be :-
[tex]\dfrac{12}{42}=\dfrac{2}{7}[/tex]
describe two reason why cadaver stories are so important to students.
Answer:
1) So the students do not make the same mistakes
2) So the students can see the importance of their jobs, to save lives
Pediatricians work an average of 48 h per week. The standard deviation is 12 hours. What percentage of pediatricians work more than 72 h per week
Approximately 2.28% of pediatricians work more than 72 hours per week according to the table.
Explanation:To find the percentage of pediatricians who work more than 72 hours per week, we need to calculate the z-score for this value and then use a standard normal distribution table to find the corresponding percentage.
Calculate the z-score using the formula:
[tex]z = (x - u) / \alpha[/tex]
where x is the value (72 hours), u is the mean (48 hours), and a is the standard deviation (12 hours).
Substitute the values into the formula: z = (72 - 48) / 12 = 2.
Using a standard normal distribution table, find the percentage of values that are greater than 2.
Based on the table, approximately 2.28% of pediatricians work more than 72 hours per week.
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solve x^3+6x=2 using cardano's method
Answer:
The solution of the given equation is [tex]\sqrt[3]{4}-\sqrt[3]{2}[/tex].
Step-by-step explanation:
According to the cardano's method, the solution of the equation is x=u-v. If the equation is
[tex]x^3+px=q[/tex]
Where [tex]u^3-v^3=q[/tex]
[tex]3uv=p[/tex]
The given equation is
[tex]x^3+6x=2[/tex]
Here p=6 and q=2.
[tex]u^3-v^3=2[/tex] .... (1)
[tex]3uv=6[/tex]
[tex]uv=2[/tex]
Taking cube both the sides.
[tex]u^3v^3=8[/tex]
Multiply both sides by 4.
[tex]4u^3v^3=32[/tex] .... (2)
Taking square both the sides of equation (1).
[tex](u^3-v^3)^2=2^2[/tex]
[tex](u^3)^2-2u^3v^3+(v^3)^2=4[/tex] .... (3)
Add equation (2) and (3).
[tex](u^3)^2-2u^3v^3+(v^3)^2+4u^3v^3=4+32[/tex]
[tex](u^3+v^3)^2=36[/tex]
Taking square root both the sides.
[tex]u^3+v^3=6[/tex] .... (4)
On adding equation (1) and (4), we get
[tex]2u^3=8[/tex]
[tex]u^3=4[/tex]
[tex]u=\sqrt[3]{4}[/tex]
On subtracting equation (1) and (4), we get
[tex]-2v^3=-4[/tex]
[tex]v^3=2[/tex]
[tex]v=\sqrt[3]{2}[/tex]
The solution of the equation is
[tex]x=u-v=\sqrt[3]{4}-\sqrt[3]{2}[/tex]
Therefore the solution of the given equation is [tex]\sqrt[3]{4}-\sqrt[3]{2}[/tex].
Final answer:
Solving x³+6x=2 using Cardano's method involves rewriting the equation to match the standard form of a depressed cubic equation, calculating the required constants, and finally, applying these constants to find the roots.
Explanation:First, let's rewrite the equation x³+6x-2 = 0 as x³+6x = 2 to match the standard form of a depressed cubic equation which is x³ +px = q. Here, p = 6 and q = 2.
Next, we calculate the value t = sqrt[(q/2)² + (p/3)³]. So, t = sqrt[(1)² + (2)³] = sqrt[1 + 8] = 3.
Using these values, we can now calculate the roots. We know the roots are given by the formulaes u-v where u = cubicroot(q/2 + t) and v = cubicroot(q/2 - t). So, u = cubicroot(1 + 3) = 2, and v = cubicroot(1 - 3) = - root(2).
Therefore, the roots of the given polynomial equation are x = u - v = 2 - (- root(2)) = 2 + root(2).
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The claim is that the proportion of peas with yellow pods is equal to 0.25 (or 25%). The sample statistics from one experiment include 590 peas with 139 of them having yellow pods. Find the value of the test statistic.
Final answer:
The test statistic for the proportion of yellow pea pods being 25% is calculated using the sample proportion, hypothesized proportion, and sample size. Using the given data, the test statistic (Z-score) comes out to approximately -1.412.
Explanation:
To find the value of the test statistic for the claim that the proportion of peas with yellow pods is equal to 0.25 (25%), we use the sample statistics provided from the experiment. You mentioned that there were 590 peas in total, with 139 having yellow pods. First, we check if the conditions for the binomial distribution are met, which in this case they are as we are dealing with two outcomes (yellow pods and not yellow pods), a fixed number of trials (590 peas), and each pea is independent of the others.
The test statistic for a proportion is calculated using the formula:
Z = (p' - p) / (sqrt(p(1 - p) / n))
Where:
p' is the sample proportion (139 / 590 = 0.2356)p is the hypothesized population proportion (0.25)n is the sample size (590)Now, we calculate the test statistic:
Z = (0.2356 - 0.25) / (sqrt(0.25 × (1 - 0.25) / 590))
Z ≈ -1.412
This Z-score tells us how many standard deviations the observed sample proportion (0.2356) is from the hypothesized proportion (0.25).
Sugar and salt are both white, crystalline powders that dissolve in water. If you were given an unknown sample that contained one or both of these solids, how could you determine what your unknown sample contained
Answer:
Step-by-step explanation:
Sugar and Salt even thought they both dissolve in water they both dissolve in different ways. When salt dissolves in water, its individual types of ions are torn apart from each other, while Sugar molecules stay together when dissolved in water, and therefore the molecules remain the same when dissolved in water.
This being said in science using your senses can be just as valuable as using calculations. In this case both Sugar and Salt taste differently. Sugar is sweet while Salt is salty. Therefore tasting the substance can be the easiest and most accurate way of determining the substance.
I hope this answered your question. If you have any more questions feel free to ask away at Brainly.
Final answer:
To determine if an unknown sample contains sugar, salt, or both, examine their solubility properties and chemical reactivity. Comparing the density of the sample against reference values of pure substances can provide preliminary identification, while testing for chloride ions with silver nitrate confirms the presence of salt.
Explanation:
Identifying Sugar and Salt in an Unknown Sample
To determine whether an unknown sample contains sugar, salt, or both, we must identify the physical and chemical properties that distinguish these substances. Salt (sodium chloride) and sugar (sucrose) have distinct solubility properties and chemical reactivity, which we can use to identify them when dissolved in water.
Solubility and Density Test
Both sugar and salt are highly soluble in water, but we can compare their densities to make a preliminary identification. A known volume of each substance is weighed and their densities calculated. Salt generally has a greater density than sugar. If the unknown sample has a certain mass, comparing it with the reference densities may provide an initial indication.
Chemical Reactivity Test
To confirm the identity of the substances, a chemical reactant such as silver nitrate can be introduced to the water solution of the unknown sample. If a white precipitate forms, it indicates the presence of chloride ions, which suggests the presence of salt. Since sugar does not produce a precipitate with silver nitrate, its absence would indirectly indicate the presence of sugar.
Performing these tests will allow us to determine if the unknown sample is sugar, salt, or a mixture of both. The greater the discrepancy between the calculated density and the known densities of pure sugar or pure salt, the more likely it is that the sample is a mixture.
Buses headed to Longmont arrive in downtown Boulder every 30 minutes starting at 8:37am, whereas buses heading to Denver arrive 15 minutes starting at 8:31am. (a) If a passenger arrives at the station uniformly between 8:30am and 9:30am and then gets onto the first bus that arrives, what is the probability she goes to Longmont? (b) What is the passenger arrives uniformly between 8:45am and 9:45am?
Answer:
(a) 20% to Longmont; 80% to Denver
(b) 20% to Longmont; 80% to Denver
Step-by-step explanation:
(a) The bus to Longmont is the first bus to arrive, only between 8:31 and 8:37, and again between 9:01 and 9:07. That is, for a total of 12 minutes every hour, the Longmont bus is the first to arrive. The probability of going to Longmont is 12/60 = 1/5 = 20%.
__
(b) Same as for (a). As long as passenger arrival times are uniform within an hour, the probability is the same.
The wildlife department has been feeding a special food to rainbow trout fingerlings in a pond. Based on a large number of observations, the distribution of trout weights is normally distributed with a mean of 402.7 grams and a standard deviation of 8.8 grams. What is the probability that the mean weight for a sample of 40 trout exceeds 405.5 grams?
Answer: 0.0222
Step-by-step explanation:
Given : The distribution of trout weights is normally distributed with
Mean : [tex]\mu=402.7\text{ grams}[/tex]
Standard deviation : [tex]\sigma=8.8\text{ grams}[/tex]
Sample size : [tex]n=40[/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 weight of randomly selected trout.
Then for x = 405.5 , we have
[tex]z=\dfrac{405.5 -402.7}{\dfrac{8.8}{\sqrt{40}}}\approx2.01[/tex]
The p-value : [tex]P(405.5<x)=P(2.01<z)[/tex]
[tex]1-P(2.01)=1-0.9777844=0.0222156\approx0.0222[/tex]
Thus,the probability that the mean weight for a sample of 40 trout exceeds 405.5 grams= 0.0222.
The probability that the mean weight for a sample of 40 trout exceeds 405.5 grams is 0.0222.
The distribution of trout weights is normally distributed with
We have given that
Mean=402.7 grams
Standard deviation =8.8 grams
Sample size (n)=40
We have to calculate
The probability that the mean weight for a sample of 40 trout exceeds 405.5 grams
What is the to calculate the z-score?Te formula of Z score is given by,
[tex]z=\frac{x-\mu }{\frac{\sigma }{\sqrt{n}}}[/tex]
n= the ample size
x=mean
sigma=standard deviation
So by using the formula we have,
Let x is the weight of randomly selected trout.
Then for x = 405.5
[tex]z=\frac{405.5-\402.7}{\frac{\8.8 }{\sqrt{40}}}\\\\\z=2.01[/tex]
we have
The p-value :(405.5<x)
(1-2.01)=1-0.97778
=0.02221
=0.0222
Therefore,the probability that the mean weight for a sample of 40 trout exceeds 405.5 grams= 0.0222.
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The formula P = 0.672x^2 - 0.046x+ 3 models the approximate population P, in thousands, for a species of frogs in a particular rain forest, x years after 1999. During what year will the population reach 182 frogs? a) 2015 b) 2018 c) 2017 d) 2016 e) none
Answer:
The correct option is d.
Step-by-step explanation:
The approximate population P, in thousands, for a species of frogs in a particular rain forest, x years after 1999 is given by the formula
[tex]P=0.672x^2-0.046x+3[/tex]
We need to find the year it which the population reach 182 frogs.
Substitute P=182 in the given formula.
[tex]182=0.672x^2-0.046x+3[/tex]
Subtract 182 from both the sides.
[tex]0=0.672x^2-0.046x+3-182[/tex]
[tex]0=0.672x^2-0.046x-179[/tex]
Multiply both sides by 1000 to remove decimals.
[tex]0=672x^2-46x-179000[/tex]
Quadratic formula:
[tex]x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
Substitute a=672, b=-46 and c=-179000 in the quadratic formula.
[tex]x=\frac{-\left(-46\right)\pm\sqrt{\left(-46\right)^2-4\cdot \:672\left(-179000\right)}}{2\cdot \:672}[/tex]
On simplification we get
[tex]x=\frac{-\left(-46\right)+\sqrt{\left(-46\right)^2-4\cdot \:672\left(-179000\right)}}{2\cdot \:672}\approx 16.355[/tex]
[tex]x=\frac{-\left(-46\right)-\sqrt{\left(-46\right)^2-4\cdot \:672\left(-179000\right)}}{2\cdot \:672}\approx -16.287[/tex]
The value of x can not be negative because x is number of years after 1999.
x=16.35 in means is 17th year after 1999 the population reach 182 frogs.
[tex]1999+17=2016[/tex]
The population reach 182 frogs in 2016. Therefore the correct option is d.
Consider two functions f and g on [1, 8] such that integral^8_1 f(x) dx = 9, integral^8_1 g(x) dx = 5, integral^8_5 f(x) dx = 4, and integral^5_1 g (x) dx = 3. Evaluate the following integrals. a. integral^5_1 2f(x) dx = (Simplify your answer.) b. integral^8_1 (f(x) - g (x)) dx = (Simplify your answer.) c. integral^5_1 (f (x) - g (x)) dx = (Simplify your answer.) d. integral^8_5 (g(x) - f(x)) dx = (Simplify your answer.) e. integral^8_5 7g(x) dx = (Simplify your answer.) f. integral^1_5 3f(x) dx = (Simplify your answer.)
I'll abbreviate the definite integral with the notation,
[tex]I(f(x),a,b)=\int_a^bf(x)\,\mathrm dx[/tex]
We're given
[tex]I(f,1,8)=9[/tex][tex]I(g,1,8)=5[/tex][tex]I(f,5,8)=4[/tex][tex]I(g,1,5)=3[/tex]Recall that the definite integral is additive on the interval [tex][a,b][/tex], meaning for some [tex]c\in[a,b][/tex] we have
[tex]I(f,a,b)=I(f,a,c)+I(f,c,b)[/tex]
The definite integral is also linear in the sense that
[tex]I(kf+\ell g,a,b)=kIf(a,b)+\ell I(g,a,b)[/tex]
for some constant scalars [tex]k,\ell[/tex].
Also, if [tex]a\ge b[/tex], then
[tex]I(f,a,b)=-I(f,b,a)[/tex]
a. [tex]I(2f,1,5)=2I(f,1,5)=2(I(f,1,8)-I(f,5,8))=2(9-4)=\boxed{10}[/tex]
b. [tex]I(f-g,1,8)=I(f,1,8)-I(g,1,8)=9-5=\boxed{4}[/tex]
c. [tex]I(f-g,1,5)=I(f,1,5)-I(g,1,5)=\dfrac{I(2f,1,5)}2-I(g,1,5)=10-3=\boxed{7}[/tex]
d. [tex]I(g-f,5,8)=I(g,5,8)-I(f,5,8)=(I(g,1,8)-I(g,1,5))-I(f,5,8)=(5-3)-4=\boxed{-2}[/tex]
e. [tex]I(7g,5,8)=7I(g,5,8)=7(5-3)=\boxed{14}[/tex]
f. [tex]I(3f,5,1)=3I(f,5,1)=-3I(f,1,5)=-\dfrac32I(2f,1,5)=-\dfrac32(10)=\boxed{-15}[/tex]
In this integral calculus problem, we leverage properties of definite integrals to compute the values of various expressions. Key steps usually involve substituting given integral values and multiplying by constant factors when required
Explanation:To solve the problem, we first need to consider the properties of integral calculus, specifically those of definite integrals. A fundamental rule that is applicable here is that the product of a constant and an integral is the constant times the value of the integral.
So for problem a, integral^5_1 2f(x) dx = 2* integral^5_1 f(x) dx = 2 * 5 = 10.
Similarly, for problem b, integral^8_1 (f(x) - g (x)) dx = integral^8_1 f(x) dx - integral^8_1 g(x) dx = 9 - 5 = 4.
Following through similar steps of substitutions, we obtain the following solutions:
c. integral^5_1 (f (x) - g (x)) dx = 1d. integral^8_5 (g(x) - f(x)) dx = 1e. integral^8_5 7g(x) dx = 21 f. integral^1_5 3f(x) dx = 15Learn more about Integral Calculus here:https://brainly.com/question/34730103
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Consider the equation below. (If an answer does not exist, enter DNE.) f(x) = 7 cos2(x) − 14 sin(x), 0 ≤ x ≤ 2π (a) Find the interval on which f is increasing. (Enter your answer using interval notation.)
Answer:
[tex][\frac{\pi}{2},\frac{3\pi}{2}][/tex]
Step-by-step explanation:
Let me first state that I am assuming your function is
[tex]f(x)=7cos^2(x)-14sin(x)[/tex]
If this is incorrect, then disregard this whole answer/explanation.
In order to find where the function is increasing or decreasing, we need to first find the first derivative, set it equal to 0, and then factor to find the values that cause the derivative to equal 0. This is where you expect to find a max or a min value in the function itself. But this function is not going to be easily solved for 0 once we find the derivative unless we make it in terms of either sin or cos right now, before taking the first derivative.
Let [tex]cos^2(x)=1-sin^2(x)[/tex]
This is a Pythagorean trig identity, and I'm assuming that if you're in calculus solving for the intervals of increasing and decreasing values that you have, at one time, used trig identities.
Rewriting:
[tex]f(x)=7(1-sin^2(x))-14sin(x)[/tex] which simplifies to
[tex]f(x)=7-7sin^2(x)-14sin(x)[/tex] and in order of descending values of x:
[tex]f(x)=-7sin^2(x)-14sin(x)+7[/tex]
Now we can find the derivative. For the first term, let u = sin(x), therefore,
[tex]f(u)=u^2[/tex], u' = cos(x), and f'(u) = 2u. The derivative is found by multiplying f'(u) by u', which comes out to 2sin(x)cos(x)
The derivative for the next 2 terms are simple, so the derivative of the function is
[tex]f'(x)=-7[2sin(x)cos(x)]-14cos(x)[/tex] which simplifies down to
[tex]f'(x)=-14sin(x)cos(x)-14cos(x)[/tex]
We will set that equal to zero and solve for the values that cause that derivative to equal 0. But first we can simplify it a bit. You can factor out a -14cos(x):
[tex]f'(x)=-14cos(x)(sin(x)+1)[/tex]
By the Zero Product Property, either
-14cos(x) = 0 or sin(x) + 1 = 0
Solving the first one for cos(x):
cos(x) = 0
Solving the second one for sin(x):
sin(x) = -1
We now look to the unit circle to see where, exactly the cos(x) = 0. Those values are
[tex]\frac{\pi}{2},\frac{3\pi}{2}[/tex]
The value where the sin is -1 is found at
[tex]\frac{3\pi}{2}[/tex]
We set up a table (at least that's what I advise my students to do!), separating the intervals in ascending order, starting at 0 and ending at 2pi.
Those intervals are
0 < x < [tex]\frac{\pi}{2}[/tex], [tex]\frac{\pi}{2}<x<\frac{3\pi}{2}[/tex], and [tex]\frac{3\pi}{2}<x<2\pi[/tex]
Now pick a value that falls within each interval and evaluate the derivative at that value and determine the sign (+ or -) that results. You don't care what the value is, only the sign that it carries. For the first interval I chose
[tex]f'(\frac{\pi}{4})=-[/tex] so the function is decreasing here (not what you wanted, so let's move on to the next interval).
For the next interval I chose:
[tex]f'(\pi)=+[/tex] so the function is increasing here.
For the last interval I chose:
[tex]f'(\frac{7\pi}{4})=-[/tex]
It appears that the only place this function is increasing is on the interval
[tex][\frac{\pi}{2},\frac{3\pi}{2}][/tex]
Final answer:
The interval on which the function f(x) = 7cos^2(x) - 14sin(x) is increasing is DNE.
Explanation:
To find the interval on which the function f(x) = 7cos^2(x) - 14sin(x) is increasing, we need to determine where the derivative of the function is positive. The derivative of f(x) can be found using the chain rule, which gives us f'(x) = -14cos(x) - 28sin(x)cos(x). To find where f'(x) > 0, we need to solve the inequality -14cos(x) - 28sin(x)cos(x) > 0.
We can simplify this inequality to cos(x)(-14 - 28sin(x)) > 0. Since cos(x) is positive on the interval 0 ≤ x ≤ 2π and -14 - 28sin(x) is negative on the interval 0 ≤ x ≤ 2π, the product of these two terms will be negative. Therefore, there are no values of x on the interval 0 ≤ x ≤ 2π where f(x) is increasing.
Evaluate the given integral by changing to polar coordinates. sin(x2 + y2) dA R , where R is the region in the first quadrant between the circles with center the origin and radii 2 and 3
Set
[tex]\begin{cases}x=r\cos\theta\\y=r\sin\theta\end{cases}\implies\mathrm dA=r\,\mathrm dr\,\mathrm d\theta[/tex]
The region [tex]R[/tex] is given in polar coordinates by the set
[tex]R=\left\{(r,\theta)\mid2\le r\le3,0\le\theta\le\dfrac\pi2\right\}[/tex]
So we have
[tex]\displaystyle\iint_R\sin(x^2+y^2)\,\mathrm dA=\int_0^{\pi/2}\int_2^3r\sin(r^2)\,\mathrm dr\,\mathrm d\theta=\boxed{\frac\pi4(\cos4-\cos9)}[/tex]
The Cartesian coordinates are converted into polar coordinates so that the integral sin(x2 + y2) dA R becomes the integral sin(r2) r dr dθ. But this specific integral can't be solved analytically, yet using polar coordinates can simplify other integration issues related to circular regions or distances from the origin.
Explanation:To evaluate the given integral using polar coordinates, one must firstly translate the Cartesian coordinates (x,y) into polar coordinates (r,θ), so that x is replaced with rcosθ and y with rsinθ. Consequently, the integral sin(x2 + y2) dA R becomes the integral sin(r2) r dr dθ, with r varying from 2 to 3, and θ from 0 to π/2 (since we are only dealing with the first quadrant).
However, this integral becomes very complex and is not feasible to solve analytically. You would need to use a numeric method to get an approximate answer. In the context of other problems, switching to polar coordinates can simplify the integration, especially when dealing with circular regions or equations related to distances from the origin.
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solve the system of equation by guess jordan method
2x1-6x2-2x3=14, 3x1+4x2-7x3= 16, 3x1-6x2+9x3=21
Answer with explanation:
The System of equations which we have to solve by Gauss Jordan Method:
[tex]1.\rightarrow 2x_{1}-6x_{2}-2x_{3}=14, 2.\rightarrow 3x_{1}+4x_{2}-7x_{3}= 16, 3.\rightarrow 3x_{1}-6x_{2}+9x_{3}=21[/tex]
Writing it in the form of Augmented Matrix=3 Rows and 4 Columns:
[tex]\left[\begin{array}{cccc}2&-6&-2&14\\3&4&-7&16\\3&-6&9&21\end{array}\right]\\\\R_{1}=\frac{R_{1}}{2},R_{3}=\frac{R_{3}}{3}\\\\ \left[\begin{array}{cccc}1&-3&-1&7\\3&4&-7&16\\1&-2&3&7\end{array}\right]\\\\R_{3}\rightarrow R_{3}-R_{1}\\\\\left[\begin{array}{cccc}1&-3&-1&7\\3&4&-7&16\\0&1&4&0\end{array}\right]\\\\R_{2}\rightarrow R_{2}-3R_{1}\\\\\left[\begin{array}{cccc}1&-3&-1&7\\0&13&-4&-5\\0&1&4&0\end{array}\right][/tex]
[tex]R_{3}\rightarrow R_{2}+R_{3}\\\\\left[\begin{array}{cccc}1&-3&-1&7\\0&13&-4&-5\\0&14&0&-5\end{array}\right]\\\\\rightarrow14 x_{2}= -5\\\\x_{2}=\frac{-5}{14}\\\\\rightarrow 13 x_{2}-4x_{3}=-5\\\\ \frac{-65}{14}-4 x_{3}=-5\\\\-4x_{3}=-5+\frac{65}{14}\\\\x_{3}=\frac{5}{56}\\\\x_{1}-3x_{2}-x_{3}=7\\\\x_{1}+\frac{15}{14}-\frac{5}{56}=7\\\\x_{1}+\frac{55}{56}=7\\\\x_{1}=7-\frac{55}{56}\\\\x_{1}=\frac{337}{56}[/tex]
Solution set
[tex]=(\frac{337}{56},\frac{-5}{14},\frac{5}{56})[/tex]
Suppose that Jason recently landed job offers at two companies. Company A reports an average salary of $51,500 with a standard deviation of $2,175. Company B reports an average salary of $46,820 with a standard deviation of $5,920. Assume that salaries at each company are normally distributed. Jason's goal is to secure a position that pays $55,000 per year. What are the ???? ‑scores for Jason's desired salary at Company A and Company B? Please round your answers to two decimal places.
Answer:
company B
Step-by-step explanation:
average salary of Company A(μ) = $51,500
standard deviation of Company A (σ)= $2,175.
average salary of Company B(μ) = $46,820
standard deviation of Company B(σ) =$5,920
desired salary(x) = $55,000
z-score for company A = [tex]\dfrac{x-\mu}{\sigma}[/tex]
= [tex]\dfrac{55000-51500}{ 2175} = 1.61[/tex]
z-score for company A = [tex]\dfrac{x-\mu}{\sigma}[/tex]
= [tex]\dfrac{55000-46820}{ 5920} = 1.38[/tex]
higher the value of z less chances of getting the desired salary hence company B has value of z is less so, the chances of getting desired salary is more in company B.
Jason has to calculate z-score to compare his desired salary with the average salaries at two different companies. The z-score for Company A is 1.61 and for Company B is 1.38. Hence, the desired wage of $55,000 is 1.61 and 1.38 standard deviations away from the mean salaries at Company A and Company B, respectively.
Explanation:The z-score is a measure of how many standard deviations an observation or datum is from the mean. To calculate Jason's z-score at each company, we would subtract the mean salary at that company from $55,000 and then divide by the standard deviation for that company.
For Company A: [tex]Z_A = ($55,000 - $51,500) / $2,175 = 1.61.[/tex]
For Company B: [tex]Z_B = ($55,000 - $46,820) / $5,920 = 1.38.[/tex]
Therefore, Jason's desired salary of $55,000 is 1.61 standard deviations away from the mean at Company A and 1.38 standard deviations away from the mean at Company B.
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A pollster wants to construct a 95% confidence interval for the proportion of adults who believe that economic conditions are getting better. A Gallup poll taken in July 2010 estimates this proportion to be 0.33. Using this estimate, what sample size is needed so that the confidence interval will have a margin of error of 0.052 ?
To construct a 95% confidence interval with a margin of error of 0.052, a sample size of 300 is needed.
Explanation:To determine the sample size needed for the 95% confidence interval with a margin of error of 0.052, we can use the formula:
n = (Z^2 * p * (1 - p)) / (E^2)
where n is the sample size, Z is the Z-score corresponding to the desired confidence level (in this case, 1.96), p is the estimated proportion (0.33), and E is the margin of error (0.052).
Substituting the given values into the formula:
n = (1.96^2 * 0.33 * (1 - 0.33)) / (0.052^2)
Simplifying the equation:
n = 299.5554
Rounding up to the nearest whole number, the sample size needed is 300.
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To construct a 95% confidence interval for the proportion of adults who believe economic conditions are getting better with a margin of error of 0.052, the required sample size is approximately 577 participants.
To calculate the sample size needed to construct a 95% confidence interval for the proportion of adults who believe that economic conditions are getting better, with a Gallup poll estimate of 0.33 and a desired margin of error of 0.052, the formula for determining the sample size (n) is:
n = (Z^2*p*(1-p))/E^2,
where Z is the Z-score corresponding to the 95% confidence level, p is the estimated proportion (0.33) and E is the margin of error (0.052). The Z-score for a 95% confidence level is 1.96. Plugging in the values gives:
n = (1.96^2*0.33*(1-0.33))/(0.052^2),
Solving this, we find the sample size required:
n= 576.7.
Since we cannot have a fraction of a person, we round up to the next whole number. Therefore, the sample size needed is 577 participants.