The given task involves analyzing the "PlantGrowth" dataset in R. The analysis includes plotting the density of weight, checking the normality assumption using QQ-plot, performing a one-way ANOVA analysis, and checking the assumptions of ANOVA.
Firstly, the density plot of weight can be generated using the ggplot2 library in R. The shape of the density plot can provide insights into the underlying distribution of the weight variable. Secondly, the QQ-plot can be used to visually assess whether the weight variable follows a normal distribution. If the points on the QQ-plot lie approximately on a straight line, it suggests that the weight variable is normally distributed. Thirdly, the mean and variance of the weight variable can be calculated using the mean() and var() functions in R, respectively. These descriptive statistics provide information about the central tendency and spread of the weight variable.
Fourthly, a boxplot of weight versus group can be created using ggplot2, which allows for visualizing the distribution of weight across different treatment groups. The boxplot can reveal differences in the median, spread, and potential outliers among the groups. Fifthly, a one-way ANOVA analysis can be performed using the aov() function in R to test whether there are significant differences in weight among the treatment groups.
The ANOVA output provides information about the F-statistic, degrees of freedom, p-value, and effect sizes, which can be used to draw conclusions about the group differences. Lastly, the assumptions of ANOVA, such as normality, homogeneity of variances, and independence, can be assessed through visualization techniques like QQ-plots and residual plots, as well as statistical tests like the Shapiro-Wilk test for normality and Levene's test for homogeneity of variances. These steps ensure the validity of the ANOVA results and interpretations.
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___ divided by 3x3 =12
Answer:
108
Step-by-step explanation:
Answer:
108 hope this helps! :)
I need some answer plssss help me
A. Find the unknown term of each proportion.
1. Answer: X=5
Step-by-step explanation:
x/7=15/21 ==> 21x=7(15) ==> 21x=105
21x/21=105/21 ==> x=5
2. Answer: X=40.5
Step-by-step explanation:
27/x=x/3 ==> 27(3)=x(x) ==> 81=2x
81/2=2x/2 ==> x=40.5
3. Answer: X=16
Step-by-step explanation:
8/x+2=4/9 ==> 8(9)=4(x+2) ==> 72=4x+8
72-8=4x+8-8 ==> 64=4x ==> 64/4=4x/4 ==> x=16
4. Answer: X=8
Step-by-step explanation:
3/x=30/80 ==> 3(80)=30x ==> 240=30x
240/30=30x/30 ==> x=8
5. Answer: M=56
Step-by-step explanation:
m/8=m+7/9 ==> 9m=8(m+7) ==> 9m=8m+56
9m-8m=8m-8m+56 ==> m=56
B. Solve the following.
1. Answer: 80 and 100
Step-by-step explanation:
4:5 ==> 4x+5x=180 ==> 9x=180
9x/9=180/9 ==> x=20
4(20)= 80 5(20)=100
2. Answer: 235.29
Step-by-step explanation:
3400/8 ==> 425
100,000/425= 235.29
please answer and explain and i’ll give brainliest
Step-by-step explanation:
I really want to help you but there's no question.
help i really need help WILL GIVE BRAINLIEST
Answer:
3 and 4
Step-by-step explanation:
A triangle has sides with lengths of 14 feet, 48 feet, and 50 feet. Is it a right triangle?
Answer:
Yes, it is a right triangle
Step-by-step explanation:
Answer: Yes it is
Note that a right triangle is a triangle with a right angle.
A random poll of 800 middle school students found that 9% are ESL (English as a Second Language) learners. Estimate the actual percentage of ESL learners by constructing a 95% confidence interval. Separately, a local school board member claimed that 15% of students are ESL learners. Is the board member's claim plausible? The 95% confidence interval is 7% to 11% and, yes the board member's claim of 15% is plausible since his claim is inside of the confidence interval The 95% confidence interval is 7% to 11% and, no the board member's claim of 15% is not plausible since his claim is outside of the confidence interval The 95% confidence interval is 9% and, no the board member's claim of 15% is not plausible since his claim is outside of the confidence interval The 95% confidence interval is 0% to 18% and, yes the board member's claim of 15% is plausible since his claim is inside of the confidence interval QUESTION 26 A stationary time series is one whose properties depend on the time the series is observed and are likely to exhibit trends and seasonality. True False
The given statement "A stationary time series is one whose properties depend on the time the series is observed and are likely to exhibit trends and seasonality" is false.
The confidence interval provides a range of values in which the actual value of a population parameter is likely to fall.
A random poll of 800 middle school students found that 9% are ESL (English as a Second Language) learners.
The confidence interval estimates the actual percentage of ESL learners by constructing a 95% confidence interval.
Let us calculate the 95% confidence interval to check the board member's claim.
The point estimate is:9%
The margin of error is calculated as:
ME=1.96×√(pˆ(1−pˆ)/n)=1.96×√(0.09(1−0.09)/800)=0.0228
Since the margin of error is small enough relative to the point estimate, we can approximate the confidence interval using:
pˆ±ME=0.09±0.0228
=(0.0672,0.1128)
The 95% confidence interval is (0.0672, 0.1128).
Now we can check if the board member's claim of 15% is plausible or not:
15% falls outside the 95% confidence interval of (0.0672, 0.1128).
Hence, we can conclude that the board member's claim of 15% is not plausible since his claim is outside of the confidence interval.
False is the correct answer to the given question.
A stationary time series is one whose properties do not depend on the time the series is observed and do not exhibit trends and seasonality.
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What is the greatest common factor of 24, 40 and 32?
A. 1
B. 2
C. 4
D. 8
Answer:
your answer is D. 8
Step-by-step explanation:
please help!!
Explain how you could figure out the formula for the surface area of a cylinder if all you knew was the formula for surface area of a right rectangular prism. k
If you know the formula for the surface area of a right rectangular prism, you can use that knowledge as a basis to derive the formula for the surface area of a cylinder.
Deriving the surface area of a cylinder from the surface area of a right rectangular prismRecall the formula for the surface area of a right rectangular prism:
SA_prism = 2lw + 2lh + 2wh
where l, w, and h represent the length, width, and height of the prism, respectively.
Consider a cylinder as a special case of a prism with a circular base and a height. The circular base can be thought of as a rectangle with a length equal to the circumference of the base (2πr) and a width equal to the height (h) of the cylinder.
The curved surface of the cylinder can be "unrolled" and flattened to form a rectangle, with the length equal to the circumference of the base (2πr) and the width equal to the height (h) of the cylinder.
Thus, the surface area of the curved part of the cylinder is equal to the surface area of the rectangular prism with dimensions 2πr and h.
Thus, the formula for the surface area of the cylinder can be derived as follows: SA_cylinder = 2πrh.
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Please can someone help me?
Answer:
weight of a water=0.5
spherical ball is filled With water=0.95.
so dear for
22\7*0.5*0.95=1.49
Just need the answer of what the surface area is
Answer:
54
Step-by-step explanation:
need help please.........
Answer:
Yes. it is 5. bc u need to trust me
the ages of all the patients in the isolation ward of the hospital are 38, 26, 13, 41, and 22. what is the population variance?
To find the population variance of the ages of patients in the isolation ward, we need to calculate the average of the squared deviations from the mean.
To find the population variance, we follow these steps:
Find the mean (average) of the ages:
Mean = (38 + 26 + 13 + 41 + 22) / 5 = 28
Calculate the deviation of each age from the mean:
Deviation1 = 38 - 28 = 10
Deviation2 = 26 - 28 = -2
Deviation3 = 13 - 28 = -15
Deviation4 = 41 - 28 = 13
Deviation5 = 22 - 28 = -6
Square each deviation:
Deviation[tex]1^2[/tex] = [tex]10^2[/tex] = 100
Deviation[tex]2^2[/tex] = [tex](-2)^2[/tex] = 4
Deviation[tex]3^2[/tex] = [tex](-15)^2[/tex] = 225
Deviation[tex]4^2[/tex] = [tex]13^2[/tex] = 169
Deviation[tex]5^2[/tex] = [tex](-6)^2[/tex] = 36
Find the average of the squared deviations:
Variance = (100 + 4 + 225 + 169 + 36) / 5 = 106 / 5 = 21.2
Therefore, the population variance of the ages of patients in the isolation ward is 21.2.
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LEGIT BEG PLEASE HELP
Answer:
help with what problem?
Step-by-step explanation:
Answer:
Sure
Step-by-step explanation:
Use the graph of the function f(x) - (x + 1)2 + 2. Identify the vertex and
the axis of symmetry.
Answer:
x = –1 AND (-1,2)
Step-by-step explanation:
The axis of symmetry is x = –1
and the vertex is (-1,2)
i know this because the graph is on -1, 2
and I also took the quiz hope it helps ;)
The vertex of the function is (-1, 2).
The axis of symmetry of the function is -1.
What is a function?A function is a relationship between inputs where each input is related to exactly one output.
Example:
f(x) = 2x + 1
f(1) = 2 + 1 = 3
f(2) = 2 x 2 + 1 = 4 + 1 = 5
The outputs of the functions are 3 and 5
The inputs of the function are 1 and 2.
We have,
f(x) = (x + 1)² + 2
f(x) = x² + 2x + 1 + 2
f(x) = x² + 2x + 3
Now it is in the form of ax² + bx + c
a = 1
b = 2
c = 3
The coordinate of the Vertex of the function is (-b/2a, f(x)) = (x, y)
-b/2a = -2/2 = -1
f(x) = (-1)² + 2 x (-1) + 3 = 1 - 2 + 3 = 4 - 2 = 2
Vertex = (-1, 2)
The axis of symmetry is the x-coordinate of the vertex.
= -1.
Thus,
Vertex is (-1, 2).
The axis of symmetry is -1.
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with 09) Let x, y be random variables joint probability density function f(x,y) = K (2x+ +y) of as 4 of y=2 443 Find K and (Hind : draw P CY
K = 1/4, fY(y) = (8 - Y)/9
Random variables X and Y have a joint probability density function f(x,y) = K(2x+y) where 0<=x<=1, 0<=y<=2 and f(x,y) = 0 elsewhere. Also, Y = 2^(-X) + 3. Let's determine the value of K.
Determination of K
The probability density function f(x,y) must satisfy the following condition:
i.e., the integral of f(x,y) over the entire range of (x,y) should be equal to 1.
f(x,y) = 0 elsewhere implies that f(x,y) = 0 for x<0 and x>1 and y<0 and y>2. Hence, the range of integration should be [0,1] for x and [0,2] for y.
The integral of f(x,y) over the entire range of (x,y) can be expressed as follows:
[tex]∫∫K(2x+y)dydx = 1[/tex]
On integrating with respect to y first, we get:
[tex]∫(2x+y)dy = [2xy + (1/2)y^2][/tex]evaluated from 0 to 2
= 4x + 2
On integrating with respect to x, we get:
[2x^2 + 2x] evaluated from 0 to 1
= 4
On equating the integral value with 1, we get:
[tex]4K = 1K = 1/4[/tex]
Determination of probability density function of Y
We have [tex]Y = 2^(-X) + 3[/tex]. Therefore, for a given value of Y, the range of X can be determined as follows:
[tex]2^(-X) = Y - 3= > X = -log2(Y-3)[/tex]
Hence, the probability density function of Y can be obtained as follows:
[tex]fY(y) = ∫f(x,y)dxfY(y) = ∫f(x,2^(-X) + 3)dx[/tex]
From the given expression, we can observe that f(x,y) = 0 elsewhere implies that f(x,2^(-X) + 3) = 0 for x<0 and x>1 and y<3 and y>2. Also, the range of integration for x can be determined as follows:
For y<=3, X>=-log2(y-3). For y=2, the minimum value of X can be obtained by taking the limit as y tends to 2 from the right. The minimum value of X is therefore equal to [tex]-∞[/tex]. Therefore, the range of integration for x is [tex][-∞,1].[/tex]
fY(y) =[tex]∫f(x,2^(-X) + 3)dx = ∫(1/4)(2x + 2^(-X) + 3)dx[/tex]
fY(y) = (1/4)(x^2 - 2^(-X)x + 3x) evaluated from [tex]x=-∞ to x=1[/tex]
fY(y) = [tex](1/4)(1 - 2^(log2(Y-3)) + 3)[/tex]= (8 - Y)/9
Let's draw the probability density function of Y. The probability density function of Y is as follows:
fY(y) = (8 - Y)/9
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with 09) Let x, y be random variables joint probability density function f(x,y) = K (2x+ +y) of as 4 of y=2 443 Find K and (Hind : draw P CY<SX ) a picture )
given the system of equations x 3y z = −2 2x 5y z = −5 x 2y 3z = 1 . the determinant of the matrix of coefficients is −3. the value of z in the solution set is:: (a) z=−2/3 (b) z=5/3 (c) z=4/3 (d) z=−2 (e) None of the above
The value of z in the solution set is approximately -8.33 for the determinant of the matrix of coefficients is −3, Option E is the correct answer.
To solve the system of equations, we can use the method of determinants. The value of z can be determined by finding the determinant of the matrix of coefficients.
The given system of equations can be represented as:
| 1 3 1 | | x | | -2 |
| 2 5 1 | × | y | = | -5 |
| 1 2 3 | | z | | 0 |
The determinant of the matrix of coefficients is -3, which is non-zero. This means that the system of equations has a unique solution.
To find the value of z, we need to calculate the determinant of the matrix obtained by replacing the z-column with the constants column:
| 1 3 -2 |
| 2 5 -5 |
| 1 2 0 |
Using the rule of determinants for a 3x3 matrix, we can calculate the determinant:
Det = (1 × (50 - -52)) - (3 × (20 - -51)) + (-2 × (2 × -5 - 51))
= (1(0 + 10)) - (3 × (0 + 5)) + (-2 × (-10 - 5))
= (110) - (35) + (-2 × -15)
= 10 - 15 + 30
= 25
Since the determinant is non-zero, the system has a unique solution. To find the value of z, we divide the determinant of the matrix obtained by replacing the z-column with the constants column by the determinant of the matrix of coefficients:
z = Detz / Det
= 25 / -3
= -8.33
Therefore, the value of z in the solution set is approximately -8.33.
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The question is -
Given the system of equations x + 3y + z = -2
2x + 5y + z = -5
x+ 2y + 3z = 0
The determinant of the matrix of coefficients is -3. The value of z in the solution set is:
(a) z=−2/3
(b) z=5/3
(c) z=4/3
(d) z=−2
(e) None of the above
Therese filled a rectangular pool with water. The pool measured 2 1/4 yards long, 2 yards wide and 1 2/3 yards tall. If the pool was filled to the top, how many cubic yards of water did Therese use?
Answer: 7 1/2 cubic yards
Step-by-step explanation: sorry if this is wrong lol
{18-12+8-16/3+...} find the sum
I think it is 8.6.. but I am not 100% sure
Step-by-step explanation:
A certain bacterium has an exponential growth rate of 25% per day. If we start with 0.5 gram and provide unlimited resources, how many bacteria can we grow in 2 weeks? Round to the nearest tenth of a gram. HINT: How many days are in a week?
Answer:
FV= 11.369 gr
Step-by-step explanation:
Giving the following information:
Growth rate (g)= 25% per day
Number of periods (n)= 14 days
Present Value (PV)= 0.5 gr
To calculate the Future Value(FV) in grams, we need to use the following formula:
FV= PV*(1+i)^n
FV= 0.5*(1.25^14)
FV= 11.369 gr
use a known maclaurin series to obtain a maclaurin series for the given function. f(x) = xe5x
The maclaurin series for the given function xe⁵ˣ = x + 5x² + (25x³)/2! + (125x⁴)/3! + ...
To find the Maclaurin series for the function f(x) = xe⁵ˣ, we can utilize the Maclaurin series expansion of the exponential function, eˣ:
eˣ = 1 + x + (x²)/2! + (x³)/3! + ...
Substituting 5x for x in the above expansion, we have:
e⁵ˣ = 1 + 5x + (5x)²/2! + (5x)³/3! + ...
Multiplying the above series by x, we get:
xe⁵ˣ = x + 5x² + (25x³)/2! + (125x⁴)/3! + ...
This is the Maclaurin series for the function f(x) = xe⁵ˣ.
The calculation involves applying the Maclaurin series expansion of the exponential function to the function f(x) = xe⁵ˣ by substituting 5x for x in the series expansion. Then, multiplying the resulting series by x gives us the desired Maclaurin series for f(x).
The series can be continued by following the pattern of increasing powers of x, with the coefficients determined by the corresponding terms in the expansion.
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Let S be any sample space, and E, F and G be any three events. Describe the event that E and F occur, and G does not occur.
a) ( E ∪ Fc ) ∩ G
b) ( E ∪ F ) ∩ G
c) ( E ∪ F ) ∩ Gc
d) ( E ∩ F ) ∪ Gc
e) E ∩ F ∩ Gc
f) None of the above.
The event that E and F occur, and G does not occur is E ∩ F ∩ Gc. So, correct option is E.
The event that E and F occur, and G does not occur can be represented by the expression (E ∩ F) ∩ Gc. Let's break down the options provided to determine the correct representation:
a) (E ∪ Fc) ∩ G: This represents the event where either E or the complement of F occurs, and G occurs. It does not capture the condition that both E and F occur.
b) (E ∪ F) ∩ G: This represents the event where either E or F occurs, and G occurs. It does not exclude the possibility of E and F occurring together.
c) (E ∪ F) ∩ Gc: This represents the event where either E or F occurs, and G does not occur. However, it does not specify that E and F occur together.
d) (E ∩ F) ∪ Gc: This represents the event where either E and F occur together or G does not occur. It does not exclude the possibility of E and F occurring without G.
e) E ∩ F ∩ Gc: This correctly represents the event where both E and F occur together, and G does not occur. This is the desired outcome as stated in the question.
Therefore, the correct representation for the event is option e) E ∩ F ∩ Gc.
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Find the area of rhombus with diagonals 6 mm and 9 mm.
A patient is given 3 mg of morphine to control pain. About 31% of any morphine in the blood is washed out every hour. (a) Construct a function that models the level M of morphine (in mg) in the blood t hours after one dose. M = (b) How much morphine (in mg) remains in the blood after 5 hours? (Round your answer to two decimal places.) mg (c) Estimate how long, in hours, it will take for the amount of morphine left to drop to 0.1 mg. (Round your answer to two decimal places.) h
a) the function that models the level M of morphine (in mg) in the blood t hours after one dose is M = [tex]3 * (0.69)^t[/tex]
b) 0.50 mg of morphine remains in the blood after 5 hours.
c) it will take approximately 14.67 hours for the amount of morphine left to drop to 0.1 mg.
(a) To construct a function that models the level M of morphine (in mg) in the blood t hours after one dose, we can use the formula for exponential decay:
M = [tex]initial amount * (1 - decay rate)^t[/tex]
Given that the patient is initially given 3 mg of morphine and 31% (0.31) of any morphine is washed out every hour, we can write the function as:
M = [tex]3 * (1 - 0.31)^t[/tex]
Simplifying further, we have:
M = [tex]3 * (0.69)^t[/tex]
Therefore, the function that models the level M of morphine (in mg) in the blood t hours after one dose is M = [tex]3 * (0.69)^t[/tex]
(b) To find out how much morphine remains in the blood after 5 hours, we can substitute t = 5 into the function and calculate the value of M:
M = 3 * (0.69)⁵
M ≈ 3 * 0.1681
M ≈ 0.5043
So, approximately 0.50 mg of morphine remains in the blood after 5 hours.
(c) To estimate how long it will take for the amount of morphine left to drop to 0.1 mg, we need to find the value of t when M = 0.1 in the function:
0.1 = [tex]3 * (0.69)^t[/tex]
Dividing both sides by 3:
0.0333 ≈ [tex](0.69)^t[/tex]
Taking the logarithm of both sides:
log(0.0333) ≈ [tex]log[(0.69)^t][/tex]
Using the logarithm properties, we can bring down the exponent:
log(0.0333) ≈ t * log(0.69)
Now, we can solve for t by dividing both sides by log(0.69):
t ≈ log(0.0333) / log(0.69)
Calculating this expression:
t ≈ -2.3859 / -0.1625
t ≈ 14.67
So, it will take approximately 14.67 hours for the amount of morphine left to drop to 0.1 mg.
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Can someone explain this step by step pls and no links
Answer: 12
Step-by-step explanation: Use the quadratic equation to solve the equation.
a = 1
b = -8
c = 12
Answer:
x=6 is the answer
Step-by-step explanation:
x squared -8 x +12=0
6 squared=36
36-8x +12=0
x= 6 so 8*6 =48
36-48+12=0
36-48= -12
-12+12=0
hope this helps
Please help me, GodBless.
Answer:
m= 5/2
Step-by-step explanation:
Points: (0, 2) and (4, 12)
Slope:
m=(y2-y1)/(x2-x1)
m=(12-2)/(4-0)
m=10/4
m= 5/2
Answer:
2.5
Step-by-step explanation:
Hello There!
We can easily calculate the slope using the slope formula
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
all we need to do is pick to ordered pairs from the table and plug in the x and y values of the selected pairs
The ordered pairs selected can vary but i chose the points (0,2) and (4,12)
now we plug in the values into the formula
(remember y values on top and x values down below)
[tex]m=\frac{12-2}{4-0} \\12-2=10\\4-0=4\\\frac{10}{4}=2.5[/tex]
so we can conclude that the slope of the function is 2.5
The riddle family income is 3400 per month. They spend 1088 per month on housing. Use information to determine the percent of the budget spent on housing
Answer:
32%
Step-by-step explanation:
[tex]\frac{1088}{3400}[/tex]=[tex]\frac{x}{100}[/tex]
(1088)(100)=(3400)(x)
108800=3400x
x=32
32%
The percent of the budget spent on housing is 31.76%.
What is percentage?
Percentage is a mathematical term which means number or ratio expressed in terms of fractions of 100. it can be calculated by dividing given value to the whole value and multiplied by 100. we represent it through ' %'.
Given that, The riddle family income is 3400 per month.
They spend 1088 per month on housing.
We know the formula for percentage which is;
Percentage = (given value/whole value)*100,
Percentage = (1080/3400)*100
Percentage = 31.76
Hence, The percentage is 31.76%.
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Find a vector with magnitude 14 in the same direction as (2,6, -3)
To find a vector with a magnitude of 14 in the same direction as (2, 6, -3), the vector (4, 12, -6) has a magnitude of 14 and is in the same direction as (2, 6, -3).
we need to scale the original vector while preserving its direction. By normalizing the vector, we can determine its unit vector and then multiply it by the desired magnitude to obtain the final vector. magnitude = √(x² + y² + z²), where x, y, and z are the components of the vector. In this case, the magnitude of the vector (2, 6, -3) is √(2² + 6² + (-3)²) = √(4 + 36 + 9) = √49 = 7.
To obtain a vector with a magnitude of 14, we need to scale the original vector by a factor of 14/7. This ensures that the new vector has the desired magnitude while maintaining the same direction. Scaling a vector involves multiplying each of its components by the scaling factor. Therefore, we can calculate the new vector as follows: New vector = (2, 6, -3) * (14/7) = (2 * 14/7, 6 * 14/7, -3 * 14/7) = (4, 12, -6).
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Aaron is making a jewelry box of wood in the shape of a rectangular prism the jewelry box will have the dimensions shown in the shape to the right the cost of painting exterior of the box is $0.50 per square inch is how much does Aaron have to spend to paint the jewelry box
A. $684
b. $342
c.$700
d. $100
need help on quiz please answer
Answer:
b. $342
Step-by-step explanation:
Base = Length * width
Base Area = 12 * 5
B = 180 square inches.
Perimeter of the prism will be :
P = 2 (length) + 2 (15)
P = 2 * 12 + 30
P = 54 inches
Height of the jewelry container will be :
X = Ph + 2B
X = 54 * 6 + 2 * 180
X = 684 square inches.
Total cost for paint of jewelry box is :
X * cost per inch
Total cost = 682 * $0.50
Total Cost = $342.
ra elaborar un pastel de 2 kg, se necesitan, ¾ de kg de harina y 4 huevos (con un peso de 100 gr cada uno), además de otros ingredientes. ¿Cuál es el peso que tienen los demás ingredientes del pastel?
Based on the above, the weight of the other ingredients in the cake is about 850 grams.
What is the weight about?To determine the weight of the other ingredients in the cake, one need to subtract the weight of the flour as well as the eggs from the total weight of the cake.
Note that from the question:
Weight of flour = ¾ kg = 750 gNumber of eggs = 4Weight of each egg = 100 gSo: Weight of flour and eggs = (Weight of flour) + (Number of eggs × Weight of each egg)
= 750 g + (4 × 100 g)
= 750 g + 400 g
= 1150 g
Hence: The Total weight of the cake = 2 kg = 2000 g
Going further:
Weight of other ingredients = Total weight of the cake - Weight of flour and eggs
= 2000 g - 1150 g
= 850 g
Therefore, the weight of the other ingredients that is seen in the cake is 850 grams.
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To make a 2 kg cake, ¾ kg of flour and 4 eggs (weighing 100 g each) are needed, in addition to other ingredients. What is the weight of the other ingredients in the cake?
Points vectors Apply the Determinant Linear Independence Test to decide whether the [5] 4 0 4 3 2 2 2 V1: 0 3 are linearly independent in R4. a) Evaluate the corresponding determinant. (b) Make your conclusion.
The determinant of the matrix formed by the given vectors is -4, indicating that the vectors are linearly independent in R⁴.
To evaluate the determinant of the given set of vectors V₁ = [5, 4, 0, 4] and V₂ = [3, 2, 2, 2] using the expanded matrix form, we can write:
| 5 3 |
| 4 2 |
| 0 2 |
| 4 2 |
Expanding the determinant along the first row, we can calculate it as follows:
det = 5 * det(| 2 2 |) - 3 * det(| 4 2 |)
| 4 2 | | 0 2 |
We can evaluate each determinant separately:
det(| 2 2 |) = (2 * 2) - (2 * 0) = 4 - 0 = 4
det(| 4 2 |) = (4 * 2) - (2 * 0) = 8 - 0 = 8
Substituting these determinants back into the expanded expression:
det = 5 * 4 - 3 * 8
= 20 - 24
= -4
Therefore, the determinant of the given matrix is -4.
Based on the determinant being nonzero (-4 ≠ 0), we can conclude that the vectors V₁ = [5, 4, 0, 4] and V₂ = [3, 2, 2, 2] are linearly independent in R⁴.
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