Answer:
[tex]P(5)=0.238[/tex]
[tex]P(x\geq 6)=0.478[/tex]
[tex]P(x<4)=0.114[/tex]
Step-by-step explanation:
In this case we can calculate the probability using the binomial probability formula
[tex]P(X=x)=\frac{n!}{x!(n-x)!}*p^x*(1-p)^{n-x}[/tex]
Where p is the probability of obtaining a "favorable outcome " x is the number of desired "favorable outcome " and n is the number of times the experiment is repeated. In this case n = 10 and p = 0.54.
(a) exactly five
This is:
[tex]x=5,\ n=10,\ p=0.54.[/tex]
So:
[tex]P(X=5)=\frac{10!}{5!(10-5)!}*0.54^x*(1-0.54)^{10-5}[/tex]
[tex]P(5)=0.238[/tex]
(b) at least six
This is: [tex]x\geq 6,\ n=10,\ p=0.54.[/tex]
[tex]P(x\geq 6)=P(6) + P(7)+P(8)+P(9) + P(10)[/tex]
[tex]P(x\geq 6)=0.478[/tex]
(c) less than four
This is: [tex]x< 4,\ n=10,\ p=0.54.[/tex]
[tex]P(x<4)=P(3) + P(2)+P(1)+P(0)[/tex]
[tex]P(x<4)=0.114[/tex]
This question is based on the probability. Therefore, the required probabilities are : (a) [tex]P(5) = 0.238[/tex], (b)[tex]P(x \geq 6) = 0.478[/tex] and (c) [tex]P(x <4) = 0.114[/tex].
Given:
54% of U.S. adults have very little confidence in newspapers. You randomly select 10 U.S. adults.
We have to find the probability that the number of U.S. adults who have very little confidence in newspapers is (a) exactly five, (b) at least six, and (c) less than four.
According to the question,
[tex]P(5) = 0.238\\P(x \geq 6) = 0.478\\P(x <4) = 0.114[/tex]
In this we have to calculate the probability using the binomial probability formula,
[tex]P(X=x) = \dfrac{n!}{x!(n-x)!} \times p^{x} \times (1-p)^{n-x}[/tex]
Where, p is the probability of obtaining a "favorable outcome ", x is the number of desired "favorable outcome " and n is the number of times the experiment is repeated. In this case n = 10 and p = 0.54.
(a) exactly five
x=5, n= 10, p = 0.54
[tex]P(X=5)= \dfrac{10!}{5!(10-5)!} \times 0.5^{x} \times (1-0.54)^{10-5}[/tex]
P(X=5) = 0.238
(b) at least six
[tex]x\geq 6, n=10, p=0.54\\P(x\geq 6) = P(6) + P(7) + P(8) + P(9)+P(10)\\P(x\geq 6) = 0.478[/tex]
(c) less than four
[tex]x< 6, n=10, p=0.54\\P(x< 4) = P(3) + P(2) + P(1) + P(0)\\P(x< 4) = 0.114[/tex]
Therefore, the answers are : (a) [tex]P(5) = 0.238[/tex], (b)[tex]P(x \geq 6) = 0.478[/tex]
and (c) [tex]P(x <4) = 0.114[/tex].
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Your company manufactures hot water heaters. The life spans of your product are known to be normally distributed with a mean of 13 years and a standard deviation of 1.5 years. What is the probability that the mean life span in a group of 10 randomly selected hot water heaters is between 12 and 15 years? (Round to the nearest ten-thousandth.)
Final answer:
Calculate the probability that the mean life span of a group of 10 hot water heaters is between 12 and 15 years using the standard normal distribution.
Explanation:
The probability that the mean life span in a group of 10 randomly selected hot water heaters is between 12 and 15 years can be calculated using the standard normal distribution.
Given: Mean = 13 years, Standard Deviation = 1.5 years.
Calculate the z-scores for 12 and 15 years using the formula z = (X - mean) / standard deviation.Look up the corresponding probabilities for these z-scores in the standard normal distribution table.Find the area between these two probabilities to get the final result.A manager at SUBWAY wants to find the total cost of 24 8 pounds of sliced turkey at S..89 a pound and 38.2 pounds of provolone cheese at $2.05 a pound. Find the fina cost. [1.4 and 1.5)
Answer:
The Total of both the sliced turkey and provolone cheese together is $46.49
Step-by-step explanation:
Hello, great question. These types are questions are the beginning steps for learning more advanced Algebraic Equations.
Assuming that the the sliced turkey costs $0.89 / pound (Since it didn't show up correctly in the question) we can create the following equation to solve for the total cost of the order.
[tex]\frac{24.8 lb}{0.89} + \frac{38.2 lb}{2.05} = Total[/tex]
[tex]27.86 + 18.63 = Total[/tex] ....rounded to nearest hundredth
[tex]46.49 = Total[/tex]
So the total of both the sliced turkey and provolone cheese together is $46.49
I hope this answered your question. If you have any more questions feel free to ask away at Brainly.
The final cost for 24.8 pounds of sliced turkey at $1.89 per pound and 38.2 pounds of provolone cheese at $2.05 per pound is $125.18.
To find the total cost of sliced turkey and provolone cheese purchased by the manager at SUBWAY, we need to calculate the cost for each item separately and then sum them up.
Given:
- Sliced turkey:
- Amount: 24.8 pounds
- Price per pound: $1.89
- Provolone cheese:
- Amount: 38.2 pounds
- Price per pound: $2.05
Calculating the cost of sliced turkey:
[tex]\[ \text{Cost of turkey} = \text{Amount of turkey} \times \text{Price per pound of turkey} \][/tex]
[tex]\[ \text{Cost of turkey} = 24.8 \, \text{pounds} \times \$1.89/\text{pound} \][/tex]
[tex]\[ \text{Cost of turkey} = \$46.87 \][/tex]
Calculating the cost of provolone cheese:
[tex]\[ \text{Cost of provolone cheese} = \text{Amount of cheese} \times \text{Price per pound of cheese} \][/tex]
[tex]\[ \text{Cost of provolone cheese} = 38.2 \, \text{pounds} \times \$2.05/\text{pound} \][/tex]
[tex]\[ \text{Cost of provolone cheese} = \$78.31 \][/tex]
Finding the final cost (total cost):
To find the total cost, we add the cost of turkey and the cost of provolone cheese:
[tex]\[ \text{Total cost} = \$46.87 + \$78.31 \][/tex]
[tex]\[ \text{Total cost} = \$125.18 \][/tex]
The complete question is
A manager at SUBWAY wants to find the total cost of 24.8 pounds of sliced turkey at $1.89 a pound and 38.2 pounds of provolone cheese at $2.05 a pound. Find the final cost.
probability of drawing two number cards from a standard deck (without replacement)
there are 52 cards in a deck, 12 of those cards are "face cards", so the remaining are number cards, namely 40.
52 = sample space
40 = favorable outcomes
P(number card | number card) = p(number) * p(number)
so the first time we pull one, there are 52 cards, the probability of a number card is 40/52, or 10/13, and we don't put it back in the deck.
the next time we pull another card, the cards are no longer 52 total, we pulled one out, they're only 51, namely 51 = sample space, and the number cards if we really pulled out before, are no longer 40, are 39, namely 39 = favorable outcomes.
probability of getting a number card the second time? 39/51 or 13/17.
[tex]\bf \stackrel{\textit{probability of getting a number card twice}}{\cfrac{10}{~~\begin{matrix} 13 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}\cdot \cfrac{~~\begin{matrix} 13 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}{17}\implies \cfrac{10}{17}~~\approx ~~ 0.59}~\hfill 59\%[/tex]
Answer:
105/221
Step-by-step explanation:
There are 52 cards in a deck.
Assuming 2-10 are the number cards
2,3,4,5,6,7,8,9,10 = 9
There are 4 suits
9*4 = 36 cards are number cards
P(1st card is a number card) = number card/ total
=36/52 = 9/13
We do not replace the card, so there are only 51 cards left, and only 35 number cards
P(2nd card is a number card) = number card/ total
=35/51
The probability of getting 2 number cards in a row is
P (number ,number) =P(1st card is a number)*P(2nd card is a number card)
= 9/13 * 35/51
Dividing the top and bottom by 3
= 3/13 * 35/17
=105/221
A recent survey found that 85 % of households had Internet access and 81 % of households had cable television. Also, it was reported that 72 % of the households in the survey had both Internet and cable television. Determine the probability of selecting a household that had either Internet access or cable television.
Answer: Probability that either had Internet access or had cable television is 94%.
Step-by-step explanation:
Since we have given that
Probability of households had internet access P(I) = 85% = 0.85
Probability of households had cable television P(C) = 81% = 0.81
Probability of households had both Internet and cable television P(I ∩ C) = 72% = 0.72
We need to find the probability that either had Internet access or had cable television.
As we know the formula:
P(I ∪ C)=P(I) + P(C) - P(I ∩ C)
[tex]P(I\cup C)=0.85+0.81-0.72\\\\P(I\cup C)=0.94=94\%[/tex]
Hence, our required probability is 94%.
To determine the probability of a household having access to either the internet or cable television, the probabilities of both events must be added then subtract the probability of both occurring simultaneously. Doing this calculation, P(Internet) + P(Cable) - P(Both), we find the probability to be 94%.
Explanation:In order to answer this question, we would need to use some principles of probability, specifically the rule for adding the probabilities of two mutually exclusive outcomes. But, because the outcomes in this case are not mutually exclusive (a household can have both Internet and cable television), we need to adjust our calculation by subtracting the probability of both outcomes occurring together.
The probability of having either Internet or cable television would be P(Internet) + P(Cable) - P(Both) which equals to [tex]85 + 81 - 72 = 94%.[/tex]Therefore, the probability of selecting a household that had either Internet access or cable television is 94%.
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Of five letters (A, B, C, D, and E), two letters are to be selected at random. How many possible selections are there?
a. 20
b. 7
c. 5!
d. 10
Answer: d. 10
Step-by-step explanation:
We know that the number of combinations of r objects selected from a group of n objects at a time is given by :-
[tex]^nC_r=\dfrac{n!}{(n-r)!r!}[/tex]
Given : The total number of letters = 5
The number of letters need to select = 2
Then , the number of combinations of 2 letters selected from a group of 5 letters at a time is given by :-
[tex]^5C_2=\dfrac{5!}{(5-2)!2!}=\dfrac{5\times4\times3!}{3!\times2}=10[/tex]
Hence, there are 10 possible selections.
The problem pertains to combinations in mathematics. When you select two letters out of five without considering the order, you use a formula of 'C(n, r) = n! / [(n-r)!r!]'. Applying this to our problem (where n=5, r=2), it gives us 10 combinations.
Explanation:The problem you're asking about is associated with combinations in combinatorial mathematics. When selecting two letters out of five (A, B, C, D, and E), we are interested in different combinations and not the order in which you select them. The standard formula to calculate combinations is C(n, r) = n! / [(n-r)!r!].
Here, n = 5 (total number of letters), and r = 2 (the number of letters you want to select). So, C(5, 2) = 5! / [(5-2)!2!] = 10. The correct answer is d. 10.
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Mrs. Alford invested $6700 in securities. Part of the money was invested at 1% and part at 9%. The total annual income was $275. How much was invested at each rate?
$ at 1%
$ at 9%
Final answer:
Mrs. Alford invested $3437.50 at a 1% interest rate and $3262.50 at a 9% interest rate. The solution involved setting up and solving a system of linear equations based on the given total investment and income.
Explanation:
The problem involves solving a system of linear equations to determine how much money Mrs. Alford invested at 1% and 9%. Let 'x' represent the amount invested at 1% and 'y' represent the amount invested at 9%. The total amount invested is $6700, so the first equation is x + y = 6700. The total annual income from these investments is $275. The income from the investment at 1% is 0.01x, and the income from the investment at 9% is 0.09y, creating the second equation: 0.01x + 0.09y = 275.
To solve the system, we can start by multiplying the second equation by 100 to get rid of the decimals, resulting in 1x + 9y = 27500. Subtracting the first equation from this gives 8y = 26100, which implies that y = 3262.5. Therefore, Mrs. Alford invested $3262.50 at 9%. Using the first equation, we find that x = 6700 - 3262.5 = 3437.5, meaning $3437.50 was invested at 1%.
Identify the lower class limits, upper class limits, class width, class midpoints, and class boundaries for the given frequency distribution. Also identify the number of individuals included in the summary. Blood Platelet Count of Males (1000 cells/muL) Frequency 0-99 3 100-199 54 200-299 76 300-399 21 400-499 0 500-599 0 600-699 1 Identify the lower class limits (in 1000 cells/muL).
Answer:
The lower limits (in 1000 cells/muL): 0,100, 200, 300, 400, 500, 600 .
The upper limits (in 1000 cells/muL): 99, 199, 299, 399, 499, 599, 699.
The class width(in 1000 cells/muL): 100.
Class midpoints (in 1000 cells/muL): 49.5, 149.5, 249.5, 349.5, 449.5, 549.5, 649.5.
Class boundaries (in 1000 cells/muL): -0.5, 99.5, 199.5, 299.5, 399.5, 499.5, 599.5, 699.5.
Individuals included in the summary: 155.
Step-by-step explanation:
For the lower class limit is needed the smallest value in each class: 0,100, 200, 300, 400, 500, 600.
For the upper-class limit is needed the biggest value in each class: 99, 199, 299, 399, 499, 599, 699.
The class width is the difference between the lower limit of one class and the lower limit of the previous class. For example, 200 is the lower limit of one class and the lower limit of the previous class is 100, so 200-100=100.
Class midpoints are the average of the limits of a class if the limits are 0 and 99 then:
[tex]Midpoint=\frac{0+99}{2}=49.5[/tex]
[tex]Midpoint=\frac{100+199}{2}=149.5[/tex]
[tex]Midpoint=\frac{200+299}{2}=249.5[/tex]
[tex]Midpoint=\frac{300+399}{2}=349.5[/tex]
[tex]Midpoint=\frac{400+499}{2}=449.5[/tex]
[tex]Midpoint=\frac{500+599}{2}=549.5[/tex]
[tex]Midpoint=\frac{600+699}{2}=649.5[/tex]
Class boundaries are the numbers than allow to separate each class, to find each one, first calculate the gap between each class (the lower limit of one class minus the upper limit of the previous one) and divide by 2:
100-99=1
1/2=0.5
Add this number to all the upper limit to find the upper boundaries:
[tex]99+0.5=99.5[/tex]
[tex]199+0.5=199.5[/tex]
[tex]299+0.5=299.5[/tex]
[tex]399+0.5=399.5[/tex]
[tex]499+0.5=499.5[/tex]
[tex]599+0.5=599.5[/tex]
[tex]699+0.5=699.5[/tex]
And subtract this number (0.5) from the lower limit to find the lower boundaries:
[tex]0-0.5=-0.5[/tex]
[tex]100-0.5=99.5[/tex]
[tex]200-0.5=199.5[/tex]
[tex]300-0.5=299.5[/tex]
[tex]400-0.5=399.5[/tex]
[tex]500-0.5=499.5[/tex]
[tex]600-0.5=599.5[/tex]
The total of individuals is equal to the sum of all the frequencies of each class:
[tex]3+54+76+21+0+0+1= 155[/tex]
The lower class limits are 0, 100, 200, etc. The class width is 100. The summation of the frequencies, which equals the total number of individuals, is 155.
Explanation:In the given data, the lower class limits are the smallest numbers in each class or group, which are 0, 100, 200, 300, 400, 500, and 600 (in 1000 cells/muL). The upper class limits, which are the highest numbers in each class, are 99, 199, 299, 399, 499, 599, and 699.
The class width, which is the difference between two consecutive lower class limits, is 100. The class midpoints can be calculated by adding the lower and upper limits of each class and divide by 2, yielding 49.5, 149.5, 249.5, etc. The class boundaries are the numbers that separate classes without leaving gaps: -0.5, 99.5, 199.5, etc.
The total number of individuals included in the summary is the sum of the frequencies, which equals 155 in this case.
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Lacinda has 120 ft of fencing to make a rectangular kennel for her dogs. The house is to be used as one side of the kennel. What length will maximize the area of the kennel?
To maximize the area of the kennel, the fencing should be arranged to form a square or closely resemble a square. The optimal length for each side would be 120/3 = 40 ft. This arrangement would provide the most amount of space for the dogs.
Explanation:In the problem highlighted, Lacinda has 120 ft of fence available to make a kennel with her house serving as one side of the rectangle. In terms of mathematics, this is an application of optimization in calculus or geometrical considerations for non-calculus level. However, to maximize the area with a given perimeter, a square or a rectangle closest to a square should be constructed.
If the length of the kennel adjacent to the house is x (ft), the length of the other two sides required would be (120 - x) / 2. Therefore, the area of the rectangle (in square feet) can be represented as (120 - x) * x / 2.
To find the maximum area, we would solve this for x. Without using calculus, we would say that for a rectangle, equality of sides (i.e., square) gives the maximum area. Therefore, the optimal length for x would be 120/3 = 40 ft. This arrangement would provide the maximum area for the kennel.
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The length that maximizes the area of Lacinda's rectangular kennel is 60 feet.
Let the length of the side parallel to the house be L and each of the two widths perpendicular to the house be W.
The total fencing used is 120 ft, so we have the equation:
2W + L = 120
We need to express the area A in terms of a single variable:
A = L * W
From the fencing equation, solve for L:
L = 120 - 2W
Substitute into the area equation:
A = (120 - 2W) * W
A = 120W - 2W²
To maximize the area, take the derivative of A with respect to W and set it to zero:
dA/dW = 120 - 4W = 0
Solve for W:
4W = 120
W = 30
Substitute W back into the fencing equation to find L:
L = 120 - 2 * 30 = 60
Thus, the length that maximizes the area of the kennel is 60 feet.
checking congruence
Check the picture below.
In the past, 25% of the town residents participated in the school board elections. The town officials recently undertook an extensive campaign to encourage registered voters to participate in the election process. They are interested in determining whether the promotional campaign actually increased the proportion of voters. The correct set of hypotheses is _____. a. H0: p ≤.25 Ha: p >.25 b. H0: p > 25 Ha: p ≤ 25 c. H0: p ≥ .25 Ha: p <.25 d. H0: p <.25 Ha: p ≥ .25
Answer: [tex]H_0:p\leq0.25[/tex]
[tex]H_a:p>0.25[/tex]
Step-by-step explanation:
Given : In the past , 25% of the town residents participated in the school board elections.
Let 'p' be the proportion of voters will participated in the school board elections.
Claim : [tex]p>0.25[/tex]
We know that the null hypothesis has equal sign.
Therefore , the null hypothesis for the given situation will be opposite to the given claim will be :-
[tex]H_0:p\leq0.25[/tex]
And the alternative hypothesis must be :-
[tex]H_a:p>0.25[/tex]
Hence, the correct set of hypotheses is
[tex]H_0:p\leq0.25[/tex]
[tex]H_a:p>0.25[/tex]
A random sample of 10 subjects have weights with a standard deviation of 11.9407 kg. What is the variance of their weights? Be sure to include the appropriate units with the result. The variance of the sample data is nothing ▼ kg cubed . kg squared . kg. (Round to four decimal places as needed.)
Answer: [tex]142.58\text{squared kg}[[/tex]
Step-by-step explanation:
Answer:
Step-by-step explanation:
Given : A random sample of 10 subjects have weights with a standard deviation of 11.9407 kg
i.e. [tex]\sigma = 11.9407[/tex]
Since we know that the value of variance is the square of standard deviation.
i.e. [tex]\text{Variance}=\sigma^2[/tex]
Therefore, to find the value of variance, we need to find the square of the given standard deviation.
i.e. [tex]\text{Variance}=(11.9407)^2=142.58031649\approx142.58\text{squared kg}[/tex]
Thus, the variance of their weights =[tex]142.58\text{squared kg}[[/tex]
Final answer:
The variance of the sample data is 142.58 kg squared, found by squaring the given standard deviation of 11.9407 kg.
Explanation:
The calculation of variance involves squaring the standard deviation. Given a sample with a standard deviation of 11.9407 kg, the variance can be found by squaring this value:
Variance (
s2) = Standard Deviation (
s)2 = 11.9407 kg2
The variance of the sample data is:
142.58 kg2 (this value has been rounded to four decimal places as instructed).
The units for variance are always the square of the units for the original data, hence the variance of weights is expressed in kilograms squared (kg2).
Given nonempty sets A and B, prove that every set in P(B − A) ⊆ P(B) − P(A).
Proof:
Let [tex]X \in P(B-A) [/tex]. As we chose [tex]X[/tex] in [tex]P(B-A) [/tex] we know that [tex]X \subseteq B-A[/tex]. Since [tex]B-A \subseteq B[/tex] by transitivity we get:
[tex]X \subseteq B \quad \implies X \in P(B)[/tex].
If [tex]X [/tex] is the empty set, we already have that [tex]X = \emptyset \in P(B) - P(A)[/tex]. But if [tex]X[/tex] is not empty, that means that it can't be subset of [tex]A[/tex], because [tex] X [/tex] is already subset of [tex]B-A[/tex], and those sets do not share any element. In other words:
[tex]X \subseteq A \cup (B-A) = \emptyset[/tex]
[tex] \Rightarrow X = \emptyset[/tex]
As [tex]X[/tex] can't be subset of [tex]A[/tex], then [tex] X\notin P(A) [/tex]. [tex]X[/tex] was an arbitrary element, and
[tex]X \in P(B)[/tex][tex] X\notin P(A) [/tex]Thus, [tex] X\in P(B)-P(A) [/tex], where we conclude that
[tex]P(B-A) \subseteq P(B) - P(A)[/tex]
A 25-ounce solution is 20% alcohol. If 50 ounces of
water are added to it, what percent of the new solution is
alcohol?
To find out what percent of the new solution is alcohol after water has been added, we need to follow these steps:
Step 1: Calculate the amount of alcohol in the original solution.
The original solution is 20% alcohol and the total volume of the original solution is 25 ounces. To find the amount of alcohol in the original solution, we multiply the total volume by the percentage of alcohol (in decimal form):
Amount of Alcohol = Total Volume * Alcohol Percentage
= 25 ounces * 0.20
= 5 ounces
So the original solution contains 5 ounces of alcohol.
Step 2: Calculate the new total volume of the solution after adding water.
We add 50 ounces of water to the original 25 ounces of the solution:
New Total Volume = Original Solution Volume + Water Added
= 25 ounces + 50 ounces
= 75 ounces
Step 3: Calculate the new percentage of alcohol in the solution.
The amount of alcohol hasn't changed; it's still the original 5 ounces. The percentage of alcohol in the new solution is the amount of alcohol divided by the new total volume:
New Alcohol Percentage = Amount of Alcohol / New Total Volume
= 5 ounces / 75 ounces
= 0.0667 (approximately)
To express this as a percentage, we multiply by 100:
New Alcohol Percentage = 0.0667 * 100
= 6.67% (approximately)
Therefore, after adding 50 ounces of water to the 25-ounce solution that was originally 20% alcohol, the new solution is approximately 6.67% alcohol.
Find f'(x) and F"(x). f(x)=9+ 3x – 3x^3
Answer:
[tex]f'(x)=3-9x^{2}[/tex] and [tex]f''(x)=-18x[/tex]
Step-by-step explanation:
In order to find the derivatives, first we need to remember that for polynomial functions:
[tex]f'(x)=(x^{n}+x^{m})'= (x^{n})'+(x^{m})'[/tex], as well as that:
[tex]f'(x)= (x^{n})' = n*(x^{n-1})[/tex]
1. First derivative of the function:
[tex]f(x)=9+3x-3x^{3}[/tex]
[tex]f'(x)=(9)'+(3x)'-(3x^{3})'[/tex] using the property [tex]f'(x)= (x^{n})' = n*(x^{n-1})[/tex] then
[tex]f'(x)=3-3*3x^{2}[/tex], remember that the derivative of a constant is equal to 0
[tex]f'(x)=3-9x^{2}[/tex]
2. Second derivative:
[tex]f'(x)=3-9x^{2}[/tex]
[tex]f''(x)=(3-9x^{2})'[/tex] using the property [tex]f'(x)= (x^{n})' = n*(x^{n-1})[/tex] then
[tex]f''(x)=(3)'-(9x^{2})'[/tex]
[tex]f''(x)=-(9*2)x^{1}[/tex]
[tex]f''(x)=-18x[/tex]
In conclusion, [tex]f'(x)=3-9x^{2}[/tex] and [tex]f''(x)=-18x[/tex]
Given the differential Equation (dy/dx)+(2/x)y=x^2y^3 ;solve this equation using the Bernoulli method; Final answer should be (1/y^2)=?
Answer:
[tex]\frac{1}{y^2}=2x^3+Cx^4[/tex].
Step-by-step explanation:
Given differential equation
[tex]\frac{\mathrm{d}y}{\mathrm{d}x}+\frac{2}{x}y=x^2y^3[/tex]
Differential equation can be write as
[tex]y^{-3}\frac{\mathrm{d}y}{\mathrm{d}x}+\frac{2}{x}y^{-2}=x^2[/tex]
By Bernoulli method
Susbstitute [tex]y^{-2}=t[/tex].....{equationI}
Differentiate equation I w.r.t x then we get
[tex]\frac{\mathrm{d}t}{\mathrm{d}x}=-2y^{-3}\frac{\mathrm{d}y}{\mathrm{d}x}[/tex]
[tex]-\frac{1}{2}\frac{\mathrm{d}t}{\mathrm{d}x}=y^{-3}\frac{\mathrm{d}y}{\mathrm{d}x}[/tex]
Susbstitute the values in the given differential equation then we get
[tex]-\frac{1}{2}\frac{\mathrm{d}t}{\mathrm{d}x}+\frac{2}{x}t=x^2[/tex]
[tex]\frac{\mathrm{d}t}{\mathrm{d}x}-\frac{4}{x}t=-2x^2[/tex]
It is first order linear differential equation and compare with the first order linear differential equation [tex]\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)[/tex]
Then we get P(x)=[tex]-\frac{4}{x}[/tex] and Q(x)=[tex]-2x^2[/tex]
Integration factor=[tex]e^\intP(x)dx[/tex]
Integration factor= [tex]e^{-\int\frac{4}{x}dx[/tex]
Integration factor= [tex]e^{-4lnx}=e^{lnx^{-4}}=x^{-4}[/tex].
Using [tex]e^{logb}=b[/tex]
[tex]t\times \frac{1}{x^4}=\int{-2x^2}\times\frac{1}{x^4}dx+C[/tex]
[tex]t=-2x^4{\intx^{-2}dx+C}[/tex]
[tex]t=2x^4\times\frac{1}{x}+Cx^4[/tex]
[tex]t=2x^3+Cx^4[/tex]
Substitute [tex]t=\frac{1}{y^2}[/tex] then we get
[tex]\frac{1}{y^2}=2x^3+Cx^4[/tex].
Answer: [tex]\frac{1}{y^2}=2x^3+Cx^4[/tex].
The functions s and t are defined as follows.
s(x) = -3x -4
t(x) = 4x - 1
Find the value of t(s(5)) .
The answer is:
[tex]t(s(5))=-77[/tex]
Why?To solve the problem, first, we need to compose the functions, and then evaluate the obtained function. Composing function means evaluating a function into another function.
We have that:
[tex]f(g(x))=f(x)\circ g(x)[/tex]
From the statement we know the functions:
[tex]s(x)=-3x-4\\t(x)=4x-1[/tex]
We need to evaluate the function "s" into the function "t", so:
[tex]t(s(x))=4(-3x-4)-1\\\\t(s(x))=-12x-16-1=-12x-17[/tex]
Now, evaluating the function, we have:
[tex]t(s(5))=-12(5)-17=-60-17=-77[/tex]
Have a nice day!
The goal for the size of the Santa on a Christmas Santa cup is 3.5 cm (T) with an acceptable tolerance of ± 0.9 cm. The grand mean of the size of the Santa from the samples that were taken is 3.4 cm (m) and the standard deviation is 0.28 cm. What is CPk? (rounded to three decimals) 1.500 0.952 0.800 0.705 0.000
Answer:
The Cpk is 0.952
Step-by-step explanation:
The formula to calculate the Cpk of a process is
[tex]Cpk = min(\frac{USL-mean}{3*sigma}, \frac{mean-LSL}{3*sigma} )[/tex]
where
USL (Upper Specification Limit) =3.5cm+0.9cm = 4.4cm
LSL (Lower Specification Limit) =3.5cm-0.9cm=2.6cm
Standard Deviation = sigma = 0.28cm
Mean = 3.4cm
So,
[tex]Cpk=min(\frac{4.4-3.4}{3*0.28} ,\frac{3.4-2.6}{3*0.28})\\\\Cpk=min(\frac{1}{0.84} ,\frac{0.8}{0.84})\\\\Cpk=min(1.190 ,0.952)\\\\\\[/tex]
The Cpk is 0.952
The process capability index or CPk is calculated using the formula min([USL - m]/3σ, [m - LSL]/3σ). In this case, USL is calculated as 4.4 cm and LSL is found to be 2.6 cm. The final CPk value is the smaller of the two resulting values, which in this case is 0.952.
Explanation:The question asks for the calculation of CPk, which is an index in statistics determining the potential capability of a process in meeting the specification limits. This index considers both the variability of the process and the target in its calculation. The formula for CPk is given by
CPk = min([USL - m]/3σ, [m - LSL]/3σ)
where:
m is the grand mean, σ is the standard deviation, USL (Upper Specification Limit) is T + tolerance, and LSL (Lower Specification Limit) is T - tolerance.
Using the given values from the question,
USL = 3.5 cm + 0.9 cm = 4.4 cm,
LSL = 3.5 cm - 0.9 cm = 2.6 cm,
[USL - m]/3σ = (4.4 cm - 3.4 cm) / (3 * 0.28 cm) = 1.19,
[m - LSL]/3σ = (3.4 cm - 2.6 cm) / (3 * 0.28 cm) = 0.952.
The CPk value will be the smaller of these two values, which is 0.952.
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How do we conduct a formal hypothesis test of a claim made about two population standard deviations or variances.
Answer:
Given:
Standard deviation or variance of two population.
We need to write a method by which a formal hypothesis test can be conducted of claim made about two population standard deviations or variances.
In General Chi-Square test and F-test are used for variance or standard deviation.
Also Chi-Square test and F-test require that the original population be normally distributed.
Now for Testing a Claim about Variance or Standard Deviation
To test a claim about the value of the variance or the standard deviation of population, then we use the test statistic which follows chi-square distribution with n − 1 degrees of freedom, and is given by the following formula.
[tex]\chi^2=\frac{(n-1)s^2}{\sigma_0^2}[/tex]
Where s is for given standard deviation and [tex][\sigma[/tex] is for claimed standard deviation.
First we make Hypothesis, then we choose the value of α ( level of significance ), after that using above formula we find value of chi-square.
then we find table value for the chosen α also known as table value or p-value. Finally we give final answer by checking relation between p-value and α.
If p-value < α then null hypothesis is rejected
If p-value > α then null hypothesis accepted.
Consider the function on the interval (0, 2π). f(x) = sin x + cos x (a) Find the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation.)
Answer:Increasing in x∈(0,π/4)∪(5π/4,2π) decreasing in(π/4,5π/4)
Step-by-step explanation:
given f(x) = sin(x) + cos(x)
f(x) can be rewritten as [tex]\sqrt{2} [\frac{sin(x)}{\sqrt{2} }+\frac{cos(x)}{\sqrt{2} } ]..................(a)\\\\\ \frac{1}{\sqrt{2} } = cos(45) = sin(45)\\\\[/tex]
Using these result in equation a we get
f(x) = [tex]\sqrt{2} [ cos(45)sin(x)+sin(45)cos(x)]\\\\= \sqrt{2} [sin(45+x)]..........(b)[/tex]
Now we know that for derivative with respect to dependent variable is positive for an increasing function
Differentiating b on both sides with respect to x we get
f '(x) = [tex]f '(x)=\sqrt{2} \frac{dsin(45+x)}{dx}\\ \\f'(x)=\sqrt{2} cos(45+x)\\\\f'(x)>0=>\sqrt{2} cos(45+x)>0[/tex]
where x∈(0,2π)
we know that cox(x) > 0 for x∈[0,π/2]∪[3π/2,2π]
Thus for cos(π/4+x)>0 we should have
1) π/4 + x < π/2 => x<π/4 => x∈[0,π/4]
2) π/4 + x > 3π/2 => x > 5π/4 => x∈[5π/4,2π]
from conditions 1 and 2 we have x∈(0,π/4)∪(5π/4,2π)
Thus the function is decreasing in x∈(π/4,5π/4)
To find the intervals of increase and decrease for f(x) = sin x + cos x, we first find the derivative f'(x) = cos x - sin x. By setting the derivative equal to zero, we find the critical points at x = π/4 + kπ. By testing these intervals in the derivative, we can identify the intervals of increase and decrease.
Explanation:The function given is f(x) = sin x + cos x, which is a combination of a sine and cosine function. To find the intervals where the function is increasing or decreasing, we need to find the derivative of the function first. The derivative of sin x is cos x, and the derivative of cos x is -sin x. So, the derivative of the function f(x) is f'(x) = cos x - sin x.
By setting the derivative equal to zero, cos x - sin x = 0, we can find the critical points where the function may change from increasing to decreasing or vice versa. The solutions to this equation are x = π/4 + kπ, where k is an integer.
Using these points, we can find the intervals of increase and decrease. For instance, if we test a number between 0 and π/4 in the derivative, we find that the function is increasing on the interval (0, π/4). Continuing this process for the rest of the intervals should provide all the intervals of increase and decrease for the function.
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Use the construction in proof of the Chinese reminder theorem to find all solutions to the system of congruence:
x ≡ 2 ( mod 3 )
x ≡ 1 ( mod 4 )
x ≡ 3 ( mod 7 )
Answer:
17,101,185, 269,.... is the solution.
i.e. x≡17 mod(84) is the solution
Step-by-step explanation:
Given that the system is
[tex]x ≡ 2 ( mod 3 )x ≡ 1 ( mod 4 )x ≡ 3 ( mod 7 )[/tex]
Considering from the last as 7 is big,
possible solutions would be 10,17,24,...
Since this should also be 1(mod4) we get this as 1,5,9,...17, ...
Together possible solutions would be 17, 45,73,121,....
Now consider I equation and then possible solutions are
5,8,11,14,17,20,23,26,29,...,47,....75, ....
Hence solution is 17.
Next number satisfying this would be 101, 185, ...
A trash company is designing an open-top, rectangular container that will have a volume of 1080 ft cubed. The cost of making the bottom of the container is $5 per square foot, and the cost of the sides is $4 per square foot. Find the dimensions of the container that will minimize total cost.
Answer:
Dimensions of the container should be 12×12×7.5 ft to minimize the making cost.
Step-by-step explanation:
A trash company is designing an open top, rectangular container having volume = 1080 ft³
Let the length of container = x ft , width of the container = y ft and height of the container = z ft.
So volume of the rectangular container = xyz = 1080 ft³
Or [tex]z=\frac{1080}{xy}[/tex] ft -----(1)
Cost of making the bottom of the container = $5 per square ft
Area of the bottom = xy
Cost of making the bottom @ $5 per square ft = 5xy
Area of all sides of the container = 2(xz + yz) = 2z(x+ y)
Now it has been given that cost of making all sides of the container is = $4 per square ft
So total cost to manufacture sides = 4[2z(x + y)]
Now cost of making bottom and sides of the container = 5xy + 8z(x + y)
We put the value of z from equation 1
Total cost A = 5xy+8(x + y)[tex](\frac{1080}{xy})[/tex]
A = 5xy +[tex]8(\frac{1080}{y})+8(\frac{1080}{x})[/tex]
Now we will find the derivative of A and equate it to the zero
[tex]\frac{dA}{dx}=0[/tex] and [tex]\frac{dA}{dy}=0[/tex]
[tex]\frac{dA}{dx}=5y+8(1080)(0)+8(1080)(-\frac{1}{y^{2}})=0[/tex]
5y =[tex]\frac{8\times1080}{y^{2} }[/tex]
5y³ = 8640
y³ =[tex]\frac{8640}{5}=1728[/tex]
y = 12 ft
For [tex]\frac{dA}{dy}=0[/tex]
[tex]\frac{dA}{dy}=5x+\frac{8(-1080)}{x^{2}}[/tex]=0
5x =[tex]\frac{8(1080)}{x^{2} }[/tex]
5x³ = 8640
x³ = 1728
x = 12
Now from equation 1
z =[tex]\frac{1080}{x}[/tex]
=[tex]\frac{1080}{144}[/tex]
z = 7.5
Therefore, dimensions of the container should be 12×12×7.5 ft to minimize the making cost.
Final answer:
To minimize the total cost, we need to minimize the cost of the bottom and the cost of the sides. We can find the dimensions of the container that will minimize the total cost by solving a system of equations.
Explanation:
Let's assume that the length of the rectangular container is x ft, the width is y ft, and the height is z ft.
The volume of the container is given as 1080 ft3.
Therefore, we have the equation:
x * y * z = 1080
The cost of making the bottom of the container is $5 per square foot and the cost of the sides is $4 per square foot.
The cost of the bottom is 5 * (x * y).
The cost of the sides is 4 * (2xy + 2xz + 2yz).
To minimize the total cost, we need to minimize the cost of the bottom and the cost of the sides.
First, let's solve the volume equation for x:
x = (1080) / (y * z)
Substituting the value of x in the cost equation, we have:
Cost = 5 * (1080) / (y * z) * y + 4 * (2 * (1080 / (y * z)) * y + 2 * (1080) / (y * z) * z + 2 * y * z)
Now, we can find the minimum cost by taking the derivative of the cost equation with respect to y and z, and setting it equal to zero.
Then, we solve the resulting system of equations to find the values of y and z that minimize the cost.
Finally, we substitute the values of y and z back into the volume equation to find the value of x.
By solving the equations, we can find the dimensions of the container that will minimize the total cost.
For a certain population, the regression equation to predict salary (in dollars) from education (in years) is y=2530x + 5200. What are the units of 2530 ? Choose the correct answer. You have only one attempt.
Answer:
2530 has no units
Step-by-step explanation:
In order to understand the units from a linear equation we need to understand the general equation of a line which is:
y=mx+b where:
m=slope of the line
b=y-intercept.
Comparing the given equation with the general line equation, we noticed that 2530 represents the slope of the line.
Since the slope can be obtained by:
m=(y2-y1)/(x2-x1) whatever the units are, the slope is dimensionless, which means that 2530 has no units.
Grace is three times as old as Hans, but in 5 years she will be twice as old as Hans is then. How old are they now? Set up an then solve a system of linear equations. please show step by step
Answer:
3x - y = 0; 2x - y = -5
Step-by-step explanation:
Let x be the present age of Hans and y be the present age of Grace,
Since, in present Grace is three times as old as Hans,
⇒ y = 3x
⇒ 3x - y = 0
Now, after 5 years,
The age of Hans = x + 5,
And, the age of Grace = y + 5
Also, in 5 years Grace will be twice as old as Hans is then,
⇒ y + 5 = 2 ( x + 5 )
⇒ y + 5 = 2x + 10
⇒ 2x - y = -5
Hence, the required system of linear equations is,
3x - y = 0; 2x - y = -5
Find LCD and solve 1/3-1/8+5/18
Answer:
The LCD is 72; the sum is 35/72
Step-by-step explanation:
Let's find the least common denominator (LCD) and find a solution.
The given expression: [tex]\frac{1}{3}-\frac{1}{8}+\frac{5}{18}[/tex] has three fractions from which their denominators can be expressed as the multiplication of prime numbers:
Fraction 1: 1/3 --> 3 is a prime number
Fraction 2: 1/8 --> 8=2*4=2*2*2
Fraction 3: 5/18 --> 18=3*6=3*2*3
Now, the next step is considering that if a number is repeated using two different fractions, one of the numbers is deleted. Notice that 'fraction 1' has a 3 and 'fraction 3' also has a 3, so we delete one '3'. Now notice that 'fraction 2' has a 2 and 'fraction 3' also has a 2, so we delete one '2'. So initially we have:
(3)*(2*2*2)*(3*2*3)
But after the previous process (erasing one '3' from the first fraction and one '2' from the second fraction) we now have:
(2*2)*(3*2*3)
Doing the math we obtain (2*2)*(3*2*3)=72, so 72 is our LCD.
Now we have to multiply each fraction in order to obtain the same denominator (LCD=72) for all fractions, so:
For fraction 1: 1/3 --> (1/3)*(24*24)=24/72
For fraction 2: 1/8 --> (1/8)*(9/9)=9/72
For fraction 3: 5/18 --> (5/18)*(4/4)=20/72
Now we can sum all the fractions (remember the correct sign for each fraction):
24/72 - 9/72 + 20/72 = (24-9+20)/72 = 35/72
What is 75percent of 300
Answer:
225
Step-by-step explanation:
To find your answer, multiply 300 by the decimal form of 75%, which is 0.75.
[tex]300 * 0.75 = 225[/tex]
Answer:
Half of 300 is 150 and half of 150 is 75, which is 25%. 75 x 3 gives us 225. 225 is the answer
Step-by-step explanation:
Solve differential equation:
y'''+4y''-16y'-64y=0 y(0)=0, y'(0)=26, y''(0)=-16
Final answer:
To solve the given differential equation y'''+4y''-16y'-64y=0 with initial conditions, we can use the characteristic equation method. By finding the roots of the characteristic equation and applying the initial conditions, the general solution is obtained as y(t) = (-16/21)e^(-8t) + (8/21)e^(2t) + (8/21)e^(-4t).
Explanation:
To solve the given differential equation, we can use the characteristic equation method. We first find the characteristic equation by substituting y = e^(mt) into the differential equation, which gives us the equation (m^3 + 4m^2 - 16m - 64)e^(mt) = 0. Since e^(mt) is never zero, we can simplify the equation to m^3 + 4m^2 - 16m - 64 = 0.
Using a numerical method or factoring, we find that the roots of the characteristic equation are m = -8, m = 2, and m = -4. Therefore, the general solution to the differential equation is y(t) = c1e^(-8t) + c2e^(2t) + c3e^(-4t), where c1, c2, and c3 are constants determined by the initial conditions.
Using the given initial conditions y(0) = 0, y'(0) = 26, and y''(0) = -16, we can solve for the constants. Substituting t = 0 into the general solution and its derivatives, we get the equations c1 + c2 + c3 = 0, -8c1 + 2c2 - 4c3 = 26, and 64c1 + 4c2 + 16c3 = -16. Solving these equations, we find c1 = -16/21, c2 = 8/21, and c3 = 8/21.
Therefore, the solution to the differential equation is y(t) = (-16/21)e^(-8t) + (8/21)e^(2t) + (8/21)e^(-4t).
Solves 7/4 =3/x Round to the nearest tenth.
Answer:
x = 12/7 or 1.7
Step-by-step explanation:
first, cross multiply to get 7x = 12. then, divide 12 by 7 to get 12/7, which can be simplified and rounded to 1.7.
Final answer:
To solve 7/4 = 3/x, use cross multiplication to get 7x = 12, then divide both sides by 7 to find x, which is approximately 1.7 when rounded to the nearest tenth.
Explanation:
To solve the equation 7/4 = 3/x, we can set up a proportion and use cross multiplication. Cross multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction, and setting the products equal to each other. In this case, we multiply 7 by x and 4 by 3 to get the equation 7x = 12.
After cross multiplying, divide both sides of the equation by 7 to solve for x. Doing this, we find that x = 12/7. To convert this to a decimal and round to the nearest tenth, we can divide 12 by 7 using a calculator or long division, resulting in approximately 1.7.
use a Venn diagram and the given information to n(union) = 103, n(A) = 35, n(B) = 42, n(C) = 45, n(A intersection B) = 8, n(A intersection C) = 8, n(B intersection C) = 6, and n(A intersection (B intersection C) = 3. Find n(A intersection (B union C)'). A) 4 B) 22 C) 3 D) 26
Answer:
The correct option is B.
Step-by-step explanation:
Given information: n(A) = 35, n(B) = 42, n(C) = 45, n(A∩B) = 8, n(A∩C) = 8, n(B∩C) = 6, and n(A∩B∩C) = 3.
We need to find the value of n(A∩(B∩C)')
Using venn diagram we get
n(A∩B∩C')=n(A∩B)-n(A∩B∩C)= 8-3 = 5
n(A∩B'∩C)=n(A∩C)-n(A∩B∩C)= 8-3 = 5
n(A'∩B∩C)=n(B∩C)-n(A∩B∩C)= 6-3 = 3
n(A∩(B∪C)')=n(A)-n(A∩B'∩C)-n(A∩B∩C')-n(A∩B∩C)
n(A∩(B∪C)')=35-5-5-3 = 22
The value of n(A∩(B∪C)') is 22. Therefore the correct option is B.
Convert the density of surface sea water to metric tons/cubic meter.
Answer: 1.029 tons/m³
Step-by-step explanation:
Density of sea water may varies for different temperature but at 25°C the density of sea water is taken as 1029 kg/m³.
Density of sea water is 1029 kg/m³
1 tons=1000 kg
1 kg=0.001 tons
Density of sea water=1029×0.001 tons/m³
=1.029 tons/m³
hence the density will be 1.029 tons/m³
Answer:
According to the internet, the density of the water is 1023.6 kg/m^ 3, but when it is converted, it is 1.0273 tons/m^3.
Step-by-step explanation:
This is what I found, hope it helps.
Emily wants to rent a cargo trailer to move her son into an apartment when he returns to college. A+ Rental charges $0.60 per mile while Rock Bottom Rental charges $70 plus $0.25 per mile. Let x be the number of miles driven, and let y be the cost of the rental. Write a linear equation for each company. DO NOT SOLVE.
Answer:
A+ Rental charges $0.60 per mile.
Rock Bottom Rental charges $70 plus $0.25 per mile.
Let 'x' be the number of miles driven. Let 'y' be the cost of the rental.
The equation for A+ Rental:
y = $0.60x
The equation for Rock Bottom Rental:
y = $0.25x + $70