Answer:
Using either method, we obtain: [tex]t^\frac{3}{8}[/tex]
Step-by-step explanation:
a) By evaluating the integral:
[tex]\frac{d}{dt} \int\limits^t_0 {\sqrt[8]{u^3} } \, du[/tex]
The integral itself can be evaluated by writing the root and exponent of the variable u as: [tex]\sqrt[8]{u^3} =u^{\frac{3}{8}[/tex]
Then, an antiderivative of this is: [tex]\frac{8}{11} u^\frac{3+8}{8} =\frac{8}{11} u^\frac{11}{8}[/tex]
which evaluated between the limits of integration gives:
[tex]\frac{8}{11} t^\frac{11}{8}-\frac{8}{11} 0^\frac{11}{8}=\frac{8}{11} t^\frac{11}{8}[/tex]
and now the derivative of this expression with respect to "t" is:
[tex]\frac{d}{dt} (\frac{8}{11} t^\frac{11}{8})=\frac{8}{11}\,*\,\frac{11}{8}\,t^\frac{3}{8}=t^\frac{3}{8}[/tex]
b) by differentiating the integral directly: We use Part 1 of the Fundamental Theorem of Calculus which states:
"If f is continuous on [a,b] then
[tex]g(x)=\int\limits^x_a {f(t)} \, dt[/tex]
is continuous on [a,b], differentiable on (a,b) and [tex]g'(x)=f(x)[/tex]
Since this this function [tex]u^{\frac{3}{8}[/tex] is continuous starting at zero, and differentiable on values larger than zero, then we can apply the theorem. That means:
[tex]\frac{d}{dt} \int\limits^t_0 {u^\frac{3}{8} } } \, du=t^\frac{3}{8}[/tex]
PLEASE HELP I WILL GIVE YOU BRAINLIEST
Your friend earns $10.50 per hour.This is 125% of her hourly wage last year. How much did your friend earn per hour last year?
Let your friend's hourly wage last year be $x.
125% of x = 10.50
[tex]\frac{125}{100}[/tex] x = 10.5
[tex]\frac{5}{4}[/tex] x = 10.5
5x = 42
x = $8.40
Your friend's hourly wage last year was $8.40.
I hope this helps! If you have any doubts, feel free to ask.
Answer:
it is $8.40
Step-by-step explanation:
sorry my answer got deleted :(
identify and interpret (explain) one other point on the graph.?
Answer:
can i getttt a pictture
Step-by-step explanation:
Find two positive numbers such that the ratio of the two numbers is 7 to 3 and the product of the two numbers is 525.
Answer:
35 and 15
Step-by-step explanation:
Process of elimination:
Try 105 and 5 - 105 is way too much and 5 is way too little. We know that they are between 5 and 105.
Try 50 and 10.5 - This is closer, but 50 is still too much and 10.5 is too little. We now know that the two numbers are between 10.5 and 50
Try 30 and 17.5 - This is very close, but now 30 is not enough and 17.5 is too much. We now know that the smaller number is between 10.5 and 17.5 as well as the larger number is between 30 and 50.
Try 40 and 13.125 - This is also very close, but 40 is too much and 13.125 is too little. We now know that the lower number is between 13.125 and 17.5 as well as the larger number is between 30 and 40.
Try 35 and 15 - Success: 35 and 5 make the simplified ratio of 7:3! They also are products of 525!
Final answer:
The two positive numbers with a ratio of 7 to 3 and a product of 525 are found to be 35 and 15, by assigning a constant to the numbers, forming and solving a quadratic equation to find the value of the constant, and then using it to find the two numbers.
Explanation:
To find two positive numbers where their ratio is 7 to 3, and their product is 525, we can assign values x and y to the numbers such that x/y = 7/3, and x*y = 525. We can solve these equations simultaneously to find the values of x and y.
Step-by-step solution:
Let the two numbers be 7k and 3k, where k is a constant. This ensures that the ratio 7 to 3 is preserved.Since the product of the two numbers is 525, we have that 7k * 3k = 525.Combining like terms, we get 21k^2 = 525.Dividing both sides by 21 gives k^2 = 25.Taking the square root of both sides, we find that k = 5.Substitute k = 5 back into 7k and 3k to find the two numbers: 7*5 = 35 and 3*5 = 15.Therefore, the two numbers are 35 and 15.
Among the following, the BEST example of qualitative data would include
 Group of answer choices
A) countywide census of speakers of more than one language.
B) average community income levels, by block.
C) ethnic composition of a community, by percentage.
D) field notes recorded during participant observation.
Answer:
Option D) field notes recorded during participant observation.
Step-by-step explanation:
We are given the following in the question:
Qualitative data and quantitative data:
Quantitative data are measures of values or counts.Quantitative data can be expressed as numbers. Quantitative data are data about numeric variablesQualitative data are measures of 'types'This type of data can be represented by a name, symbol, or a number code.A) countywide census of speakers of more than one language.
This is a quantitative data as it can be expressed in numbers. The census would give a numerical value of speakers of more than one language.
B) average community income levels, by block.
This is a quantitative data. Average income levels are expressed in numbers.
C) ethnic composition of a community, by percentage.
Again it is quantitative data as expressed in numbers.
D) field notes recorded during participant observation.
Thus is an example of qualitative data as field notes are expressed in name, text.
Thus, the best example of qualitative data is
Option D) field notes recorded during participant observation.
The best example of qualitative data among the choices is field notes recorded during participant observation. These notes are subjective descriptions of experiences, offering insights into the observed party's behaviors and interactions, which are qualitative in nature.
Explanation:Qualitative data is a type of data that is non-numerical and used to understand concepts, thoughts, or experiences. It goes beyond counts or numbers to capture the nature, quality, or meaning of a thing. Out of the options provided, D) field notes recorded during participant observation are the best example of qualitative data. The field notes are subjective descriptions of observations and experiences, rather than precise measurements or calculations. The other options (A to C) represent quantitative data, which is information about quantities and hard numbers.
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The image of the lemon is at point I. What is the size of the image compared to the size of the lemon?
same size...hope this helps :)))))
Which expression could be used to determine the area of the triangle shown? One-half + 12 and one-third + 3 One-half (3) (12) (one-third) 12 and one-third times 3 One-half (12 and one-third) (3)
Answer:
The answer is D.
Step-by-step explanation:
store sold combined totalof 412 biology and math books in a week. the number of biology textbooks sold was three times the number of math books sold. how many textbooks of each type were sold?
Step-by-step explanation:
Total books sold = 412
So if there are 103 maths books then biology books should be three times = 103 × 3 = 309
So when we add 103 + 309 = 412
It means the number of biology books is 309 and maths book is 103
Order the decimals from least to greatest 7.508,5.2161,7.5,7.58
Answer:
5.2161,7.5,7.508,7.58
Step-by-step explanation:
Answer:
5.2161; 7.5; 7.508; 7.58
Step-by-step explanation:
-First you will look at the first digit of each number given, obviously 5 is the smallest so it will go first
-Then you will check the following number after the first digit to see which one is smaller of the "7" values. In this case, they all have the number "5" has their second value, so we move on to the third digit.
-Since "7.5" does not have a third digit, it will be the smallest of the number "7" values.
-We then have a remaining of two values, 7.508 and 7.58. Still looking at the third numbers of each of the two, we see "0" and "8," and obviously 0 is smaller than 8, so 7.508 is smaller than 7.58
A real-valued function f is said to be periodic with period T ≠ 0 if f(x + T) = f(x) for all x in the domain of f. If T is the smallest positive value for which f(x + T) = f(x) holds, then T is called the fundamental period of f. Determine the fundamental period T of the given function. f(x) = sin(2x) + cos(4x)
Answer:
Period T of the given function f(x) = sin(2x) + cos(4x)
= π
Step-by-step explanation:
Given that y(x) is a sum of two trigonometric functions. The period T of sin 2x would be (2π÷2) = π. Period T of cos4x would be (2π÷4) that is π/2
Find the LCM of π and π/2 . That would be π. Hence the period of the given function would be π
The fundamental period ( T ) of the function [tex]\( f(x) = \sin(2x) + \cos(4x) \) is:\[ T = \pi \][/tex].
To determine the fundamental period ( T ) of the function [tex]\( f(x) = \sin(2x) + \cos(4x) \)[/tex], we need to find the smallest positive value of ( T ) such that [tex]f(x + T) = f(x) \) for all \( x \).\\[/tex]
Let's start by analyzing the periods of the individual components of [tex]\( f(x) \).[/tex]
1. Period of [tex]\( \sin(2x) \):[/tex]
The standard period of [tex]\( \sin(x) \) is \( 2\pi \). For \( \sin(2x) \),[/tex] the argument ( 2x ) scales the period. To find the period of [tex]\( \sin(2x) \)[/tex], we set:
[tex]\[ 2x = 2x + 2\pi \]\\[/tex]
Solving for the period, we get:
[tex]\[ x = x + \frac{2\pi}{2} \][/tex]
Thus, the period of [tex]\( \sin(2x) \)[/tex] is:
[tex]\[ \frac{2\pi}{2} = \pi \][/tex]
2. Period of [tex]\( \cos(4x) \):[/tex]
The standard period of [tex]\( \cos(x) \) is \( 2\pi \). For \( \cos(4x) \)[/tex], the argument ( 4x ) scales the period. To find the period of [tex]\( \cos(4x) \),[/tex] we set:
[tex]\[ 4x = 4x + 2\pi \][/tex]
Solving for the period, we get:
[tex]\[ x = x + \frac{2\pi}{4} \][/tex]
Thus, the period of [tex]\( \cos(4x) \)[/tex]is:
[tex]\[ \frac{2\pi}{4} = \frac{\pi}{2} \][/tex]
Next, we need to find the smallest positive ( T ) such that both [tex]\( \sin(2x) \)[/tex] and [tex]\( \cos(4x) \)[/tex] have the same period ( T ). This means that ( T ) must be a common multiple of the periods of the two components, [tex]\( \pi \)[/tex] and [tex]\( \frac{\pi}{2} \).[/tex]
To find the fundamental period ( T ), we determine the least common multiple (LCM) of [tex]\( \pi \) and \( \frac{\pi}{2} \):[/tex]
- [tex]\( \pi \)[/tex]can be written as [tex]\( \pi \times 1 \).[/tex]
- [tex]\( \frac{\pi}{2} \)[/tex] can be written as [tex]\( \pi \times \frac{1}{2} \).[/tex]
The LCM of [tex]\( 1 \) and \( \frac{1}{2} \) is \( 1 \) since \( 1 \)[/tex] is the smallest number that both [tex]\( 1 \) and \( \frac{1}{2} \)[/tex] can divide without leaving a remainder.
Thus, the LCM of [tex]\( \pi \) and \( \frac{\pi}{2} \) is:[/tex]
[tex]\[ \text{LCM}\left(\pi, \frac{\pi}{2}\right) = \pi \][/tex]
Therefore, the fundamental period ( T ) of the function [tex]\( f(x) = \sin(2x) + \cos(4x) \) is:\[ T = \pi \][/tex]
The following 5 questions are based on this information. An economist reports that 47% (p¯=0.47p¯=0.47) of a random sample of 1200 middle-income American households actively participate in the stock market. The goal is to construct a 95% confidence interval of the proportion (pp) of all middle-income Americans who actively participate in the stock market. The standard error (SE) of p¯p¯ is Select one:
a. 0.47
b. 0.047
c. 0.021
d. 0.014
Answer:
Option D) 0.014
Step-by-step explanation:
We are given the following in the question:
Sample size, n = 1200
Sample proportion =
[tex]\hat{p} = 0.47[/tex]
We have to make a 95% confidence interval.
Formula for standard error:
[tex]S.E = \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]
Putting the values, we get:
[tex]S.E = \sqrt{\dfrac{0.47(1-0.47)}{1200}} = 0.014[/tex]
Thus, the correct answer is
Option D) 0.014
If you put $6.57 into a savings account that earns 4%, how much interest
will you receive at the end of eight years?
The interest for 8 years is $2.10
Step-by-step explanation:
Principal amount (p) = $6.57
Rate of interest (r) = 4%
Time (t) = 8 years
Interest = (p x r x t) /100
= (6.57 x 4 x 8) /100
= 210.24/100
= 2.10
The interest for 8 years is $2.10
What is the value of the discriminant, b2 − 4ac, for the quadratic equation 0 = x2 − 4x + 5, and what does it mean about the number of real solutions the equation has?
The discriminant is −4, so the equation has 2 real solutions.
The discriminant is −4, so the equation has no real solutions.
The discriminant is 35, so the equation has 2 real solutions.
The discriminant is 35, so the equation has no real solutions.
The correct option is B because the discriminant is [tex]-4[/tex], so the equation has no real solutions.
Given:
The equation is:
[tex]0=x^2-4x+5[/tex]
To find:
The discriminant of the given equation.
Explanation:
In a quadratic equation [tex]ax^2+bx+c=0[/tex], the discriminant is:
[tex]D=b^2-4ac[/tex]
If [tex]D>0[/tex], then the equation has 2 real solutions.
If [tex]D=0[/tex], then the equation has 1 real solution.
If [tex]D<0[/tex], then the equation has no real solutions.
In the given equation, we have [tex]a=1,b=-4,c=5[/tex].
[tex]D=(-4)^2-4(1)(5)[/tex]
[tex]D=16-20[/tex]
[tex]D=-4[/tex]
Since [tex]D<0[/tex], therefore the equation has no real solutions.
Hence, the correct option is B.
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A rePrism measures 6.7 cm by 3.2 cm by 9 cm what is the volume of the rectangle prism
Answer:
The volume is 192.96
Step-by-step explanation:
You multiply length x width x height to find the volume.
Find the Laplace transform of the given function: f(t)=(t−5)u2(t)−(t−2)u5(t), where uc(t) denotes the Heaviside function, which is 0 for t
[tex]f(t)=(t-5)u_2(t)-(t-2)u_5(t)[/tex]
The Laplace transform is
[tex]F(s)=\displaystyle\int_0^\infty f(t)e^{-st}\,\mathrm dt=\int_2^\infty(t-5)e^{-st}\,\mathrm dt-\int_5^\infty(t-2)e^{-st}\,\mathrm dt[/tex]
Integrate by parts; in the first integral, take
[tex]u=t-5\implies\mathrm du=\mathrm dt[/tex]
[tex]\mathrm dv=e^{-st}\,\mathrm dt\implies v=-\dfrac{e^{-st}}s[/tex]
[tex]\implies\displaystyle\int_2^\infty(t-5)e^{-st}\,\mathrm dt=-\frac{e^{-st}}s(t-5)\bigg|_2^\infty+\frac1s\int_2^\infty e^{-st}\,\mathrm dt[/tex]
[tex]=-\dfrac{3e^{-2s}}s-\dfrac{e^{-st}}{s^2}\bigg|_2^\infty[/tex]
[tex]=-\dfrac{3e^{-2s}}s+\dfrac{e^{-2s}}{s^2}=-\dfrac{(3s-1)e^{-2s}}{s^2}[/tex]
For the second integral, take
[tex]u=t-2\implies\mathrm du=\mathrm dt[/tex]
[tex]\mathrm dv=e^{-st}\,\mathrm dt\implies v=-\dfrac{e^{-st}}s[/tex]
[tex]\implies\displaystyle\int_5^\infty(t-2)e^{-st}\,\mathrm dt=-\dfrac{(t-2)e^{-st}}s\bigg|_5^\infty+\frac1s\int_5^\infty e^{-st}\,\mathrm dt[/tex]
[tex]=\dfrac{3e^{-5s}}s-\dfrac{e^{-st}}{s^2}\bigg|_5^\infty[/tex]
[tex]=\dfrac{3e^{-5s}}s+\dfrac{e^{-5s}}{s^2}=\dfrac{(3s+1)e^{-5s}}{s^2}[/tex]
So we have
[tex]F(s)=\dfrac{(3s+1)e^{-5s}-(3s-1)e^{-2s}}{s^2}[/tex]
The manufacturer of a machine to package soap powder claimed that her machine could load cartons at a given weight with a range of no more than 0.4 ounce. The mean and variance of a sample of eight 3-pound boxes were found to equal 3.5 and 0.015, respectively. Test the hypothesis that the variance of the population of weight measurements is σ2 = 0.01 against the alternative that σ2 > 0.01.
Answer:
Please see the attached file for the complete answer.
Step-by-step explanation:
Please use mean = 3.5 instead of 3.1 and varience = 0.015 instead of 0.018 in the solution.
f(x) = 2x+3
g(x)=x^2
(a) Find fg(6)
Answer:
g(6)= 6^2= 36
2(36)+3= 72+3= 75
Suppose Julio is a veterinarian who is doing research into the weight of domestic cats in his city. He collects information on 131 cats and finds the mean weight for cats in his sample is 10.87 lb with a standard deviation of 4.31 lb. What is the estimate of the standard error of the mean (SE)
Answer:
Estimate of the standard error of the mean = 0.38 lb
Step-by-step explanation:
We are given the following in the question:
Sample mean, [tex]\bar{x}[/tex] = 10.87 lb
Sample size, n = 131
Standard deviation, σ = 4.31 lb
We have ti find the estimate of the standard error of the mean.
Formula for standard error:
[tex]S.E= \dfrac{\sigma}{\sqrt{n}}[/tex]
Putting values, we get,
[tex]S.E = \dfrac{4.31}{\sqrt{131}} = 0.3766 \approx 0.38[/tex]
0.38 lb is the standard error of the mean.
For a large supermarket chain in a particular state, a women's group claimed that female employees were passed over for management training in favor of their male colleagues. The company denied this claim, saying it picked the employees from the eligible pool at random to receive this training. Statewide, the large pool of more than 1000 eligible employees who can be tapped for management training is 30% female and 70% male. Since this program began, 39 of the 50 employees chosen for management training were male and 11 were female. Complete parts a through d. LOADING... Click the icon to view the results of using technology. a. The company claims that it selected employees for training according to their proportion in the pool of eligible employees. Define a parameter of interest and state this claim as a hypothesis. Explain why this hypothesis is a no-effect hypothesis. Define a parameter of interest. Choose the correct answer below.
Answer:
Step-by-step explanation:
Hello!
Tha claim is those female employees have less probability of being selected for management training than the males employees of a supermarket chain.
The company denies this claim and assures that the employees are picked at random from an eligible pool, i.e. that there is no gender bias in the manager selection
If you consider the variables:
X₁: Number of female employees selected for management training
X₂: Number of male employees selected for management training
The parameter of interest would be
p₁: population proportion of female employees selected for management training
p₂: population proportion of male employees selected for management training
If the company claims that there is no gender bias in the management selection is true, then the proportion of female employees selected for management training and the proportion of male employees selected for management training should be the same, symbolically: p₁ = p₂
If the company's claim is incorrect, then these proportions should be different: p₁ ≠ p₂
I hope this helps!
Multiply the polynomials. (2x+1)(x−5)
Answer:
2x^2-9x-5
Step-by-step explanation:
Just multiply
Answer:
2x^2-9x-5
Step-by-step explanataion:
multiply
How can you best describe a stop sign using
polygons?
The sign has 8 sides, so it is an octagon
It appears to be
because the sides and
angles appear to be congruent.
STOP
Dene
Answer:
The answer to the last one is regular
Step-by-step explanation:
A polygon is a planar figure defined by a finite number of straight-line segments. The polygon that best describes a stop sign is a regular polygon.
What is a polygon?A polygon is a planar figure defined by a finite number of straight-line segments that are joined to form a closed polygonal chain in geometry. A polygon can be defined as a bounded planar region, a bounding circuit, or both.
The polygon that best describes a stop sign is a regular octagon because a stop sign has 8 sides and the length of each side is the same, therefore, the polygon will be a regular polygon.
Thus, the polygon that best describes a stop sign is a regular polygon.
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The quality-control manager at a compact fluorescent light bulb (CFL) factory needs to determine whether the mean life of a large shipment of CFLs is equal to 7 comma 5137,513 hours. The population standard deviation is 1 comma 080 hours1,080 hours. A random sample of 8181 light bulbs indicates a sample mean life of 7 comma 2137,213 hours. a. At the 0.050.05 level of significance, is there evidence that the mean life is different from 7 comma 513 hours question mark7,513 hours? b. Compute the p-value and interpret its meaning. c. Construct a 9595% confidence interval estimate of the population mean life of the light bulbs. d. Compare the results of (a) and (c). What conclusions do you reach?
Answer:
Step-by-step explanation:
We would set up the hypothesis test. This is a test of a single population mean since we are dealing with mean
For the null hypothesis,
µ = 7463
For the alternative hypothesis,
µ ≠ 7463
This is a 2 tailed test
Since the population standard deviation is given, z score would be determined from the normal distribution table. The formula is
z = (x - µ)/(σ/√n)
Where
x = sample mean
µ = population mean
σ = population standard deviation
n = number of samples
From the information given,
µ = 7463 hours
x = 7163 hours
σ = 1080 hours
n = 81
b) z = (7163 - 7463)/(1080/√81) = - 2.5
Looking at the normal distribution table, the probability corresponding to the z score is 0.02
Recall, population mean is 7463
The difference between sample sample mean and population mean is 7463 - 7163 = 300
Since the curve is symmetrical and it is a two tailed test, the x value for the left tail is 7463 - 300 = 7163
the x value for the right tail is 7463 + 300 = 7763
These means are higher and lower than the null mean. Thus, they are evidence in favour of the alternative hypothesis. We will look at the area in both tails. Since it is showing in one tail only, we would double the area. We already got the area below z as 0.02
We would double this area to include the area in the right tail of z = 2.5. Thus
p = 0.02 × 2 = 0.04
It means that in a sample of size 81 light bulbs, we would observe a sample mean of 300 hours or more away from 7463 about 4% of the time by chance alone.
c) Confidence interval is written in the form,
(Sample mean - margin of error, sample mean + margin of error)
The sample mean, x is the point estimate for the population mean.
Margin of error = z × σ/√n
To determine the z score, we subtract the confidence level from 100% to get α
α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025
This is the area in each tail. Since we want the area in the middle, it becomes
1 - 0.025 = 0.975
The z score corresponding to the area on the z table is 1.96. Thus, z score for 95% confidence level is 1.96
Margin of error = 1.96 × 1080/√81
= 235.2
Confidence interval = 7163 ± 23.2
a) Since alpha, 0.05 > than the p value, 0.04, then we would reject the null hypothesis. Therefore, at a 5% level of significance, there is evidence that the mean life is different from 7463 hours
Comparing the results of a and c, it is true that the population mean life is not 7463 hours.
Which best describes the graph of
f(x) = log2(x + 3) + 2 as a transformation of the
graph of g(x) = log2x?
O
O
o
o
a translation 3 units right and 2 units up
a translation 3 units left and 2 units up
a translation 3 units up and 2 units right
a translation 3 units up and 2 units left
Answer:
Step-by-step explanation:
A translation 3 units left and 2 units up
A translation 3 units left and 2 units up best describes the graph of f(x) = log2(x + 3) + 2 as a transformation of the graph of g(x) = log2x
How to solve this problem?f(x) = log2(x + 3) + 2 (given)g(x) = log2x (given)We need to describe the best statement for the graphThe graph is shown in the imageThe following steps are shown to describe the graph.
The general equation of f(x) = log2(x-h)+k
When h > 0 (positive)The graph of the base of the function shift to the right
When h < 0 (Negative)The graph of the base function shifts to the left.
When k > 0 (Positive)The graph of the base function shifts upward.
When k < 0 (Negative)The graph of the base function shifts downward
Here h = 3 , k = 2
Hence , a translation 3 units left and 2 units up describes the graph.
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In 2012, Gallup asked participants if they had exercised more than 30 minutes a day for three days out of the week. Suppose that random samples of 100 respondents were selected from both Vermont and Hawaii. From the survey, Vermont had 65.3% who said yes and Hawaii had 62.2% who said yes. What is the value of the population proportion of people from Hawaii who exercised for at least 30 minutes a day 3 days a week? Group of answer choices 0.6375 Unknown 0.653 0.622
The population proportion of people from Hawaii who exercised for at least 30 minutes a day three days a week, according to the 2012 Gallup survey, is 0.622.
Explanation:The question is seeking to find the population proportion of people from Hawaii who said they exercised for at least 30 minutes a day for at least three days in a week. This is given directly in the data from the survey by Gallup in 2012. The population proportion is a fundamental concept in statistics and refers to the fraction of individuals in a group, or population, possessing a certain characteristic. In this case, the characteristic is exercising for at least 30 minutes a day for three days in a week.
From the data given, we know that from a random sample of 100 respondents from Hawaii, 62.2% said yes to the question posed by the survey. In terms of population proportion, this can be expressed as 0.622. This means that if we were to select one person at random from the population of Hawaii, the probability that this person exercises for at least 30 minutes a day, three days a week is estimated to be 0.622.
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The arch beneath a bridge is semi-elliptical, a one-way roadway passes under the arch. The width of the roadway is 38 feet and the height of the arch over the center of the roadway is 12 feet. Two trucks plan to use this road. They are both 8 feet wide. Truck 1 has an overall height of 11 feet and Truck 2 has an overall height of 12 feet. Draw a rough sketch of the situation and determine which of the trucks can pass under the bridge.
Answer:
Only truck 1 can pass under the bridge.
Step-by-step explanation:
So, first of all, we must do a drawing of what the situation looks like (see attached picture).
Next, we can take the general equation of an ellipse that is centered at the origin, which is the following:
[tex] \frac{x^2}{a^2}+\frac{y^2}{b^2}[/tex]
where:
a= wider side of the ellipse
b= shorter side of the ellipse
in this case:
[tex] a=\frac{38}{2}=19ft[/tex]
and
b=12ft
so we can go ahead and plug this data into the ellipse formula:
[tex] \frac{x^2}{(19)^2}+\frac{y^2}{(12)^2}[/tex]
and we can simplify the equation, so we get:
[tex] \frac{x^2}{361}+\frac{y^2}{144}[/tex]
So, we need to know if either truk will pass under the bridge, so we will match the center of the bridge with the center of each truck and see if the height of the bridge is enough for either to pass.
in order to do this let's solve the equation for y:
[tex] \frac{y^{2}}{144}=1-\frac{x^{2}}{361}[/tex]
[tex] y^{2}=144(1-\frac{x^{2}}{361})[/tex]
we can add everything inside parenthesis so we get:
[tex] y^{2}=144(\frac{361-x^{2}}{361})[/tex]
and take the square root on both sides, so we get:
[tex] y=\sqrt{144(\frac{361-x^{2}}{361})}[/tex]
and we can simplify this so we get:
[tex] y=\frac{12}{19}\sqrt{361-x^{2}}[/tex]
and now we can evaluate this equation for x=4 (half the width of the trucks) so:
[tex] y=\frac{12}{19}\sqrt{361-(8)^{2}}[/tex]
y=11.73ft
this means that for the trucks to pass under the bridge they must have a maximum height of 11.73ft, therefore only truck 1 is able to pass under the bridge since truck 2 is too high.
Truck 1 and Truck 2 can both pass under the bridge.
Explanation:To determine which truck can pass under the bridge, we need to compare the height of the truck to the height of the arch over the center of the roadway. The height of the arch is given as 12 feet. Truck 1 has an overall height of 11 feet, so it can pass under the bridge. Truck 2 has an overall height of 12 feet, which is equal to the height of the arch, so it can also pass under the bridge.
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The Shady Farm Milk Company can process milk at a fixed rate of 7500 gallons/hour. The company’s clients request 100,000 gallons of milk over the course of one day. This demand is spread out uniformly from 8 a.m. to 6 p.m. The company starts producing at 8 a.m. and continues to work until all of the demand has been satisfied. At noon, how many gallons of milk are in the queue to be processed?
At noon, the Shady Farm Milk Company has 10,000 gallons of milk in the queue to be processed given the demand and the processing rate.
Explanation:The Shady Farm Milk Company can process 7500 gallons of milk per hour. Given that the company operates from 8 a.m. to 6 p.m., this is a total of 10 hours of operation in a day. Therefore, in 10 hours, the company can process 7500 × 10 = 75,000 gallons of milk.
However, the demand for milk is 100,000 gallons over the course of the day. Therefore, by noon, the company has been operating for 4 hours, meaning they can process 7500 × 4 = 30,000 gallons.
The demand over the same 4 hours period (from 8 a.m. to noon) is calculated by dividing the total demand over the entire course of the day (which is evenly spread) by the number of operating hours. Thus: 100,000 / 10 = 10,000 gallons/hour.
Consequently, the demand from 8 a.m. to noon is: 10,000 × 4 = 40,000 gallons. So, the amount of milk in the queue at noon would be the demand minus what the company has processed at that time.
Hence: 40,000 (demand from 8 a.m. to noon) - 30,000 (processed milk from 8 a.m. to noon) = 10,000 gallons. Therefore, at noon, the company has 10,000 gallons of milk in the queue to be processed.
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At noon, there are 20,000 gallons of milk in the queue to be processed.
To find out how many gallons of milk are in the queue to be processed at noon, we first need to calculate how many gallons of milk have been processed by noon.
The company can process milk at a fixed rate of 7500 gallons per hour. From 8 a.m. to noon, there are 4 hours.
Total gallons processed by noon = Rate of processing Time
[tex]\[ = 7500 \, \text{gallons/hour} \times 4 \, \text{hours} = 30000 \, \text{gallons} \][/tex]
Now, we need to find out how many gallons of milk are still in demand by noon. The total demand over the course of the day is 100,000 gallons, and it is spread out uniformly from 8 a.m. to 6 p.m.
This means that by noon, half of the day has passed.
So, the total demand by noon = Total demand / 2
[tex]\[ = \frac{100000}{2} = 50000 \, \text{gallons} \][/tex]
Now, to find out how many gallons are in the queue to be processed at noon, we subtract the gallons already processed from the total demand:
Gallons in the queue at noon = Total demand by noon - Gallons processed by noon
[tex]\[ = 50000 \, \text{gallons} - 30000 \, \text{gallons} = 20000 \, \text{gallons} \][/tex]
So, at noon, there are 20,000 gallons of milk in the queue to be processed.
Suppose we have a collection of the heights of all students at your college. Each of the 250 people taking statistics randomly takes a sample of 40 of these heights and constructs a 95% confidence interval for mean height of all students at the college. Which of the following statements about the confidence intervals is most accurate?
A. About 95% of the heights of all students at the college will be contained in these interval
B. About 95% of the time, a student’s sample mean height will be contained in his or her interval.
C. About 95% of the intervals will contain the population mean height.
D. About 95% of the intervals will be identical.
Answer:
Correct option: (C)
Step-by-step explanation:
The (1 - α)% confidence interval for population parameter implies that there is a (1 - α) probability that the true value of the parameter is included in the interval.
Or, the (1 - α)% confidence interval for the parameter implies that there is (1 - α)% confidence or certainty that the true parameter value is contained in the interval.
Or, if n such (1 - α)% confidence intervals are constructed then (1 - α)% of these interval will consist of the true parameter value.
It is provided that each of the 250 people taking statistics randomly takes a sample of 40 of these heights and constructs a 95% confidence interval for mean height of all students at the college.
So there sill be n = 250, 95% confidence intervals for mean height of all students at the college.
Then 95% of these 250 confidence intervals for mean will consist of the true mean height of the college students.
Thus, the correct option is (C).
A bread recipe calls for 1 teaspoon of yeast for every 2
cups of flour.
Write an equation that represents the number of cups of
flour, c, for every teaspoon of yeast, t.
Equation representing the cups of flour for every teaspoon of yeast will be → c = 2t
Proportional relation between two variables: If a variable 'y' is directly proportional to another variable 'x', expression will be,
y ∝ x
y = kx [Here, k = proportionality constant]
It has been given in the question,
"A bread recipe calls for 1 teaspoon of yeast for every 2 cups of flour."
If the number of teaspoons of yeast is represented by 't' and number of cups of flour by 'c',
c = kt
From the given statement,
2 = k × 1
k = 2
Therefore, equation for the given statement will be → c = 2t
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Really confused can someone please help me with this?
Step-by-step explanation:
4x + 7° = 5x - 1°
5x - 4x = 7° + 1°
x = 8°
As OR~=OS
Select each polynomial that is a perfect square.
Question 5 options:
x2−10x−25
4x2−12x+9
9x2+12x+16
16x2+16x+1
Answer:
4x^2−12x+9
Step-by-step explanation:
The form of a perfect square trinomial is ...
(a +b)² = a² +2ab +b²
The first and last terms must be positive and perfect squares. The middle term must be twice the product of their roots (possibly with a minus sign).
x^2 -10x -25 . . . . -25 is not a positive perfect square
4x^2 -12x +9 . . . . 12x = 2√(4x^2·9) = 2·6x . . . . perfect square
9x^2 +12x +16 . . . 12x ≠ 2√(9x^2·16) = 2·12x
16x^2 +16x +1 . . . 16x ≠ 2√(16x^2·1) = 2·4x
The shares of the U.S. automobile market held in 1990 by General Motors, Japanese manufacturers, Ford, Chrysler, and other manufacturers were, respectively, 36%, 26%, 21%, 9%, and 8%. Suppose that a new survey of 1,000 new-car buyers shows the following purchase frequencies:
GM Japanese Ford Chrysler Other
193 384 170 90 163
(a) Show that it is appropriate to carry out a chi-square test using these data.
(b) Determine whether the current market shares differ from those of 1990. Use α = .05.
Answer:
Step-by-step explanation:
Hello!
The objective is to test if the purchase frequencies of new-car buyers follow the distribution of the shares of the U.S. automobile market for 1990.
You have one variable of interest:
X: Brand a new-car buyer prefers, categorized: GM, Japanese, Ford, Chrysler and Other
n= 1000
Observed frequencies
GM 193
Japanese 384
Ford 170
Chrysler 90
Other 163
a) The test to use to analyze if the observed purchase frequencies follow the market distribution you have to conduct a Goodness to Fit Chi-Square test.
The conditions for this test are:
- Independent observations
In this case, we will assume that each buyer surveyed is independent of the others.
- For 3+ categories: each expected frequency (Ei) must be at least 1 and at most 20% of the Ei are allowed to be less than 5.
In our case we have a total of 5 categories, 20% of 5 is 1, only one expected frequency is allowed to have a value less than 5.
I'll check this by calculating all expected frequencies using the formula: Ei= n*Pi (Pi= theoretical proportion that corresponds to the i-category)
E(GM)= n*P(GM)= 1000*0.36= 360
E(Jap)= n*P(Jap)= 1000*0.26= 260
E(Ford)= n*P(Ford)= 1000*0.21= 210
E(Chrys)= n*P(Chrys)= 1000*0.09= 90
E(Other)= n*P(Other)= 1000*0.08= 80
Note: If all calculations are done correctly then ∑Ei=n.
This is a quick way to check if the calculations are done correctly.
As you can see all conditions for the test are met.
b) The hypotheses for this test are:
H₀: P(GM)= 0.36; P(Jap)= 0.26; P(Ford)= 0.21; P(Chrys)= 0.09; P(Other)= 0.08
H₁: At least one of the expected frequencies is different from the observed ones.
α: 0.05
[tex]X^2= sum \frac{(Oi-Ei)^2}{Ei} ~~X^2_{k-1}[/tex]
k= number of categories of the variable.
This test is one-tailed right this mean you'll reject the null hypothesis to high values of X²
[tex]X^2_{k-1;1-\alpha }= X^2_{4;0.95}= 9.488[/tex]
Decision rule using the critical value approach:
If [tex]X^2_{H_0}[/tex] ≥ 9.488, reject the null hypothesis
If [tex]X^2_{H_0}[/tex] < 9.488, don't reject the null hypothesis
[tex]X^2_{H_0}= \frac{(193-360)^2}{360} + \frac{(384-260)^2}{260} + \frac{(170-210)^2}{210} + \frac{(90-90)^2}{90} + \frac{(163-80)^2}{80} = 230.34[/tex]
The value of the statistic under the null hypothesis is greater than the critical value, so the decision is to reject the null hypothesis.
Using a 5% level of significance, there is significant evidence to conclude that the current market greatly differs from the preference distribution of 1990.
The chi-square test can be used to determine if there is a significant difference between the observed frequencies and the expected frequencies based on the shares of the U.S. automobile market in 1990. A calculated chi-square statistic greater than the critical value would indicate a significant difference between the current market shares and those of 1990.
Explanation:The first step in the process is to calculate the expected counts for each category based on the shares of the U.S. automobile market in 1990. The expected counts for GM, Japanese manufacturers, Ford, Chrysler, and other amounts to 360, 260, 210, 90, and 80, respectively.
Given the observed frequencies, the chi-square statistic X² can be calculated. The formula for the chi-square test is X² = Σ[(O-E)²/E], where O represents the observed frequency and E represents the expected frequency. Using this formula, the chi-square statistic is calculated.
Finally, using a chi-square distribution table with 4 degrees of freedom (5 categories - 1), and an alpha level of 0.05, you can compare the calculated chi-square statistic to the critical value (9.488). If the chi-square statistic is larger than the critical value, you would reject the null hypothesis and conclude that the current market shares do differ from those of 1990. If the chi-square statistic is smaller than the critical value, you would not reject the null hypothesis and conclude that the current market shares do not significantly differ from those of 1990.
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