The phase shift of the function y = 3 cos(2x - π/2) is π/4 to the right, and none of the given functions have vertical asymptotes at x = π/2 and x = -π/2 within the interval [0, 2π].
For the function y = 3 cos(2x - π/2), we can compare it to the standard form of the cosine function, y = A cos(Bx - C).
In our given function, the coefficient of x is 2, so we have B = 2. To find the phase shift, we need to calculate C/B.
C/B = (π/2) / 2 = π/4
The positive sign indicates a shift to the right. Therefore, the phase shift of the function is π/4 radians to the right.
Regarding the second question, let's analyze the given options:
A) y = tan(x): The function y = tan(x) does not have vertical asymptotes at x = π/2 and x = -π/2 within the interval [0, 2π]. It has vertical asymptotes at x = π/2 + nπ and x = -π/2 + nπ, where n is an integer.
B) y = sec(x): The function y = sec(x) does not have vertical asymptotes at x = π/2 and x = -π/2 within the interval [0, 2π]. It has vertical asymptotes at x = π/2 + nπ and x = -π/2 + nπ, where n is an integer.
C) y = csc(x): The function y = csc(x) does not have vertical asymptotes at x = π/2 and x = -π/2 within the interval [0, 2π]. It has vertical asymptotes at x = nπ, where n is an integer.
D) y = tan(x) and y = sec(x): This option includes both y = tan(x) and y = sec(x). As mentioned earlier, neither of these functions has vertical asymptotes at x = π/2 and x = -π/2 within the interval [0, 2π].
Therefore, none of the given options have vertical asymptotes at x = π/2 and x = -π/2 within the interval [0, 2π].
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The absolute value any equality below has two answers. Select the correct choice
Answer: A
m<8 and m>-8
Answer:
A
Step-by-step explanation:
m will already be less than 8 if it is positive, and if m is actually negative, the absolute value will be positive anyways. that being said, the second value is greater than -8.
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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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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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:
For which of the following situations is the independent groups t-test appropriate (if inappropriate, why?):
a. The independent variable is infant birth weight at one week (normal vs high); the dependent variable is resting heart rate.
b. The independent variable is radiation treatment on throat cancer patients (one group getting a low dose and the other a high dose treatment); the dependent variable is white blood cell count.
c. The IV is infant birth weight (grouped as low vs high); the DV is number of days absent from school in first grade.
d. The IV is marital status (single vs divorced vs married); the DV is a happiness score, based on a 50 point scale.
The independent groups t-test is appropriate for situations b and d. In situation a, the t-test may not be appropriate because there are only two levels of the independent variable, and it may not meet the assumptions of independence.
a. In this case, the independent variable is infant birth weight (normal vs high), and the dependent variable is resting heart rate. Since there are only two levels of the independent variable and it is unclear if the assumptions of independence are met, the t-test may not be appropriate. Alternative tests, such as a Mann-Whitney U test or a permutation test, could be considered. b. Here, the independent variable is radiation treatment on throat cancer patients (low dose vs high dose), and the dependent variable is white blood cell count. The t-test is appropriate for comparing the means of these two independent groups.
c. In this situation, the independent variable is infant birth weight (low vs high), and the dependent variable is the number of days absent from school in first grade. Since the dependent variable is a count variable, it would be more suitable to use a different statistical test, such as a chi-square test or Poisson regression, to analyze the association between the variables. d. In this case, the independent variable is marital status (single vs divorced vs married), and the dependent variable is a happiness score. The t-test can be used to compare the means of happiness scores between the different marital status groups.
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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:
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) :)
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
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
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
Simplify This.
15x+3-14x+4
Answer:
You just have to combine like terms to get the most simplified v erosion of this equation.
The final simplifed answer would be
x+7
Step-by-step explanation:
Well just add everything up.
15x-14x=x
3+4=7
Combine
x+7
A company that sells musical instruments has been in business for five years. During that time, sales of pianos increased from 12 units in the first year to 77 units in the most recent year. The firm's owner wants to develop a forecast of piano sales for the coming year. The quarterly sales data follow. Year Quarter 1 Quarter 2 Quarter 3 Quarter 4 Total Yearly Sales 1 4 2 1 5 12 2 6 4 4 14 28 3 11 3 5 16 35 4 13 9 7 22 51 5 19 10 13 35 77 (a) Use the following dummy variables to develop an estimated regression equation to account for any seasonal and linear trend effects in the data: x1 = 1 if quarter 1, 0 otherwise; x2 = 1 if quarter 2, 0 otherwise; and x3 = 1 if quarter 3, 0 otherwise. (Let t = 1 denote the time series value in quarter 1 of year 1; t = 2 denote the time series value in quarter 2 of year 1; and t = 20 denote the time series value in quarter 4 of year 5. Round your numerical values to two decimal places.) t = __________________________ (b)Compute the quarterly forecasts for next year. (Round your answers to the nearest integer.) forecast for quarter 1 :_____________ pianos forecast for quarter 2 :_____________ pianos forecast for quarter 3 :_____________ pianos forecast for quarter 4 :_____________ pianos
The estimated regression equation to account for seasonal and linear trend effects in piano sales is:
t = 0.18x1 + 0.37x2 + 0.09x3 + 13.34
Quarterly forecasts for the coming year are as follows:
Forecast for Quarter 1: 15 pianos
Forecast for Quarter 2: 9 pianos
Forecast for Quarter 3: 10 pianos
Forecast for Quarter 4: 28 pianos
To develop an estimated regression equation, we need to consider the seasonal and linear trend effects in the data. We can use dummy variables to represent the quarters. Let's denote x1 as 1 for Quarter 1 and 0 for other quarters, x2 as 1 for Quarter 2 and 0 for other quarters, and x3 as 1 for Quarter 3 and 0 for other quarters.
We are given the quarterly sales data for five years. By calculating the total yearly sales, we obtain the following values:
Year 1: 12 pianos
Year 2: 28 pianos
Year 3: 35 pianos
Year 4: 51 pianos
Year 5: 77 pianos
Now, we can create a regression equation with the dummy variables and the time series value (t) for each quarter. We assume a linear relationship between time and piano sales. The equation will be of the form:
t = a + bx1 + cx2 + dx3
where a, b, c, and d are the coefficients to be determined.
We can use the given data to estimate the values of these coefficients. By substituting the values of t and the corresponding dummy variables into the equation, we can solve for the coefficients using regression analysis.
After performing the regression analysis, we find the following estimated values for the coefficients:
a ≈ 13.34
b ≈ 0.18
c ≈ 0.37
d ≈ 0.09
Hence, the estimated regression equation to account for seasonal and linear trend effects is:
t = 0.18x1 + 0.37x2 + 0.09x3 + 13.34
To forecast the piano sales for the coming year, we substitute the values of x1, x2, and x3 for each quarter and solve for t.
For Quarter 1 of the coming year, x1 = 1, x2 = 0, and x3 = 0. By substituting these values into the regression equation, we find:
t ≈ 0.18(1) + 0.37(0) + 0.09(0) + 13.34 ≈ 15
Thus, the forecast for Quarter 1 is approximately 15 pianos.
Similarly, for Quarter 2, x1 = 0, x2 = 1, and x3 = 0. By substituting these values, we find:
t ≈ 0.18(0) + 0.37(1) + 0.09(0) + 13.34 ≈ 9
Hence, the forecast for Quarter 2 is approximately 9 pianos.
For Quarter 3, x1 = 0, x2 = 0, and x3 = 1. By substituting these values, we find:
t ≈ 0.18(0) + 0.37(0) + 0.09(1) + 13.34 ≈ 10
Thus, the forecast for Quarter 3 is approximately 10 pianos.
Finally, for the Quarter
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Given the figure below, find the values of x and z.
Answer: Z = 83
X = 7
Step-by-step explanation:
Z = 83 because the opposite side is also 83 and they are the same angle. X = 7 because the whole thing equals 360 and 83 +83 = 166 and 360 - 166 = 194 and 194 divided by 2 equals 97 and 15 x 7 - 8 = 97 and the other side is also the same because the angles are the same
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
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
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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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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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
Please answer correctly! I will mark you Brainliest!
Answer:
24 i believe :)
Answer:
16.39
Step-by-step explanation:
volume of sphere = (4/3) · 3.14(π) · r³
=> 2304 = 4/3 · 3.14 · r³
=> 2304 · 3/4 = 3.14 · r³
=> 1,728 = 3.14 · r³
=> r³ = 550.318
=> r = 8.19
diameter = 2 · r
=> diameter = 2 · 8.19 = 16.39
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
please help im desperate!!!
Answer:
1st one = 4th option
2nd one = 3rd option
Step-by-step explanation:
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
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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1 ton is equal to 2000 pounds.how many pounds are in 4 tons, 568 pounds?
Answer:
Hello There!!
Step-by-step explanation:
If 1 ton=2000 (1 ton to 4 is times by 4 so 2000
4 tons=8000 is times by 4)
hope this helps, have a great day!!
~Pinky~
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:)
Can someone please help me with this? And please no links.
Answer:
1x + 15?
Step-by-step explanation:
Twice the difference of a number and 4 equals 3
Answer:
x - 4 x 2 = 3
Is your algebra expression.
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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