The integral / 5√1-4x² dx is to be evaluated directly and using a series approximation. (Give all your answers rounded to 3 significant figures.) a) Evaluate the integral exactly, using a substitution in the form ax = sin 0 and the identity cos²x = (1 + cos2x). Enter the value of the integral b) Find the Maclaurin Series expansion of the integrand as far as terms in x6. Give the coefficient of x4 in your expansion: C) Integrate the terms of your expansion and evaluate to get an approximate value for the integral. Enter the value of the integral: d) Give the percentage error in your approximation i.e. calculate.
a) the value of the integral 1.664. b) the coefficient of x4 is (3/1280). c) the value of the integral 2.14 d) the percentage error in the approximation is -28.67%.
a) To evaluate the integral exactly, we make a substitution in the form of ax = sin 0 where a = (1/2).
Substitute x = (sin θ)/2, dx = (cos θ)/2 dθ, and 1 - 4x² = cos² θ in the integral to get it in terms of θ.
∫5√(1-4x²)dx = ∫(5/2) cos²θ dθApply the identity cos²θ = (1 + cos 2θ)/2 to simplify the integrand as shown.∫(5/2) cos²θ dθ = (5/4)∫(1 + cos2θ) dθ = (5/4)θ + (5/8)sin 2θ
Evaluate the above expression from 0 to π/2 to get the value of the integral.
(5/4)θ + (5/8)sin 2θ = (5/4) π/2 + (5/8) sin π = 1.664
b) The integrand is f(x) = 5√(1-4x²).We can write it as shown below, using the binomial series. f(x) = 5(1 - 4x²)^(1/2) = 5∑_(n=0)^∞〖(1/2)_n (2n)!/n! (1/16)^n x^(2n) 〗
The above expression is the Maclaurin Series expansion of f(x) as required.In the expansion, the coefficient of x4 is (1/2)_2 (2.4)/(2!) (1/16)^2 = (3/1280)
c) Integrating each term of the expansion, we obtain the following expression.(5/4)θ + (5/8)sin 2θ = (5/4) π/2 + (5/8) sin π = 1.664
We approximate the value of the integral using the first three terms of the series expansion, and then add the values of the integrated terms.
The terms up to x4 are included in this calculation. I=5[1+(1/2) (-4x²) +(1/2)_2 (-4x²)²]I=5[1-2x²+3x^4/4] = (5/4) (π/2 + (5/16)π²) ≈ 2.14
d) The percentage error in the approximation is given by:%Error = [(Exact value - Approximation value)/ Exact value] x 100Substitute the appropriate values to calculate.%Error = [(1.664 - 2.14)/1.664] x 100 = -28.67% (correct to 3 significant figures)Thus, the percentage error in the approximation is -28.67%.
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7 of possible A survey conducted in a small business yielded the results shown in the table. Test the claim that health care coverage is independent of gender. Use a 0.05 significance level. What is the value of the test statistic? Men Women Health insurance 50 20 No health insurance 30 10 O A. x² = 5.821 O B.x²=3.841 c. x² = 8.263 O D. x² = 0.1637
The value of the test statistic is x² = 3.841.
Therefore option B is correct.
How do we calculate?Expected frequency for "Health insurance" and "Men"
= (80 * 70) / 110= 50.909
Expected frequency for "Health insurance" and "Women"
= (30 * 70) / 110 = 19.091
Expected frequency for "No health insurance" and "Men"
= (80 * 40) / 110
= 29.091
Expected frequency for "No health insurance" and "Women"
= (30 * 40) / 110 = 10.909
We find the chi-square statistic
χ² = Σ [(O - E)² / E]
χ² = [(50 - 50.909)² / 50.909] + [(20 - 19.091)² / 19.091] + [(30 - 29.091)² / 29.091] + [(10 - 10.909)² / 10.909]
= (0.081² / 50.909) + (0.909² / 19.091) + (0.909² / 29.091) + (0.081² / 10.909)
= 0.00131 + 0.04537 + 0.02784 + 0.00604
= 0.08056
We find the critical value for a 0.05 significance level and 1 degree of freedom = 3.841.
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complete question:
A survey conducted in a small business yielded the results shown in the table. Test the claim that health care coverage is independent of gender. Use a 0.05 significance level. What is the value of the test statistic?
Men Women
Health insurance 50 20
No health insurance 30 10
OA. x^2=0.1637
OB. x^2=8.263
OC. x^2 =3.841
OD. x^2 =5.821
Joe has 3 1/4 gallons of punch. He adds 1 1/2 quarts of juice to the punch . He drinks 1/4 quarts of punch . Which expression can be used to fine the number of quarts of punch joe has left
Answer:
13 + 1 1/2 - 1/4
Step-by-step explanation:
1 gallons = 4 quarts
3 1/4 = 13/4 x 4 = 13
Quarts left = original quart + quart added - quart drunk
13 + 1 1/2 - 1/4
5/6 + 7/8 + 3/4
i need help
Answer:2 11/24
Step-by-step explanation:
Answer:
2.4583
Step-by-step explanation:
I helped you.
First
5÷6 and 7÷8 and again similarly 3÷4 and sum the all outcome come from these eqations
consider a normally distributed population with mean =10 and standard deviation σ=2.5. suppose a random sample of size is selected from this population. Find the distribution of X and the indicated probability in each of the following cases. a. n = 7 P(X < 9)
b. n = 12, P(X> 11.5). c. n = 15, P(9.5 10.25). e. n=100, P(X <9.8 UX >0.2)
The probability P(Z < -1.06) is approximately 0.142. The probability P(Z > 2.386) is about 0.008. The probability P(-0.777 < Z < 0.777) is approximately 0.456.
The probability P(X < 9.8) ≈ 0.211. The probability P(X > 10.2) = = 0.212. The probability P(X < 9.8 or X > 10.2) = 0.423.
To locate the distribution of X and the indicated possibilities for the given instances, we need to use the residences of the everyday distribution. Given that the populace has a median (μ) of 10 and a widespread deviation (σ) of 2.5, we will continue as follows:
a. N = 7, P(X < 9):
For a pattern size of seven, the distribution of X follows a normal distribution with the equal mean (10) however a trendy deviation of σ/sqrt(n) = 2.5/[tex]\sqrt{7}[/tex] ≈ 0.944.
To discover P(X < nine), we need to standardize the cost of 9 with the use of the Z-rating formula: Z = (X - μ) / σ.
Substituting the values, we get Z = (9 - 10) / 0.944 ≈ -1.06.
Using a standard regular distribution table or calculator, we are able to locate that the chance P(Z < -1.06) is approximately 0.142.
B. N = 12, P(X > 11.5):
For a sample length of 12, the distribution of X follows a regular distribution with the same suggestion (10) but a well-known deviation of σ/[tex]\sqrt{n}[/tex] = 2.5/[tex]\sqrt{12}[/tex] ≈ 0.7217.
To discover P(X > 11.5), we standardize the value of 11.5 for the usage of the Z-rating method: Z = (X - μ) / σ.
Substituting the values, we get Z = (11.5 - 10) / 0.7217 ≈ 2.386.
Using a trendy everyday distribution table or calculator, we will locate that the chance P(Z > 2.386) is about 0.008.
C. N = 15, P(9.5 < X < 10.25):
For a sample size of 15, the distribution of X follows a normal distribution with identical implies (10) however a popular deviation of σ/sqrt(n) = 2.5/[tex]\sqrt{15}[/tex]≈ 0.6455.
To discover P(9.5 < X < 10.25), we need to standardize the values using the Z-score components.
Z1 = (9.5 - 10) / 0.6455 ≈ -0.777, and Z2 = (10.25 - 10) / 0.6455 ≈ 0.777.
Using a widespread ordinary distribution desk or calculator, we can locate that P(-0.777 < Z < 0.777) is approximately 0.456.
D. N = 100, P(X < 9.8 or X > 10.2):
For a sample size of 100, the distribution of X follows a regular distribution with the equal implies (10) however a general deviation of σ/sqrt(n) = 2.5/[tex]\sqrt{100}[/tex] = 0.25.
To find P(X < 9.8 or X > 10.2), we need to calculate the probabilities for each person's case and subtract them from 1.
P(X < 9.8) = P(Z < (9.8 - 10) / 0.25) ≈ P(Z < -0.8) ≈ 0.211.
P(X > 10.2) = P(Z > (10.2 - 10) / 0.25) ≈ P(Z < -0.8) ≈ 1 - P(Z < 0.8) ≈ 1 - 0.788 = 0.212.
Therefore, P(X < 9.8 or X > 10.2) ≈ P(X < 9.8) + P(X > 10.2) ≈ 0.211 + 0.212 = 0.423.
Remember to consult a trendy everyday distribution desk or use a calculator to locate the possibilities associated with the Z-scores.
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Consider the sets of natural numbers, whole numbers, integers, rational numbers, and real numbers. Identify from the list above the first set that describes the given number. 8.7104 Choose the correct answer below. O Natural numbers O Integers Whole numbers Rational numbers Real numbers
The number 8.7104 belongs to the set of real numbers. The sets of natural numbers, whole numbers, integers, rational numbers, and real numbers are ordered from most specific to most inclusive.
Natural numbers: Also known as counting numbers, they include positive whole numbers starting from 1 (1, 2, 3, 4, ...).
Whole numbers: Similar to natural numbers, they include all positive integers starting from 0 (0, 1, 2, 3, ...).
Integers: This set includes both positive and negative whole numbers, including zero (-∞, ..., -3, -2, -1, 0, 1, 2, 3, ..., +∞).
Rational numbers: These are numbers that can be expressed as fractions, where the numerator and denominator are both integers. Rational numbers can be written as terminating or repeating decimals.
Real numbers: This set includes all rational and irrational numbers. Real numbers can be represented on the number line and include all possible decimal values, including non-terminating and non-repeating decimals.
In the case of the number 8.7104, it is a decimal number that can be expressed as a terminating decimal. Therefore, it falls within the set of real numbers. Real numbers encompass all possible decimal values, both terminating and non-terminating, making them the broadest set in terms of representation on the number line.
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Twice the difference of a number and 4 equals 3
Answer:
x - 4 x 2 = 3
Is your algebra expression.
The Broadway Theatre in NYC houses 1761 seats. In order to maximize revenues, the theatre has decided to have two fare classes. High (H) fare class tickets sell for $100 and the Low (L) fare class tickets sell at a discounted price of $75. There is ample demand for the low fare class, but high fare demand is random. Furthermore, the customers who buy low fares, buy their tickets well in advance before high fare customers. Assume the demand for high fare tickets is normally distributed with a mean of 1200 and a standard deviation of 150.
1. What would be the theatre’s revenues without yield management?
$132,075
$157,275
$176,100
Cannot be determined from the information provided.
Without yield management, the theatre's revenues would be $176,100.
To calculate the theatre's revenues without yield management, we need to determine the number of tickets sold for each fare class and multiply it by the corresponding ticket price. The number of low fare class tickets sold is not provided in the information, so we cannot determine the exact revenue for that fare class. However, since there is ample demand for the low fare class, it is reasonable to assume that all 1761 seats are sold at the discounted price of $75. Therefore, the revenue from the low fare class would be 1761 seats × $75 = $132,075. For the high fare class, the demand is normally distributed with a mean of 1200 and a standard deviation of 150. To maximize revenues, the theatre should sell as many high fare class tickets as possible. Assuming all remaining seats (1761 - number of low fare class tickets) are sold at the high fare of $100, the revenue from the high fare class would be (1761 - number of low fare class tickets) × $100. To calculate the exact number of high fare class tickets sold, we need to know the cutoff point for the demand distribution that separates high fare demand from low fare demand. Since this information is not provided, we cannot determine the exact revenue from the high fare class. However, the only answer option that is close to the estimated revenue considering all high fare class tickets sold is $176,100.
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Consider the following research title: "Seasonal distribution of endemic typhus in the Southern United States, 1922-1925". What type of research design would this study most likely utilize? Cross-sectional ecologic study Longitudinal ecologic study o Qualitative study Cross-sectional survey
The research title, "Seasonal distribution of endemic typhus in the Southern United States, 1922-1925" would most likely utilize a longitudinal ecologic study.
The correct option is: b. Longitudinal ecologic study
A longitudinal study is a research design that follows participants over an extended period to establish patterns of change or stability in development. The same group of people is monitored at various stages in their life. For example, a group of people may be followed from the age of 5 to the age of 50, and their health status, life choices, and other variables are monitored at each age interval.
The research title, "Seasonal distribution of endemic typhus in the Southern United States, 1922-1925," would most likely utilize a longitudinal ecologic study since the research is centered on an illness and covers four years. Therefore, researchers are likely to follow up with participants and monitor their health status over an extended period to obtain data on the seasonal distribution of endemic typhus.
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The radius of a circle is 3 meters what's the circles area Use 3.15 for^
Answer:
The area of the circle is 28.26 m
Step-by-step explanation:
A = π r^2
A = (3.14) × 3^2
A = 28.26
Area of the circle = 28.26 m
(I'm sorry but I think you made a mistake, PI is 3.14 not 3.15) :)
Which sets of shapes can be used as the net of a three-dimensional figure? Select two options.
1 hexagon and 6 rectangles
2 pentagons and 5 triangles
6 rectangles, none of which are squares
4 squares and two rectangles that are not squares
4 triangles, only 3 of which are congruent triangles
Answer:
C and D
Step-by-step explanation:
Got it right on the test.
1 hexagon and 6 rectangles and 4 squares and two rectangles that are not squares are sets of shapes which can be used as the net of a three-dimensional figure. Option A and option D are correct.
Hexagon and 6 rectangles: This combination of shapes can be used to create the net of a hexagonal prism.
The hexagon will be used as the base of the prism, and the six rectangles can be attached to the sides of the hexagon to form the remaining faces of the prism.
When folded along the edges, the net will result in a hexagonal prism.
4 squares and two rectangles that are not squares: This combination can be used to create the net of a rectangular prism or a cuboid.
The four squares will be used as the top, bottom, and side faces of the prism, while the two rectangles (which are not squares) can be attached to the remaining sides to complete the net.
When the net is folded along the edges and assembled, it will form a rectangular prism or cuboid.
Hence, Option A and option D are correct. 1 hexagon and 6 rectangles and 4 squares and two rectangles that are not squares are sets of shapes which can be used as the net of a three-dimensional figure.
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Can someone answer one of these rows. It’s volume of prisms.
Answer:
1.
Step-by-step explanation:
base formula
is length × height so that would be 11×4=44
base= 44
height=4
volume= 132
2.
base formula
since its a cube it would be length× hight = 36
base= 36
hight=6
volume =216
3.
base formula
pie×radius^2 so that would be 3.14×1.5^2= 14.80
base 14.80
hight 4
volume= 59.2
4
According to a college survey, 22% of all students work full time. Find the mean for the number of students who work full time in samples of size 16. Round to the nearest tenth. A. 4.0 B. 3.5 C. 2.8 D. 0.2
The average number of full-time students in samples of size 16 is B) 3.5.
Because of the extraordinarily huge population, this can be regarded a binomial distribution if all students globally are considered. A normal distribution is commonly used to approximate the binomial distribution. As a result, the mean equals the expectation:
E[x] = np = (16)(0.22) = 3.52
μ = 3.52
The likelihood of success raised to the power of the number of successes is multiplied by the probability of failure raised to the power of the difference between the number of successes and the number of trials. The product is then multiplied by the sum of the number of trials and successes.
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XA ) -
Calculate the volume of the prism by first finding the total number of half-unit
cubes that will fill it. There are 8 half-unit cubes in every unit cube.
23
A. Number of half-unit cubes = 10
V = 2 cubic units
B. Number of half-unit cubes = 20
V=2 cubic units
O C. Number of half-unit cubes = 20
V = 10 cubic units
D. Number of half-unit cubes = 10
V = 5 cubic units
Answer:
Option B will be your answer
Step-by-step explanation:
V=2 1/2×1/2×2
=2.5 cube
volume of 1/2 cubic units =)(1/2×1/2×1/2=0.125cube^3
2.5/0.125=20
Hope it helps...
v = 2½
number of half unit cubes = 20
The sum of the lengths of the sides of triangle ABC is 25 in . The lengths of sides overline AB and overline BC are 9 inches and 8 inches . Find the length of side overline AC and classify the triangle.
Answer:
AC = 8 The classification is isosceles
Step-by-step explanation:
What is the length of Line segment A C? Round to the nearest tenth. 3.0 cm. 9.8 cm. 10.5 cm. 12.8 cm.
Answer:
10.5 cm
Step-by-step explanation:
A line segment is defined as a part of a line. It is a straight line having two end points. The only difference between a line and line segment is that a line is continuous with indefinite length whereas a line segment has a definite length to it between the two points.
In the figure, the length of the line segment is 10.5 cm (rounded to the nearest tenth).
Answer:
10.5
Step-by-step explanation:
ok so-
{[Math from BIM]}
ss below
Answer:
2 real solutions
Step-by-step explanation:
x=6i, -6i
Daphne wants to buy a coat that costs $80. The store has a sales tax of 6.5%. How much tax will Daphne be charged if she buys the coat?
Answer:
$85.2
Step-by-step explanation:
Answer:
85.2 dollar
Step-by-step explanation:
Tax is 5.2 because 80 times 0.065 is that, then you add them up
3
Which conical container holds more water?
1
1
-
1
1
k
8 cm
8 cm
or
1
4 cm
6 cm
Cone 1
Cone 2
Answer:
A bridge combines two cities X and Y. Cars cross this bridge according to a Poisson process of rate µ= 600/hour. Independently each car travels from X to Y with probability p = 0.XX XX is the last two digits of your student ID after the decimal point (for example: if your student ID is 63171234, you should use p = 0.34) and from Y to X with probability 1 - p. a. What is the probability that during a one minute period at noon 2 cars cross the bridge? b. What is the probability that during a one minute period at noon 2 cars travel from X to Y?
The probability of 2 cars crossing the bridge from X to Y in a 1-minute period is:
P(X = 2) = ((λ^x) / x!) * e^(-λ)
P(X = 2) = ((XX/100 * 10)^2 / 2!) * e^(-XX/100 * 10)
P(X = 2) = (XX/500) * e^(-XX/100 * 10)
P(X = 2) = (XX/500) * 0.000045
a) Let X be the number of cars that cross the bridge in a 1-minute period.
Since the Poisson process with a rate of µ = 600/hour, the rate of cars crossing the bridge is
λ = 600/60
= 10/min.
The probability that two cars will cross the bridge during the 1-minute period can be calculated by the following formula:
P(X = 2) = ((λ^x) / x!) * e^(-λ)
P(X = 2) = ((10^2) / 2!) * e^(-10)
P(X = 2) = 0.0045
b) Let Y be the number of cars that travel from X to Y in a 1-minute period.
Since the probability that a car travels from X to Y is p = 0.XX,
The probability that a car travels from Y to X is 1 - p.
As per the Poisson process, the probability of
λ = 10/min.
Let X be the number of cars that cross the bridge from X to Y in a 1-minute period.
Then X follows a Poisson distribution with a rate of
µ = 10*p
= XX/100 * 10.
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Can someone please help me with this? And please no links.
Answer:
1x + 15?
Step-by-step explanation:
person who answers this WITH complete explanation, formula, AND correct answer will be marked BRAINLIEST...... pls answer:
2/5 x -3/7 - 1/6 x 3/2 - 1/4 x 2/5
Please help me out! You can just give me your answer! PLEASE AND THANK YOU!
Answer:
15°, 16°, 46°, 59°
Step-by-step explanation:
1.) so the entire angle is 89°, then you take off the 44° and the 30°
89° - 44° - 30° = 15°
So the answer to 1 is x = 15°
2.) 90° - 74° = 16°
3.) 180° - 134° = 46°
4.) the angles are vertical, so they're equal:
x = 59°
hope this helps:)
A die is rolled 100 times. A 1 is rolled 20 times, a 2 is rolled 14 times, a 3 was rolled 20 times, a 4 was rolled 15 times a 5 was rolled 19 times, and a 6 was rolled 12 times.
a) What is the experimental probability of rolling a 6?
b) What is the theoretical probability of rolling a 6?
Answer:
A) Experimental probability = 0.12
B) Theoretical probability = 1/6
Step-by-step explanation:
A) Experimental probability is based on the total number of times an event occurs with respect to the total outcome of the experiment in question.
Now, we are told that the die was rolled 100 times and that 6 was gotten 12 times for the 100 rolls.
Thus;
Experimental probability = 12/100
Experimental probability = 0.12
B) Theoretical probability is the number of ways that an event can occur in relation to the total outcomes.
Here, the number of ways 6 can occur is 1 and the total outcome is 6 possible due numbers.
Thus,
Theoretical probability = 1/6
HURRY WILL MARK BRAINLIEST subtract 2x from both sides = x+2=8
Then subtract 2 from both sides and you'll get x=6.
Answer:
x=6
Step-by-step explanation:
x+2=8
subtract 2 from both sides
x=6
help pleaseeeeeeeeeee
During his shift at Pennant Sports Shop, Sam compared the items sold during last weekend's sale. There were 25 lacrosse sticks sold on Friday, 10 of which were defense sticks. There were 50 lacrosse sticks sold on Saturday, 15 of which were defense sticks. Did Pennant Sports Shop sell the same ratio of defense sticks to total lacrosse sticks on both days?
Answer: No. Pennant Sports Shop did not sell the same ratio of defense sticks to total lacrosse sticks on both days.
Step-by-step explanation:
On Friday, there were 25 lacrosse sticks sold, 10 of which were defense sticks. The ratio of defense sticks to lacrosse will be:
= 10/25 = 2/5 = 2:5
On Saturday, there were 50 lacrosse sticks sold, 15 of which were defense stick. The ratio of defense sticks to lacrosse will be:
= 15/50 = 3/10 = 3:10
Based on the calculation above, Pennant Sports Shop did not sell the same ratio of defense sticks to total lacrosse sticks on both days
Find the distance between the points ( -5,1) and (4,0) round to the nearest tenth
Answer:
9.1 units
Step-by-step explanation:
formula of a distance of two points:
[tex]\sqrt{(x1-x2)^2+(y1-y2)^2}[/tex] where x and y indicate the coordinates of the points
[tex]\sqrt{(4+5)^2 + (-1)^2} = \sqrt{81 + 1} = \sqrt{82} = 9.1 units[/tex]
Factor 30-24x
6(15-12x)
6(5x-4x)
x(5-4x)
6(5 - 4x)
Answer:
the second option
Step-by-step explanation:
pretty sure it's the second option
hope this helped ;)
Answer:
6 (5 - 4x)Step-by-step explanation:
[tex]\sf 30-24x[/tex]
30 → 6 * 5
24 → 6 * 4
[tex]\sf 6* \:5-6* \:4x[/tex]
[tex]\sf 6\left(5-4x\right)[/tex]
A: y=5x+7 b:y= 3x+5 c: y=2x+5
Answer: A: y=5x+7
Step-by-step explanation: Hope this help :D