Final answer:
To construct a 99% confidence interval for the proportion of people who accept the theory, you need to calculate the point estimate, standard error, margin of error, lower bound, and upper bound.
Explanation:
The psychologist wants to construct a 99% confidence interval for the proportion of people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain. Let's calculate the confidence interval:
Calculate the point estimate: 64 out of 708 people agreed with the theory, so the estimated proportion is 64/708 = 0.0904.
Calculate the standard error: SE = sqrt((0.0904 * (1 - 0.0904)) / 708) = 0.0091.
Calculate the margin of error: ME = 2.57 * 0.0091 ≈ 0.0234.
Calculate the lower bound: Lower bound = 0.0904 - 0.0234 ≈ 0.067.
Calculate the upper bound: Upper bound = 0.0904 + 0.0234 ≈ 0.113.
So, with 99% confidence, the proportion of all people who accept the theory is between 0.067 and 0.113.
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.
Learn more about Area Maximization here:https://brainly.com/question/34713449
#SPJ3
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.
UESTION 2 120 MARKS Underground cable breakdown often occur due to the unpredictable deterioration rate of the cable insulation. Cable replacement can be very costly without regular mainte A 9-month study on an underground cable insulation had been conducted to check for the length of the cable insulation that have been deteriorated using fluorescence microscope in order to estimate the maintenance cycle. Table 1 shows the data measured every 3 months during the study Table 1 Time, t (month) Deteriorated cable insulation length, I 1.35 3.72 7.28 15.45 (a) Generate a third order polynomial using Newton's interpolation method to describe the variable of the deteriorated cable insulation length, (t) with respect to time (in month). Write the polynomial in the form of At3BtCt D, where [10 marks] A, B, C and D are constants. (b) According to regulation, the cable insulation requires replacement if it is degraded by 6 mm. Calculate the replacement time using Newton Raphson method with initial guess of 6 months from the polynomial generated in part (a). Perform THREE iterations only and calculate true percent relative error for the final iteration answer [10 marks if the true value of the replacement time is 5 months.
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.
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.
Learn more about Intervals of Increase and Decrease here:https://brainly.com/question/11051767
#SPJ3
checking congruence
Check the picture below.
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%.
Learn more about probability here:https://brainly.com/question/32117953
#SPJ3
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.
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
1) For A = {a, b, c, d, e} and B = {yellow, orange, blue, green, white, red, black}. a) Define a relation R from A to B that is a function and contains at least 4 ordered pairs. b) What is the domain of this function? c) What is the range of this function?
Answer: The required function is R = {(a, blue), (b, green), (c, green), (d, white), (e, black)}, domain, D = {a, b, c, d, e} and range, R = {blue, green, white, black} .
Step-by-step explanation: We are given the following two sets :
A = {a, b, c, d, e}
and
B = {yellow, orange, blue, green, white, red, black}.
We are to define a relation R from A to B that is a function and contains at least 4 ordered pairs. Also, to find the domain and range of the function.
(a) Let the function R be defined as follows :
R = {(a, blue), (b, green), (c, green), (d, white), (e, black)}.
Since R contains five ordered pairs, so this will fulfill our criterion. Also, since each first element is associated with one and only one second element, so R defines a function.
(b) We know that
the domain of a function is the set of all the first elements in the ordered pairs, so the domain of the function R will be
D = {a, b, c, d, e}.
(c) We know that
the range of a function is the set of all the second elements in the ordered pairs, so the domain of the function R will be
R = {blue, green, white, black}.
Thus, the required function is R = {(a, blue), (b, green), (c, green), (d, white), (e, black)}, domain, D = {a, b, c, d, e} and range, R = {blue, green, white, black} .
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, ...
State the domain and range of each relation. Then determine whether the relation is a function.
(6,3)
(10,3)
(-1,3)
(0,3)
Answer:
Domain {-1,0,6,10}
Range: {3}
Yes it's a function.
Step-by-step explanation:
Domain is all the x's used by your relation. In case of a set of points all you have to do is your list x's. Domain: {-1,0,6 ,10}.
Range is all the y's being used by your relation. In case of a set of points all you have to do is list your y's. Range {3}.
Function: For it be a function no x can be paired with more than one y. Basically all the x's have to be different. In they are in this case so it is a function.
Answer:
See below.
Step-by-step explanation:
The domain is the set of x-values = {0, -1, 6, 10}.
The range is {3}.
The relation is a function because each element of the domain maps on to only one value of the range.
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%.
Betty paints twice as fast as Dan. Working together, Dan and Betty can paint 2, 400 square feet in 4 hours. Another employee, Sue, joined their painting team. Working together, Dan, Betty, and Sue can paint 3, 600 square feet in 3 hours. If Sue works alone, how many square feet can she paint in 4 hours and 27 minutes? a 600 square feet b 1, 570 square feet c 1, 700 square feet d 2, 530 square feet e 2, 670 square feet
Answer:
2670 square feet. Option e.
Step-by-step explanation:
Dan and Betty can paint 2,400 square feet in 4 hours.
They can paint in one hour [tex]\frac{2400}{4}[/tex] = 600 square feet.
Since given that Betty paints twice as fast as Dan. Let us take an equation:
Let Betty = B, Dan = D and Sue = S
B = 2D
4(B+D) = 2400
4B + 4D = 2400
12D = 2400
D = 200 sq. ft.
B = 2D = 400 sq. ft.
Therefore, Dan can paint 200 square feet in 1 hour and Betty paints twice 400 square feet in 1 hour.
Now given three of them can paint 3,600 square feet in 3 hours.
3( B+D+S) = 3600
3B + 3D + 3S = 3600
3(400) + 3(200) + 3(S) = 3600
1200 + 600 + 3S = 3600
S = 600 Sq. ft.
Sue can paint 600 square feet in one hour.
So sue can paint in 4 hours and 27 minutes.
[tex](\frac{4+27}{60})[/tex] × 600
= 2670 square feet. Option e.
Sue's painting rate is 600 square feet per hour. For a total of 4 hours and 27 minutes, which is 4.45 hours when converted, she can paint 2,670 square feet.
Explanation:The question is asking for Sue's rate of painting when she works alone given the painting rates when they all work together. From the given information, we know that Dan and Betty together can paint 2,400 square feet in 4 hours, which means their combined painting rate is 2,400 ÷ 4 = 600 square feet per hour. Additionally, we know that Dan, Betty, and Sue together can paint 3,600 square feet in 3 hours. This means their combined rate is 3,600 ÷ 3 = 1,200 square feet per hour.
Because Sue's rate is the only variable that changes between these two situations, we can determine her rate by subtracting the combined rate of Dan and Betty from the combined rate of the whole team. This gives us 1,200 - 600 = 600 square feet per hour for Sue.
To find out how many square feet she can paint in 4 hours and 27 minutes, we need to convert 27 minutes into hours, which is 27 ÷ 60 = 0.45 hours. Adding this to the 4 hours, we get 4.45 hours. Multiplying Sue's rate by this time gives us 600 × 4.45 = 2,670 square feet, which matches answer option (e).
Learn more about solving rates here:
https://brainly.com/question/29623648
#SPJ3
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
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.
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.
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 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.
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].
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.
Solve the following congruence equations for X a) 8x = 1(mod 13) b) 8x = 4(mod 13) c) 99x = 5(mod 13)
Answer:
a) 5+13k where k is integer
b) 20+13k where k is integer
c)12+13k where k is integer
Step-by-step explanation:
(a)
[tex]8x \equiv 1 (mod 13) \text{ means } 8x-1=13k[/tex].
8x-1=13k
Subtract 13k on both sides:
8x-13k-1=0
Add 1 on both sides:
8x-13k=1
I'm going to use Euclidean Algorithm.
13=8(1)+5
8=5(1)+3
5=3(1)+2
3=2(1)+1
Now backwards through the equations:
3-2=1
3-(5-3)=1
3-5+3=1
(8-5)-5+(8-5)=1
2(8)-3(5)=1
2(8)-3(13-8)=1
5(8)-3(13)=1
So compare this to:
8x-13k=1
We see that x is 5 while k is 3.
Anyways 5 is a solution or 5+13k is a solution where k is an integer.
b)
[tex]8x \equiv 4 (mod 13)[/tex]
8x-4=13k
Subtract 13k on both sides:
8x-13k-4=0
Add 4 on both sides:
8x-13k=4
We got this from above:
5(8)-3(13)=1
If we multiply both sides by 4 we get:
8(20)-13(12)=4
So x=20 and 20+13k is also a solution where k is an integer.
c)
[tex]99x \equiv 5 (mod 13)[/tex
99x-5=13k
Subtract 13k on both sides:
99x-13k-5=0
Add 5 on both sides:
99x-13k=5
Using Euclidean Algorithm:
99=13(7)+8
13=8(1)+5
Go back through the equations:
13-8=5
13-(99-13(7))=5
8(13)-99=5
99(-1)+8(13)=5
Compare this to 99x-13k=5 and see that x=-1 or -1+13=12 or 12+13k is a solution where k is an integer.
Answer:
a) x = 5 mod 13.
b) x = 7 mod 13.
Step-by-step explanation:
a) 8x = 1 mod 13
x = 2, 16 = 3 mod 13
x = 3, 24 = 11 mod 13
x = 4, 32 = 6 mod 13
x = 5 , 40 = 1 mod 13
8x = 40
x = 5 mod 13.
b) 8x = 4 mod 13
x = 7, 56 = 4 mod 13.
7 = 4 mod 13
x = 7 mod 13.
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).
Use Newton's Method to approximate the zero(s) of the function. Continue the iterations until two successive approximations differ by less than 0.001. Then find the zero(s) to three decimal places using a graphing utility and compare the results. f(x) = x5 + x − 6
Answer:
There is only one real zero and it is located at x = 1.359
Step-by-step explanation:
After the 4th iteration the solution was repeating the first 3 decimal places. The formula for Newton's Method is
[tex]x_{n}-\frac{f(x_{n}) }{f'(x_{n}) }[/tex]
If our function is
[tex]f(x)=x^5+x-6[/tex]
then the first derivative is
[tex]f'(x)=5x^4+1[/tex]
I graphed this on my calculator to see where the zero(s) looked like they might be, and saw there was only one real one, somewhere between 1 and 2. I started with my first guess being x = 1.
When I plugged in a 1 for x, I got a zero of 5/3.
Plugging in 5/3 and completing the process again gave me 997/687
Plugging in 997/687 and completing the process again gave me 1.36976
Plugging in 1.36976 and completing the process again gave me 1.359454
Plugging in 1.359454 and completing the process again gave me 1.359304
Since we are looking for accuracy to 3 decimal places, there was no need to go further.
Checking the zeros on the calculator graphing program gave me a zero of 1.3593041 which is exactly the same as my 5th iteration!
Newton's Method is absolutely amazing!!!
Using Newton's Method with an initial guess of 1.5, approximations for f(x) = 0 are x ≈ 1.189, close to the actual zeros.
To approximate the zero(s) of the function f(x) = x^5 + x - 6 using Newton's Method, we start with an initial guess, x0. The formula for the iterative step is:
x_(n+1) = x_n - f(x_n) / f'(x_n),
where f'(x) is the derivative of f(x). In this case, f'(x) = 5x^4 + 1.
1. Choose an initial guess, say x0 = 1.5.
2. Iterate using the formula: x_(n+1) = x_n - (x_n^5 + x_n - 6) / (5x_n^4 + 1).
Continue these iterations until two successive approximations differ by less than 0.001. It may take several iterations to reach this level of accuracy.
After reaching a sufficiently accurate result, use a graphing utility to confirm the zero(s) to three decimal places. This will help ensure the accuracy of the approximation from Newton's Method. The actual zeros of the function are approximately x ≈ 1.189, x ≈ -1.187, and x ≈ 1.999. Compare the results from Newton's Method to these values.
For such more questions on initial
https://brainly.com/question/30684257
#SPJ2
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]
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
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.
https://brainly.com/question/31538429
#SPJ3
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!
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]
Emily was going to sell all of her stamp collection to buy a video game. After selling half of them she changed her mind. She then bought seventeen more. Write an expression for how many she has now.
i think you have to write an expression to this...
Answer:
s/2 +17
Step-by-step explanation:
If s represents the number of stamps Emily started with, then the number she had after selling half of them is ...
s/2
After purchasing 17 more, she had ...
s/2 +17
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.