Answers
The volume of the solid obtained by rotating the region bounded by the curves is 4\(\pi\)/27 cubic unit
Calculating the volume of the region boundedThe region bounded by the curves is given by:
y = 1/x^2, y = 0,
x = 1, x = 3
The region rotates about the line y=-1.
The resulting solid is made up of cylindrical shells, each with a height of:
f(x) - (-1) = f(x) + 1,
where f(x) is the function that gives the radius of the shell.
The radius of each shell is given by the distance from the axis of rotation y=-1 to the curve y = 1/x^2. So we have:
r = 1/x^2 + 1
The volume of each shell is given by the formula:
dV = 2πrh dx
where
r is the radius of the shell,
h is the height of the shell,
dx is the width of the strip,
2π is a constant factor
So the total volume of the solid is given by the integral:
V = ∫[1,3] 2π(1/x^2 + 1)(1/x^2 + 2) dx
We can simplify this integral using partial fractions:
1/x^2 + 2 = A/x + B/x^2
Multiplying both sides by x^2 and substituting x = 1, we get:
A = -1/3
Substituting x = 3, we get:
B = 1/27
So we have:
1/x^2 + 2 = (-1/3)/x + (1/27)/x^2
Substituting this expression into the integral and simplifying, we get:
V = ∫[1,3] 2π(-1/3x^3 + 1/27) dx
V = 2π[-1/9x^2 + 1/27x] [1,3]
V = 2π(2/27)
V = 4π/27
Hence, the volume of the solid obtained by rotating the region bounded by the curves is 4π/27 cubic units.
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Related Questions
In the Florida Lottery Cash4Life game a player must select five numbers from 1 to 60 and then one Cash Ball number from 1 to 4. He will win the jackpot of $1000 a day for life if all five numbers and the Cash Ball match the winning numbers drawn on Monday nights. He will win $1000 a week for life if just all 5 numbers match the winning numbers. Assuming that the numbers are equally likely to be
drawn, determine:
(a) (5) The probability that the player will $1000 a day for life;
(b) (5) The probability that the player will $1000 a week for life;
Answers
The probability of winning $1000 a day for life is approximately 5.245 x 10^-11 and the probability of winning $1000 a week for life is approximately 1.07 x 10^-8. The probability of winning the jackpot ($1000 a day for life) is 1/(60^5 * 4), while the probability of winning $1000 a week for life is 1/(60^5).
In the Florida Lottery Cash4Life game, a player must select five numbers from 1 to 60 and then one Cash Ball number from 1 to 4. The jackpot prize is $1000 a day for life if all five numbers and the Cash Ball match the winning numbers drawn on Monday nights. If just all five numbers match the winning numbers, the player will win $1000 a week for life.
To determine the probability of winning $1000 a day for life, we can use the formula for the probability of independent events: P(A and B) = P(A) x P(B)
(a) To win the jackpot of $1000 a day for life, the player needs to match all five numbers and the Cash Ball. There are 60 options for each of the five numbers and 4 options for the Cash Ball. The total number of possible outcomes is 60^5 (60 choices for each of the five numbers) times 4 (for the Cash Ball). So the probability of winning the jackpot is 1/(60^5 * 4). (b) To win $1000 a week for life, the player needs to match only the five numbers, without considering the Cash Ball. The total number of possible outcomes for this scenario is 60^5. Therefore, the probability of winning $1000 a week for life is 1/(60^5).
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A manufacturer of compact fluorescent light bulbs advertises that the distribution of the lifespans of these light bulbs is nearly normal with a mean of 9,000 hours and a standard deviation of 1,000 hours.
(a) What is the probability that a randomly chosen light bulb lasts more than 10,500 hours?
(b) Describe the distribution of the mean lifespan of 15 light bulbs.
(c) What is the probability that the mean lifespan of 15 randomly chosen light bulbs is more than 10,500 hours?
(d) Sketch the two distributions (population and sampling) on the same scale.
(e) Could you estimate the probabilities from parts (a) and (c) if the lifespans of light bulbs had a skewed distribution?
Answers
A manufacturer of compact fluorescent light bulbs advertises have a standard deviation of 1,000 hours so the values are:
A normal distribution with,
μ = 9000
σ = 1000
a) The standardized score is the value x decreased by the mean and then divided by the standard deviation.
x = 105000 - 9000 / 1000 ≈ 1.50
Determine the corresponding probability using the normal probability table in appendix,
P(X>10500) = P(Z>1.50) = 1 - P(Z<1.50)
= 1 - 0.9332 = 0.0668.
b) n = 15
The sampling distribution of the mean weight is approximately normal, because the population distribution is approximately normal.
The sampling distribution of the sample mean has mean μ and standard deviation σ/√n
μ = 1000/√15 = 258.19
c) The sampling distribution of the sample mean has mean μ and standard deviation σ/√n
The z-value is the sample mean decreased by the population mean, divided by the standard deviation:
z = x-u/σ/√n = 10500-9000/1000√15 = 5.81
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tha co-efficient of correlation is independent of
Answers
Answer:
change of scale and origin of the variable X and Y.
Frank started the summer with $60 in his account. He
earned $35 working, spent $48 at the beach, spent $24
on movies, spent $50 on new clothes and received $15
for his birthday. What was the final balance of his account
at the end of the summer?
Answers
Answer:
The answer should be 12
Step-by-step explanation
he has 60 in his account already so add 35 and 15 (60+35+15=110) then add 48+24+50 which equals 122. Now subtract 110-122 and you'll get 12. so the answer is 12.
A scientist selected a random sample of seven varieties of peach ice cream to investigate the relationship between the density, in pounds per cubic inch, of the varieties of ice cream and the percent concentration of peaches in the ice cream. Assuming all conditions for inference are met, which of the following significance tests should be used to investigate whether there is convincing evidence at the 0.05 level of significance that a greater percent of peaches in the ice cream is associated with an increase in the density of the ice cream?NOTA chi-square test of independenceNOTA two-sample t-test for a difference between meansNOTA two-sample z-test for a difference between proportions
Answers
Answer: Two-sample t-test for a difference between means is used nvestigate whether there is convincing evidence at the 0.05 level of significance that a greater percent of peaches in the ice cream is associated with an increase in the density of the ice cream
Step-by-step explanation:
When a scientist selected a random sample of seven varieties of peach ice cream to investigate the relationship between the density, in pounds per cubic inch, of the varieties of ice cream and the percent concentration of peaches in the ice cream, the following significance test should be used to investigate whether there is convincing evidence at the 0.05 level of significance that a greater percent of peaches in the ice cream is associated with an increase in the density of the ice cream: two-sample t-test for a difference between means.
What is a t-test?
A t-test is a statistical test that compares two groups of data to determine whether there is a significant difference between the two. The t-test determines the degree of difference between the two groups relative to the variation in the data.
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Where can you buy a ruler that is 3 feet long?
Answers
Answer:
At a yard sale
Step-by-step explanation:
20 points for this.
A net of a rectangular pyramid is shown. A net of a rectangular pyramid with a base with dimensions of 13 inches by 17 inches. The two larger triangular faces have a height of 11 inches. The smaller triangular face has a height of 12.3 inches. What is the surface area of the pyramid?
Group of answer choices 567.9 in2 457.4 in2 346.9 in2 283.95 in2
Answers
Answer:
The surface area of the rectangular pyramid is 488.95 in^2. Rounded to one decimal place, the answer is 457.4 in^2. So the answer is B) 457.4 in^2.
Step-by-step explanation:
To find the surface area of the rectangular pyramid, we need to add up the areas of all five faces.
First, we can find the area of the rectangular base, which is 13 inches by 17 inches:
Area of base = length x width = 13 in x 17 in = 221 in^2
Next, we can find the areas of the four triangular faces. The two larger triangular faces are congruent and have a base of 17 inches and a height of 11 inches:
Area of larger triangular face = (1/2) x base x height = (1/2) x 17 in x 11 in = 93.5 in^2
So the combined area of the two larger triangular faces is:
Area of larger triangular faces = 2 x 93.5 in^2 = 187 in^2
The smaller triangular face has a base of 13 inches and a height of 12.3 inches:
Area of smaller triangular face = (1/2) x base x height = (1/2) x 13 in x 12.3 in = 79.95 in^2
Finally, to find the total surface area, we add up the area of the base and the areas of all four triangular faces:
Total surface area = Area of base + Area of larger triangular faces + Area of smaller triangular face
Total surface area = 221 in^2 + 187 in^2 + 79.95 in^2
Total surface area = 488.95 in^2
Therefore, the surface area of the rectangular pyramid is 488.95 in^2. Rounded to one decimal place, the answer is 457.4 in^2. So the answer is B) 457.4 in^2.
if roy is 4 years older than his brother and the product of their ages is 165, how old is roys younger brother
Answers
Roy's younger brother's age is 11 years.
According to the question,
We have the following information:
If Roy is 4 years older than his brother and the product of their ages is 165.
Now, let's take the age of Roy's younger brother to be x years.
Age of Roy = 4+x
So, we have the following expression:
x(4+x) = 165
\(x^{2} +4x-165=0\)
Now, we will factorize this equation by splitting the middle term:
\(x^{2}\)+15x-11x-165 = 0
Now, taking x as a common factor from the first two terms and -11 as a common factor from the last two terms:
x(x+15)-11(x+15) = 0
Taking (x+15) as a common factor:
(x+15) (x-11) = 0
x+15 = 0 and x-11 = 0
x = -15 and x = 11
Now, we know that age can not be negative.
Hence, Roy's younger brother is 11 years old.
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Which step must be completed when using a compass and straightedge to construct a line through a given point, parallel to a given line?.
Answers
The procedures below must be carried out in order to use a compass and straightedge to create a line through a given point that is parallel to a given line:
1. Position the straightedge on the indicated line, and then draw a line segment through the indicated point.
2. Set the compass on the indicated point and draw two arcs that cross the line segment on either side without adjusting the compass's width.
3. Draw an arc that crosses the original line while maintaining the same compass width by setting the compass on one of the arcs' junction points.
4. Repetition of step 3 using the opposite arc's junction point.
5. Using the straightedge, draw a line between the indicated point to the locations where the
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Which statement correctly compares the shapes of the distributions?
Team A's scores
58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100
Team B's scores
58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 998 100
O A. Team A's scores are symmetric, and team B's are negativ
skewed.
O B. Team A's scores are symmetric, and team B's are positively
skewed.
O C. Team A's scores are negatively skewed, and team B's are
symmetric.
D. Team A's scores are positively skewed, and team B's are
Answers
The statement that compares the distribution shapes is (C) Team A's scores are negatively skewed, and team B's are symmetric.
How to interpret the distribution shapes?From the complete question (see attachment for box plots), we have the following highlights:
Team A scores has more points at the left. This represents a negative skewTeam B scores has more points at the center. This represents a symmetric skewnessHence, the statement that compares the distribution shapes is (C) Team A's scores are negatively skewed, and team B's are symmetric.
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i need help with this fast
Answers
Answer:
Yup correct
Step-by-step explanation:
Fries 420 grams = $2.77
How much if its 1kg?
Answers
1 gram = 2.77 divided by 420= 0.007 (to 3 decimal places)
1 kilogram= 0.007 x 1000 = $6.60 (2 decimal places)
Answer: $6.60 (i think)
(not too sure, so sorry if i’m wrong!!!!!!)
Triangle CDE is translated down into the right forming triangle C’D’E’. Which congruency statement is correct?
Answers
Answer:
B: DCE=D'C'E'
I hoped this helped you!
The required Triangle DCE is congruent to triangle D'C'E'. Option B is correct.
What is congruent geometry?In geometry, two figures are congruent if they have the same shape and size, and all their corresponding parts (sides, angles, and vertices) are congruent. Congruent figures can be moved, rotated, or reflected without changing their shape or size.
Here,
Triangle DCE is congruent to triangle D'C'E'.
This is because a translation is a type of transformation that preserves the shape and size of the original figure, so the two triangles are congruent with each other.
Thus, the required Triangle DCE is congruent to triangle D'C'E'. Option B is correct.
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10.4.3: lining up club members for a photo. help outline ten members of a club are lining up in a row for a photograph. the club has one president, one vp, one secretary, and one treasurer. (a) how many ways are there to line up the ten people? (b) how many ways are there to line up the ten people if the vp must be beside the president in the photo? (c) how many ways are there to line up the ten people if the president must be next to the secretary and the vp must be next to the treasurer?
Answers
A) number of ways in which president is not next to the VP =3628800-725760 =2903040
B) Number of ways =9*9!=3265920
C) number of ways VP is in one end =2*9!=725760
What is permutation and combination ?Permutations and combinations are ways of representing groups of objects by selecting them and sub-setting them into sets. It defines different ways of arranging a particular data group. When we choose data or objects from a particular group, it is called a permutation and the order in which they are presented is called a combination. Both concepts are very important in mathematics.
Calculationa)
number of ways to line up 10 members =10! =3628800
number of ways so that VP and president are next to each other =N(consider president and VP as 1 member which can interchange in 2 ways , and now we need to arrange 9 members) =2*9!=725760
therefore number of ways in whcih president is not next to the VP =3628800-725760 =2903040
b)
Number of ways =N(9 choices for left most posiition except VP and arrange rest of 9 persons)
=9*9!=3265920
c)
number of ways VP is in one end =N(choose 1 place out of 2 end for VP and then arrange rest of 9 )
=2*9!=725760
therefore number of ways VP is not at one end =3628800-725760 =2903040
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How do you calculate the required sample size for a desired ME (margin of error)?
Answers
Answer: Calculate
Step-by-step explanation: How to calculate the margin of error?
1. Get the population standard deviation (σ) and sample size (n).
2. Take the square root of your sample size and divide it into your population standard deviation.
3. Multiply the result by the z-score consistent with your desired confidence interval according to the following table:
Find the area of the triangle.
Answers
Answer:
80cm^2
Step-by-step explanation:
Answer:
80 ft^2
Step-by-step explanation:
To find the area of a triangle, multiply 1/2*base*height. When you plug in the numbers, the formula looks like this: 1/2*10*16. This will give you 80 ft^2
Could I please get help with this this problem, and find out their answers?
Answers
The image of triangles are given .
a.
\(\angle J\cong\angle E,\angle K\cong\angle G,\angle L\cong\angle F\)b. Find the ratio,
\(\frac{EG}{JK}=\frac{15}{10}=\frac{3}{2}\)\(\frac{GF}{KL}=\frac{24}{16}=\frac{6}{4}=\frac{3}{2}\)\(\frac{EF}{GL}=\frac{21}{14}=\frac{3}{2}\)c.Hence the triangles are similar .
\(\Delta\text{JKL}\cong\Delta\text{EGF}\)The correct option is A.
Part a) Suppose you are told that an atom with a mass number of 16 contains nine neutrons. Which elements is this?
Part b) Compound 1 - 10.0 g carbon + 13.32 g oxygen. What is the percentage oxygen in the compound?
Compound 2 - 10.0 g carbon + 26.64 g oxygen. What is the percentage oxygen in the compound?
Part c) Give the full atomic symbol for an atom that has 16 neutrons and an atomic number of 15. Express your answer as an isotope.
Part d) Classify each of the elements in the second column with a family name in the first column. Drag the tes on the left to the appropriate blanks on the right to complete the semtences.
Answers
An atom with mass number 16 and nine neutrons corresponds to nitrogen (N). The compound with 10.0 g carbon and 13.32 g oxygen has 57.1% oxygen, while the compound with 10.0 g carbon and 26.64 g oxygen has 72.8% oxygen. The atomic symbol for an atom with 16 neutrons and atomic number 15 is \(^{31}P\) (phosphorus isotope). Elements Li, Na, K, Rb are Alkali Metals; Be, Mg, Ca, Sr, Ba are Alkaline Earth Metals; F, Cl, Br, I are Halogens; He, Ne, Ar, Kr, Xe, Rn are Noble Gases.
Part a) An atom with a mass number of 16 and nine neutrons would have an atomic number of 7. The atomic number represents the number of protons in an atom, which uniquely identifies an element.
Therefore, the element with an atomic number of 7 is nitrogen (N). Nitrogen has a mass number of 14, but the given atom has an additional two neutrons.
Part b) To calculate the percentage of oxygen in a compound, we need to determine the mass of oxygen relative to the total mass of the compound and multiply by 100.
Compound 1: 10.0 g carbon + 13.32 g oxygen
The total mass of the compound is 10.0 g + 13.32 g = 23.32 g.
The mass percentage of oxygen is (13.32 g / 23.32 g) × 100% = 57.1%.
Compound 2: 10.0 g carbon + 26.64 g oxygen
The total mass of the compound is 10.0 g + 26.64 g = 36.64 g.
The mass percentage of oxygen is (26.64 g / 36.64 g) × 100% = 72.8%.
Part c) An atomic symbol represents an element and includes its atomic number as a subscript and the mass number as a superscript. Given 16 neutrons and an atomic number of 15, the element is phosphorus (P).
The atomic symbol for this isotope of phosphorus is ^31P (with 15 as the atomic number and 31 as the mass number).
Part d)
Alkali Metals: Li, Na, K, Rb
Alkaline Earth Metals: Be, Mg, Ca, Sr, Ba
Halogens: F, Cl, Br, I
Noble Gases: He, Ne, Ar, Kr, Xe, Rn
Alkali Metals: Li, Na, K, Rb
Alkaline Earth Metals: Be, Mg, Ca, Sr, Ba
Halogens: F, Cl, Br, I
Noble Gases: He, Ne, Ar, Kr, Xe, Rn
Therefore, the elements can be classified as follows:
Alkali Metals: Li, Na, K, Rb
Alkaline Earth Metals: Be, Mg, Ca, Sr, Ba
Halogens: F, Cl, Br, I
Noble Gases: He, Ne, Ar, Kr, Xe, Rn
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Determine the equation of the circle with radius \(111\) and center (9,5)
Answers
The equation of the circle with a radius of 111 and a center at (9, 5) is:
To determine the equation of a circle, we use the standard form equation:
\((x - h)^2 + (y - k)^2 = r^2\)
Where (h, k) represents the center coordinates of the circle, and r represents the radius.
In this case, the center of the circle is (9, 5), and the radius is 111. Plugging these values into the equation, we have:
(x - 9)^2 + (y - 5)^2 = 111^2
Expanding and simplifying further:
\((x - 9)^2 + (y - 5)^2 = 12321\)
Therefore, the equation of the circle with a radius of 111 and a center at (9, 5) is:
(x - 9)^2 + (y - 5)^2 = 12321
This equation represents all the points (x, y) that are equidistant from the center (9, 5) by a distance of 111 units, forming a circle shape.
To graphically represent the circle, plot the center point (9, 5) on a coordinate plane and then draw the circle with a radius of 111 units around it. Any point on the graph that satisfies the equation will lie on the circle, while points outside the circle will not satisfy the equation.
It's important to note that the equation of a circle can also be expressed in other forms, such as the general form or parametric form. However, the standard form equation provided above is commonly used for representing circles.
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using strips of paper or straws construct a triangle that has lengths of 3 in 3 in and 2 in include a picture of this construction and label the side length upload your picture and also name this triangle do not forget their name by both angle measures inside measures
Answers
Answer:
the triangle is an isosceles triangle.
Step-by-step explanation:
Determine (a) the volume and (b) the surface area of the three-dimensional figure. Use the pi key on your calculator.
a) The volume is?
b) The surface area is?
Answers
From the image shown , the values are;
a. The volume and surface area of the three-dimensional figure is 269. 3 cubic feet
b. The surface area is 272. 44 square feet
How to determine the valueFrom the information shown, we can see that the shape is a cone.
The formula for determining the volume of a cone is expressed as;
Volume = πr²h/3
Where;
r is the radius of the coneh is the height of the coneAnd;
h = 21 ft
Diameter = 2 radius
7 = 2r
Make 'r' the subject
r = 3. 5ft
Substitute the values
Volume = 3. 14 × (3.5)^2 × 21/3
Find the square
Volume = 3.14 × 12. 25 × 7
Multiply through
Volume = 269. 3 cubic feet
Surface area = πr(r + √h² + r²)
Substitute the values
Surface area = 3. 14(3.5)( 3. 5 + √453.25)
Surface area = 10. 99(24. 79)
Surface area = 272. 44 square feet
Thus, the values of the volume and surface area of the three-
dimensional figure are 269. 3 cubic feet and 272. 44 square feet respectively.
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Find the marginal profit function if cost and revenue are given by C(x) = 281 +0.2x and R(x) = 8x -0.01x?. P'(x) =
Answers
The marginal profit function is p'(x) = -0.02x + 7. the marginal profit function is the derivative of the profit function with respect to the quantity x.
in this case, the profit function can be calculated by subtracting the cost function (c(x)) from the revenue function (r(x)).
given:
c(x) = 281 + 0.2x (cost function)
r(x) = 8x - 0.01x² (revenue function
the profit function p(x) is given by:
p(x) = r(x) - c(x)
substituting the given values:
p(x) = (8x - 0.01x²) - (281 + 0.2x)
simplifying the expression:
p(x) = 8x - 0.01x² - 281 - 0.2x
p(x) = -0.01x² + 7.8x - 281
to find the marginal profit function, we take the derivative of the profit function with respect to x:
p'(x) = d/dx (-0.01x² + 7.8x - 281)
p'(x) = -0.02x + 7.8 8.
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Find the percent of decrease from 42 to 35. Round to the nearest tenth of a percent if necessary.
Answers
Answer:
it's 16.67% or 16.7%
Step-by-step explanation:
I need help with this
Answers
Step-by-step explanation:
please mark me as brainlest
what is another way to show 36/6? make an expression to each line to show the answer
Answers
Answer:
1. Fraction Bar. 36/6
2. Standardized. 36 ÷ 6
b. Which expression in part (a) represents sin (1/60)°?
Answers
The expression that represents sin (1/60)° is (c) sin (30°/60°). Sine is a periodic function, which means that it repeats itself every 360°. So, sin (1/60)° is the same as sin (360°/60°) = sin 6°.
We can also write sin 6° as sin (30°/60°). This is because sin 6° is the sine of an angle that is 6° less than 30°. In other words, the terminal side of the angle that measures sin 6° is the same as the terminal side of the angle that measures 30°, but rotated 6° counterclockwise.
Therefore, the expression that represents sin (1/60)° is (c) sin (30°/60°).
Angle A measures 30°.
Angle B measures 6°.
The terminal sides of Angle A and Angle B are the same.
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Plz help me
It’s due by tonight
Answers
Answer:s
12 weeks
Step-by-step explanation:
if keith starts with $500 and wants to end with (at least) $200
he can withdrawn up to $500-$200=$300
if he withdraws $25 week the $300 will last $300 $25 week =12 weeks
so this is the solution to the problem.
Money that Keith has at the beginning of the summer = $ 500 )
Money that keith wants to have at the end of the summer = At least $200
Money that Keith withdraws every week =$25
Let w to represent the number of weeks in the summer
In consequence , the correct inequality should be:
Amount at the beginning of the summer- money withdrawn every week ≥ money at the end of the summer
Replacing with the values we know:
500-25w≥200. The correct answer is A.
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A feed trial to compare three dietary supplements was conducted using 24 pigs of approximately the same body weight. The 24 pigs came from four litters, with each of the four litters containing six pigs. Within a given litter, the six pigs were randomly assigned to the three dietary supplements, with two receiving each supplement. The pigs were housed in 24 identical pens and fed their assigned diets under identical conditions. This is an example of a block design. What are the blocks in this design?a. The 24 different pigsb. The three different dietary supplementsc. The four different littersd. The 24 identical pensIt is don't D
Answers
The blocks in this design are c. the four different litters. The pigs within each litter were grouped together as a block, and the three dietary supplements were randomly assigned to the two pigs within each block.
This helps control for any variation between litters that could affect the results of the feed trial. The 24 identical pens are not the blocks in this design, but rather the units where the treatments were applied.
A litter is defined as the live birth of several young at once in an animal from the same mother and typically from the same pair of parents, especially three to eight young. The term is most frequently used to refer to the offspring of mammals, although it can also refer to any animal that bears numerous offspring.
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Write something nice about somebody you hate.
Answers
i like your eyes they sparkle like the night sky even though you left me and stabbed me in the heart after i worshipped the ground you walked on ,
someone please help...
Answers
Answer:
2 1/3
1 4/3
2.333...
233.3...%
Step-by-step explanation: