The probability of at least 33 male lambs is approximately 74.2%, which rounds to 25.9% to the nearest tenth of a percent.
To solve this problem, we can use the binomial distribution formula:
P(X≥33) = 1 - P(X<33)
where X is the number of male lambs born out of 60, and P(X<33) is the probability that less than 33 male lambs are born.
The probability of getting a male lamb is 0.5, assuming that males and females are equally probable. So, the probability of getting exactly x male lambs out of 60 is:
P(X=x) = (60 choose x) * 0.5^60
where (60 choose x) is the number of ways to choose x items out of 60, which is calculated by the binomial coefficient formula:
(60 choose x) = 60! / (x! * (60-x)!)
Using a binomial distribution calculator or a spreadsheet program like Excel, we can find the probability of getting less than 33 male lambs:
P(X<33) = sum(P(X=x), x=0 to 32) ≈ 0.0459
Therefore, the probability of getting at least 33 male lambs is:
P(X≥33) = 1 - P(X<33) ≈ 1 - 0.0459 ≈ 0.9541
To convert this to a percentage, we can multiply by 100 and round to the nearest tenth:
P(X≥33) ≈ 95.4%
So, the answer is not one of the options given. The closest option is D) 25.9%, which is the probability of getting exactly 30 male lambs.
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A random sample of 18 observations taken from a normally distributed population produced the following data:
28. 4 27. 3 25. 5 25. 5 31. 1 23. 0 26. 3 24. 6 28. 4
37. 2 23. 9 28. 7 27. 9 25. 1 27. 2 25. 3 22. 6 22. 7
What is the point estimate of μ?
Make a 99% confidence interval for μ.
What is the margin of error of estimate for μ in part b?
Point estimate of μ [tex]26.15[/tex]
99% confidence interval: [tex](24.213, 28.087)[/tex]
Margin of error:[tex]1.437[/tex]
How to estimate μ and create a 99% confidence interval ?The point estimate of (population mean) is calculated by finding the sample mean , which is the average of the given sample data. Adding up all the observations and dividing by the sample size (18), we get:
To make a 99% confidence interval for we can use the t-distribution and the sample mean. Since the sample size is 18, we have (18 - 1) = 17 degrees of freedom. Using a t-distribution table or a calculator, we find the critical value for a 99% confidence level and 17 degrees of freedom to be approximately 2.898.
Next, we calculate the standard error (SE) using the sample standard deviation (s) divided by the square root of the sample size .Assuming the population standard deviation is unknown, we estimate it using the sample standard deviation, which is approximately 3.858.
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u = 28.8
The 99% confidence interval for μ is (27.01, 30.59).
How to solve the data28.4 + 27.3 + 25.5 + 25.5 + 31.1 + 23.0 + 26.3 + 24.6 + 28.4 + 37.2 + 23.9 + 28.7 + 27.9 + 25.1 + 27.2 + 25.3 + 22.6 + 22.7 = 518.3
The sample mean (or the point estimate of μ) is:
518.3 / 18 = 28.8
Calculate each data point's deviation from the mean, square it, and then sum up these values.
(28.4-28.8)² + (27.3-28.8)² + (25.5-28.8)² + (25.5-28.8)² + (31.1-28.8)² + (23.0-28.8)² + (26.3-28.8)² + (24.6-28.8)² + (28.4-28.8)² + (37.2-28.8)² + (23.9-28.8)² + (28.7-28.8)² + (27.9-28.8)² + (25.1-28.8)² + (27.2-28.8)² + (25.3-28.8)² + (22.6-28.8)² + (22.7-28.8)²
= 116.4
Divide by (n - 1), where n is the number of observations. In this case, n = 18, so n - 1 = 17.
So, the variance (s²) is 116.4 / 17 = 6.85 (approximately)
s = √6.85 = 2.62
CI = x ± (t * (s/√n))
CI = 28.8 ± (2.898 * (2.62/√18))
CI = 28.8 ± 1.79
So, the 99% confidence interval for μ is (27.01, 30.59).
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A triangle has side lengths of (7m – 2) centimeters, (9m - 5) centimeters, and
(6n – 1) centimeters. which expression represents the perimeter, in centimeters, of
the triangle?
The expression that represents the perimeter, in centimeters, of the triangle is (7m - 2) + (9m - 5) + (6n - 1).
How to find the perimeter of a triangle?The perimeter of a triangle is the sum of the lengths of its sides. Therefore, to find the perimeter of the triangle with side lengths (7m - 2) cm, (9m - 5) cm, and (6n - 1) cm, we need to add these lengths together.
Thus, the expression that represents the perimeter of the triangle is (7m - 2) cm + (9m - 5) cm + (6n - 1) cm.
Simplifying this expression, we get 16m + 6n - 8 cm, which is the final answer. Therefore, the perimeter of the triangle is 16m + 6n - 8 cm.
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Evaluate the integral 4 cos x2 dx dy dz, 2y by changing the order of integration in an appropriate way. CILLS
To evaluate the integral 4 cos x^2 dx dy dz, 2y by changing the order of integration, we first need to determine the limits of integration.
Integrating with respect to x first, we have:
∫∫∫ 4 cos x^2 dx dy dz, 2y
= ∫∫ [2 sin x^2]y=0 to y dz, 2y
= ∫ [2 sin x^2]z=0 to z dz, 2y
= 2y ∫ [sin x^2]z=0 to z dz
Next, integrating with respect to z, we have:
2y ∫ [sin x^2]z=0 to z dz
= 2y [cos x^2]z=0 to z
= 2y [cos z^2 - 1]
Finally, integrating with respect to y, we have:
∫∫∫ 4 cos x^2 dx dy dz, 2y
= ∫∫ 2y [cos z^2 - 1] dy dz
= ∫ [y^2(cos z^2 - 1)]z=0 to z dz
= ∫ y^2(cos z^2 - 1) dz
Therefore, we have changed the order of integration from dx dy dz, 2y to dz dy dx, and the new limits of integration are:
0 ≤ z ≤ √(π/2)
0 ≤ y ≤ √(π/2 - z^2)
0 ≤ x ≤ √(π/2 - y^2 - z^2)
We can now evaluate the integral using these new limits and the equation we derived earlier:
∫∫∫ 4 cos x^2 dx dy dz, 2y
= ∫∫∫ 4 cos x^2 dz dy dx, 2y
= ∫ from 0 to √(π/2) ∫ from 0 to √(π/2 - z^2) ∫ from 0 to √(π/2 - y^2 - z^2) y^2(cos z^2 - 1) dx dy dz
= -4/3π
Therefore, the value of the integral is -4/3π.
To evaluate the integral by changing the order of integration, first, we need to rewrite the given integral in the correct notation. Unfortunately, the provided integral seems to have some errors or missing information, making it difficult to give a complete answer.
Please provide the correct integral and the limits of integration for each variable (x, y, and z) so I can assist you better.
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You can only make four different cuboids with 12 cubes complete the table to show the dimensions
Each cuboid has a total of 12 cubes, but they have different shapes and sizes. The table is attached below.
What is the cube?A cube is a three-dimensional geometric shape that has six equal square faces, 12 equal edges, and eight vertices (corners). All the angles between the faces and edges of a cube are right angles (90 degrees), and all the edges are of equal length. A cube is a special type of rectangular prism where all the sides are equal in length, making it a regular polyhedron.
Sure, here's a table showing the possible dimensions of the four different cuboids that can be made with 12 cubes:
Cuboid Length Width Height
A 1 2 6
B 1 3 4
C 2 2 3
D 1 1 12
Note that the dimensions are given in terms of the number of cubes in each direction. For example, cuboid A has a length of 1 cube, a width of 2 cubes, and a height of 6 cubes.
Therefore, Each cuboid has a total of 12 cubes, but they have different shapes and sizes.
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Malia had 15 lb of birdseed. She fed her birds 5 lb of birdseed every day until all the birdseed was gone. For how many days did Malia feed the birdseed to her birds? A.20 days B. 3 days C.90 days D.75 days
Answer:
B
Step-by-step explanation:
15 pounds and 5 pounds per day so to figure out how many days you do division. The equation is 15÷5=3 so the answer is 3 days.
Given square ertn, what is the length of nt?
In the given square ERTN, the length of the side NT is found to be equal to 25.
The square ERTN has ER = 5x and RT = 10x - 25. Because it is a square, the sides will be equal in magnitude. So, we can write,
Length of ER = Length of RT
5x = 10x - 25
5x = 25
x = 25/5
x = 5
So, the length of ER will be 5(5) = 25 and the length of RT will be 10(5)-25 = 25.
So, finally the length of NT would also be equal to 25.
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A sphere has a radius of 12 cm. a cylinder has the same radius and has a height of 12 cm. what is the difference in their volumes in cubic cm? record your answer to the nearest hundredth. use 3.14
The difference in volumes between the sphere and the cylinder is approximately 3619.14 cubic cm.
We need to find the difference in volumes between a sphere and a cylinder with the same radius (12 cm) and the cylinder has a height of 12 cm.
Step 1: Find the volume of the sphere.
The formula for the volume of a sphere is V_sphere = (4/3)πr³.
V_sphere = (4/3) × 3.14 × (12 cm)³
V_sphere = (4/3) × 3.14 × 1728 cm³
V_sphere ≈ 9047.78 cm³
Step 2: Find the volume of the cylinder.
The formula for the volume of a cylinder is V_cylinder = πr²h.
V_cylinder = 3.14 × (12 cm)² × 12 cm
V_cylinder = 3.14 × 144 cm² × 12 cm
V_cylinder ≈ 5428.64 cm³
Step 3: Find the difference in volumes.
Difference = V_sphere - V_cylinder
Difference = 9047.78 cm³ - 5428.64 cm³
Difference ≈ 3619.14 cm³
The difference in volumes between the sphere and the cylinder is approximately 3619.14 cubic cm.
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Your friend purchased a medium pizza for 10. 32 with a 30% off coupon what is the price of the pizza with out a coupon
The price of the medium pizza without the coupon is approximately $14.74.
To find the original price of the pizza without the coupon, we can use the following formula:
Original Price = Discounted Price / (1 - Discount Percentage)
In this case, the discounted price is $10.32 and the discount percentage is 30% or 0.3. Plugging the values into the formula:
Original Price = $10.32 / (1 - 0.3)
Original Price = $10.32 / 0.7
Original Price ≈ $14.74
So, the price of the medium pizza without the coupon is approximately $14.74.
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The vector
u
u has magnitude
2
2 and direction
5
5
∘. 55
∘. If vector
v
=
−
2
u
,
v=−2u, then what is the magnitude and direction of vector
v
?
v? Write your direction in degrees in the interval
0
∘
≤
θ
<
36
0
∘. 0
∘
≤θ<
The magnitude of vector v is 4 and the direction is 125 degrees.
Given that vector u has a magnitude of 2 and a direction of 55 degrees, we can determine the magnitude and direction of vector v.
To find the magnitude of vector v, we can use the equation:
|v| = |-2u|
Since u has a magnitude of 2, we can substitute it into the equation:
|v| = |-2 * 2|
|v| = |-4|
|v| = 4
The magnitude of vector v is 4.
To find the direction of vector v, we can note that multiplying a vector by -1 (in this case, multiplying u by -2) reverses its direction. So the direction of v is the exact opposite of the direction of u.
Since the direction of u is 55 degrees, the direction of v is 55 degrees in the opposite direction. In the interval of 0 degrees ≤ θ < 360 degrees, the direction of v can be expressed as:
θ = 180 - 55
θ = 125 degrees
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The dot plot shows the number of pencils each boy has at his desk in class.
+++ hhh
1 2 3 4 5 6
number of pencils at desk
find the median for the number of pencils.
The median for the number of pencils can be found by locating the middle value on the dot plot, which appears to be 3 pencils.
What is the middle value of the number of pencils on the dot plot?The dot plot displays the number of pencils each boy has at his desk in class. By locating the middle value of the data on the dot plot, the median number of pencils can be determined to be 3.
This indicates that half of the boys have 3 pencils or fewer, while the other half have 4 pencils or more.
The dot plot provides a visual representation of the distribution of the data and allows for easy identification of the median value.
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What is 7-12 problems. The image is attached.
The number of bases, faces, edges and vertices are;
7. 2, 4, 6, 6
8. 6, 5, 10, 10
9. 1, 4, 5, 5
10. 1, 5, 7 , 7
11. 1, 5, 7, 7
12. 1, 3, 4 , 4
How to determine the numberTo determine the number, we need to take the following into considerations;
An edge is a line segment that is between faces. Vertices are angular corners where two lines or edges and more meet between faces.A face is any of the sole flat surfaces of a solid objectBase is the surface the shape stands onLearn about edges at: https://brainly.com/question/22735873
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A prospector graphed the locations of gold vein and all of the gold dust strikes in the vicinity. She positioned the gold vein at (-3,9) and the farthest gold dust strikes at (33,9). If each unit on the graph represents 1 mile then how far away from the gold vein is the farthest gold dust strike?
Answer:
Step-by-step explanation:
The gold vein is at (-3,9) and the farthest gold dust strike is at (33,9). The distance between the two points is 36 miles.
helpppp me please……….
Answer:
45°
Step-by-step explanation:
sin∠U = 5√2 / 10 = √2/2
m∠U = sin⁻¹(√2/2) = 45°
1. write the equation for each line that passes through (1, 2) and has a) slope 2/3 b) undefined slope c) m = 0 d) point (2, 3) also on the line
The equation of the line that passes through (1, 2) and (a) having a slope of 2/3 is y = (2/3)x + 4/3, (b) Having an undefined slope is x = 1, (c) having slope m = 0 is y = 2 (d) point (2, 3) also lies on the line is y = x + 1.
a) To find the equation of a line with slope 2/3 passing through (1,2), we can use the point-slope formula:
y - y1 = m(x - x1)
Substituting in the values we know, we get:
y - 2 = (2/3)(x - 1)
Expanding and simplifying:
y = (2/3)x + 4/3
So the equation of the line is y = (2/3)x + 4/3.
b) To find the equation of a line with an undefined slope passing through (1,2), we know that this line must be vertical. The equation of a vertical line passing through (1,2) can be written as:
x = 1
So the equation of the line is x = 1.
c) To find the equation of a line with slope m=0 passing through (1,2), we know that this line must be horizontal. The equation of a horizontal line passing through (1,2) can be written as:
y = 2
So the equation of the line is y = 2.
d) To find the equation of a line passing through (1,2) and (2,3), we can use the point-slope formula again:
y - y1 = m(x - x1)
Substituting in the values we know, we get:
y - 2 = (3 - 2)/(2 - 1)(x - 1)
Simplifying:
y - 2 = 1(x - 1)
y - 2 = x - 1
y = x + 1
So the equation of the line is y = x + 1.
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The table shows the part of the students in
each grade that participated in a sport this
year. Which grade had the greatest rate of participation? The least?
we see that the greatest rate of participation was in Grade 8, and the least rate of participation was in Grade 6.
How to find the grade had the greatest rate of participation?To compare the rates of participation in sports among the three grades, we can convert each percentage or fraction to a decimal and then compare the values.
Grade 6: 0.872
Grade 7: 0.87 (87% converted to decimal)
Grade 8: 0.875 (7/8 converted to decimal)
Therefore, we see that the greatest rate of participation was in Grade 8, and the least rate of participation was in Grade 6.
Answer:
Grade 8 had the greatest rate of participation.
Grade 6 had the least rate of participation.
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Suppose you have a bag of m&ms 4 green 6 yellow 7 purple 3 red. what is the probability you select a brown m&m
The probability of selecting a brown M&M from this bag is 0.
Since there are no brown M&Ms mentioned in your bag, the probability of selecting a brown M&M is 0. In probability terms, we can express this as:
Probability of selecting a brown M&M = Number of brown M&Ms / Total number of M&Ms
There are 0 brown M&Ms, and there are a total of 4 green + 6 yellow + 7 purple + 3 red = 20 M&Ms in the bag. So the probability is:
Probability = 0 / 20 = 0
Thus, the probability of selecting a brown M&M from this bag is 0, meaning it's impossible with the given information.
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Optimize: e-222-4xy-2y2 (1 point) Consider the optimization problem: Subject to 0x2 – 4xy + ly2 = 10 The method of Lagrange gives a system of three equations in three unknowns that you must solve to find the critical points. Write out those three equations (in any order). Use an l' in place of the usual .. =
To optimize the given function e - 222 - 4xy - 2y^2 subject to the constraint 0x^2 - 4xy + ly^2 = 10, we'll first set up the Lagrange function L(x, y, λ) as follows:
Langrange- L(x, y, λ) = e - 222 - 4xy - 2y^2 + λ(0x^2 - 4xy + ly^2 - 10)
Now, we'll get the partial derivatives of L with respect to x, y, and λ and set them equal to zero:
Step:1. ∂L/∂x = -4y - λ(-4y) = 0
Step:2. ∂L/∂y = -4x - 4y - 2λy = 0
Step:3. ∂L/∂λ = 0x^2 - 4xy + ly^2 - 10 = 0
These three equations represent the system of equations we need to solve to find the critical points. To reiterate, the equations are:
1. -4y + 4yλ = 0
2. -4x - 4y - 2λy = 0
3. -4xy + ly^2 - 10 = 0
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Need help on question b- the question is attached in the photo.
The height of the new player is given as follows:
210 cm.
How to calculate the mean of a data-set?The mean of a data-set is given by the sum of all observations in the data-set divided by the number of observations, which is also called the cardinality of the data-set.
Considering the frequencies and the height of the new player of x, the sum of the observations is given as follows:
198 x 1 + 199 x 3 + 200 x 2 + 201 x 5 + 202 x 2 + x = 2604 + x.
The total number of players, considering the new player, is given as follows:
1 + 3 + 2 + 5 + 2 + 1 = 14.
The mean is of 201 cm, hence the height of the new player is obtained as follows:
(2604 + x)/14 = 201
2604 + x = 14 x 201
x = 14 x 201 - 2604
x = 210 cm.
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The figure below is not drawn to scale. The height of the triangle ABC is 2 cm shorter than its base. Find the area of the shaded portions
The area of the shaded portion in the given triangle of the attached diagram with given measurements is equal to 30 square centimeters.
In triangle ABC ,
Base 'AB' = 3+ 6 + 3
= 12cm
In the attached figure.
Let the perpendicular line passing through C intersect on AB at point D.
Height of the triangle ABC 'CD' = 12 -2
= 10 cm
Area of triangle ABC = ( 1/2) × AB × CD
= ( 1/ 2) × 12 × 10
= 60 cm²
Area of triangle excluding shaded portion = ( 1/2 ) × 6 × 10
= 30cm²
Area of the shaded portion
= Area of triangle ABC - Area of triangle excluding shaded portion
= 60 - 30
= 30 square centimeters.
Therefore, the area of the shaded portion of the triangle in the attached figure is equal to 30 square centimeters.
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The above question is incomplete, the complete question is:
The figure below is not drawn to scale. The height of the triangle ABC is 2 cm shorter than its base. Find the area of the shaded portions.
Attached figure.
A normal distribution curve, where x = 70 and o = 15,
was created by a teacher using her students' grades.
What information about their performances can be obtained by analyzing the curve?
The normal distribution curve gives useful information regarding the distribution of students' grades, such as the average grade, the dispersion of grades, the likelihood of receiving a specific grade, and the presence of any outliers.
Obtainable Information's from the Distribution Curve?The normal distribution curve generated by the educator offers multiple insights into the distribution of academic achievement among her pupils. The analysis of the curve yields various informative data such as:
The parameter of central tendency for this distribution is represented by the mean, denoted as x=70, signifying that the students' mean grade is at 70.The normal distribution's degree of dispersion is represented by the standard deviation, denoted by o=15 in this context, conveying the extent of variability among grades. In this particular instance, the presence of a higher standard deviation denotes a greater degree of variability in the distribution of grades from the central tendency.Probability theory allows for the utilization of the normal distribution curve to determine the likelihood of a student achieving a particular grade. An illustration of this notion can be depicted by estimating the likelihood of an individual receiving a marks within the range of 55 and 85 through the computation of the region beneath the curve that occupies those particular values.The utilization of the normal distribution in statistical analysis is instrumental in recognizing any possible outliers within the dataset. An outlier refers to a data point that deviates considerably from the rest of the data. In the present scenario, grades whose values extend beyond two standard deviations from the mean, calculated as 70 + 215 = 100 or 70 - 215 = 40, may be considered as outliers.Learn more about distribution curve here: https://brainly.com/question/23418254
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2 Decide if each statement about r = 6 is true or false. Choose True or False for each statement. a. The equation has the same solution as fr = 3. True False
Answer:
Step-by-step explanation:
True
A gardening club records the number of new plants each member planted in a
month. Create a histogram to show the data distribution for the number of new
plants. Show your work.
2
6
8
10
New Plants This Month
.
12
.
14
16
A histogram of the data distribution for the number of new plants is shown in the image below.
How to create a histogram to show the data distribution?In this scenario and exercise, you are required to create a histogram to show the data distribution with respect to the number of new plants. First of all, we would determine the midpoint, absolute frequency, relative frequency, and cumulative frequency;
Midpoint Absolute frequency Rel. frequency
[0, 2] = (0 + 2)/2 = 1 3 + 2 = 5 0.128205
[2, 4] = (2 + 4)/2 = 3 2 + 3 = 5 0.128205
[4, 6] = (4 + 6)/2 = 5 2 + 3 = 5 0.128205
[6, 8] = (6 + 8)/2 = 7 2 0.051282
[8, 10] = (8 + 10)/2 = 9 3 + 4 = 7 0.179487
[10, 12] = (10 + 12)/2 = 11 4 + 3 = 7 0.179487
[12, 14] = (12 + 14)/2 = 13 1 + 2 = 3 0.076923
[14, 16] = (14 + 16)/2 = 15 3 + 2 = 5 0.128205
Mathematically, the relative frequency of a data set can be calculated by using this formula:
Relative frequency = absolute frequency/total frequency × 100
Relative frequency = 5/39 × 100 = 0.128205
For the cumulative frequency, we have:
0.128205
0.128205 + 0.128205 = 0.25641
0.25641 + 0.128205 = 0.384615
0.384615 + 0.051282 = 0.435897
0.435897 + 0.179487 = 0.615385
0.615385 + 0.179487 = 0.794872
0.794872 + 0.076923 = 0.871795
0.871795 + 0.128205 = 1
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Missing information:
The question is incomplete and the complete question is shown in the attached picture.
the figure above, AB is parallel to DE; (ABC = 800 and (CDE = 280. Find (DCB.(3mks)
Answer:
Step-by-step explanation:
Since AB is parallel to DE, we know that:
(ABC + BCD) = (CDE + EDC)
Substituting the given values, we get:
800 + BCD = 280 + EDC
Simplifying, we get:
BCD = EDC - 520
We also know that:
(BCD + CDE + DCE) = 180
Substituting BCD = EDC - 520 and CDE = 280, we get:
(EDC - 520 + 280 + DCE) = 180
Simplifying, we get:
EDC + DCE - 240 = 0
EDC + DCE = 240
Now we can solve for DCE in terms of BCD:
DCE = 240 - EDC
DCE = 240 - (BCD + 520)
DCE = 760 - BCD
Substituting this expression for DCE into the equation (BCD + CDE + DCE) = 180, we get:
BCD + 280 + (760 - BCD) = 180
Simplifying, we get:
1040 - BCD = 180
BCD = 860
Therefore, (DCB) = 180 - (BCD + CDE) = 180 - (860 + 280) = -960. However, since angles cannot be negative, we can add 360 degrees to this value to get:
(DCB) = -960 + 360 = -600
Therefore, (DCB) = -600 degrees.
Work out without a calculator.
√
45
×
√
1
20
Answer:
3
√
10
√
5
Step-by-step explanation:
Find the measure of each labeled angles in the rhombus below:
The measure of each labeled angles in the rhombus are ∠1 = 49°, ∠2 = 90°, ∠3 = 49° and ∠4 = 41°
Finding the measure of each labeled angles in the rhombusFrom the question, we have the following parameters that can be used in our computation:
The rhombus
By corresponding angles, we have
∠3 = 49°
∠1 = 49°
The diagonals of a rhombus bisect each other at right angles
So, we have
∠2 = 90°
Ths sum fo angles in a triangle is 180
So, we have
∠4 = 180 - 90 - 49°
Evaluate
∠4 = 41°
Hence, the measure of the angles in the rhombus are ∠1 = 49°, ∠2 = 90°, ∠3 = 49° and ∠4 = 41°
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can someone help me answer #17 using square roots?
Answer:
13, 2x^2 - 98 = 0 ........ given
2x^2= 98 ........ take to tge left side no.98
x^2 = 98/2 = 49 ..... multiple both side by radical
x = √49 = 7 ........ simplify
17, 4x^2 + 10 = 11
4x^2 + 10 = 11 4x^2 = 11- 10 = 1
4x^2 + 10 = 11 4x^2 = 11- 10 = 1 x^2 = 1/4
4x^2 + 10 = 11 4x^2 = 11- 10 = 1 x^2 = 1/4 x = √1/4 = 1/2
An exclusive Yoghurt manufacturer sells 4,000 gallons per month at a price of GHS 40 each. When the price is reduced to GHS 30 sales increase to 6,000 gallons per month.
a. Calculate the price elasticity of demand for the Yoghurts over this price range.
b. Is demand elastic, unit elastic or inelastic?
c. Calculate the change in revenue due to the change in price
a. The price elasticity of demand for the Yoghurts over this price range is -2.5
b. The demand is elastic
c. The change in revenue due to the change in price is GHS 20,000
a. The price elasticity of demand is given by the formula:
Price elasticity of demand = (Percentage change in quantity demanded) / (Percentage change in price)
The percentage change in quantity demanded is (6000 - 4000) / 4000 * 100% = 50%
The percentage change in price is (30 - 40) / 40 * 100% = -20%
Therefore, the price elasticity of demand = 50% / (-20%) = -2.5
b. Since the price elasticity of demand is greater than 1,-2.5. This means that the percentage change in quantity demanded is greater than the percentage change in price.
c. The revenue is given by the formula:
Revenue = Price x Quantity
At a price of GHS 40, the revenue is 4000 x 40 = GHS 160,000
At a price of GHS 30, the revenue is 6000 x 30 = GHS 180,000
Therefore, the change in revenue is GHS 20,000, which is an increase of 12.5%.
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"Reduce the quadratic form 2yz^2+2xz+2xy to canonical form by an
orthogonal transformation and also find rank, index and
signature."
To reduce the quadratic form 2yz^2+2xz+2xy to canonical form, we need to complete the square.
First, we factor out the coefficient of z^2 from the yz^2 term:
2yz^2 = 2z(yz)
Next, we add and subtract the square of half the coefficient of z from the resulting expression:
2z(yz + (x/y)^2 - (x/y)^2)
= 2z((y + x/y)^2/4 - (x/y)^2)
= z(y + x/y)^2/2 - zx^2/y
Now, we can see that the quadratic form can be written in the canonical form:
q(x,y,z) = (y + x/y)^2/2 - x^2/y
To find the rank, we need to count the number of non-zero eigenvalues. In this case, we have two non-zero eigenvalues, so the rank is 2.
To find the index, we need to count the number of positive, negative, and zero eigenvalues. We can see that there is one positive eigenvalue and one negative eigenvalue, so the index is 1.
Finally, to find the signature, we subtract the index from the rank. In this case, the signature is 1.
To reduce the quadratic form 2yz^2 + 2xz + 2xy to canonical form by an orthogonal transformation, we first find the matrix representation of the form. The given quadratic form can be written as Q = [x, y, z] * A * [x, y, z]^T, where A is a symmetric matrix:
A = | 0 1 1 |
| 1 0 1 |
| 1 1 2 |
Now, we find the eigenvalues and eigenvectors of A. The eigenvalues are λ₁ = 3, λ₂ = -1, and λ₃ = 0, with corresponding eigenvectors:
v₁ = [1, 1, 1]
v₂ = [-1, 1, 0]
v₃ = [-1, -1, 2]
Normalize the eigenvectors to form an orthogonal matrix P:
P = | 1/√3 1/√2 -1/√6 |
| 1/√3 -1/√2 -1/√6 |
| 1/√3 0 2/√6 |
Now, we can transform A to its canonical form using the orthogonal matrix P:
D = P^T * A * P
D = | 3 0 0 |
| 0 -1 0 |
| 0 0 0 |
So, the canonical form of the quadratic form is:
Q canonical = 3x'^2 - y'^2
The rank of the quadratic form is the number of non-zero eigenvalues in the diagonal matrix D. In this case, the rank is 2.
The index of the quadratic form is the number of positive eigenvalues in D, which is 1 in this case.
The signature of the quadratic form is the difference between the number of positive and negative eigenvalues in D. In this case, the signature is 1 - 1 = 0.
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A spring gun at ground level fires a golf ball at an angle of 45 degrees. The ball lands 10 m away.
a) What was the ball's initial speed?
b) For the same initial speed, find the two firing angles that make the range 6 m.
Recall that the Ideal Projectile Motion Equation is
r=(vo*cos(theta))ti+((vo*sin(theta)t-1/2*g*t^2)j.
Answer: a) vo=sqrt(10g)
b) theta=1/2*arcsin(3/5),
theta=pi-1/2*arcsin(3/5).
And is arcsin the same thing as sin^-1?
Yes, arcsin and sin^-1 both represent the inverse sine function.
process of finding inital speed:
a) To find the ball's initial speed, we can use the range formula for projectile motion:
R = (v₀² * sin(2θ)) / g
where R is the range (10 m),
v₀ is the initial speed,
θ is the launch angle (45 degrees), and
g is the acceleration due to gravity (9.81 m/s²).
We can solve for v₀:
10 = (v₀² * sin(90)) / 9.81
10 = (v₀²) / 9.81
v₀² = 10 * 9.81
v₀ = sqrt(10 * 9.81)
The ball's initial speed is sqrt(10 * 9.81) m/s.
b) For the same initial speed, we can find the two firing angles that make the range 6 m:
6 = (v₀² * sin(2θ)) / 9.81
Now, we can use the initial speed found in part (a):
6 = (10 * 9.81 * sin(2θ)) / 9.81
0.6 = sin(2θ)
To find the two angles, we can use the arcsin function:
θ₁ = 1/2 * arcsin(0.6)
θ₂ = π - 1/2 * arcsin(0.6)
The two firing angles are 1/2 * arcsin(0.6) and π - 1/2 * arcsin(0.6).Yes, arcsin is the same as sin^(-1);
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if you pay $ for a 20-year zero coupon bond with a face value of $, what is your annual compound rate of return?
The annual compound rate of return on this 20-year zero coupon bond is 6%. To calculate the annual compound rate of return, we need to use the following formula:
Annual Compound Rate of Return = (Face Value / Purchase Price)^(1/Number of Years) - 1
Here, the face value of the bond is $1000, the purchase price is $500, and the bond has a term of 20 years. Substituting these values in the above formula, we get:
Annual Compound Rate of Return = (1000/500)^(1/20) - 1
Simplifying this expression, we get:
Annual Compound Rate of Return = 1.06 - 1
Annual Compound Rate of Return = 0.06 or 6%
Therefore, the annual compound rate of return on this 20-year zero coupon bond is 6%.
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