Answer:
The volume is [tex]\frac{490\pi}{39}[/tex] cubic units.
Step-by-step explanation:
The given curve is
[tex]y=7x^6[/tex]
The given line is
[tex]y=7x[/tex]
Equate both the functions to find the intersection point of both line and curve.
[tex]7x^6=7x[/tex]
[tex]7x^6-7x=0[/tex]
[tex]7x^6-7x=0[/tex]
[tex]7x(x^5-1)=0[/tex]
[tex]7x=0\rightarrow x=0[/tex]
[tex]x^5-1=0\rightarrow x=1[/tex]
According to washer method:
[tex]V=\pi \int_{a}^{b}[f(x)^2-g(x)^2]dx[/tex]
Using washer method, where a=0 and b=1, we get
[tex]V=\pi \int_{0}^{1}[(7x)^2-(7x^6)^2]dx[/tex]
[tex]V=\pi \int_{0}^{1}[49x^2-49x^{12}]dx[/tex]
[tex]V=49\pi \int_{0}^{1}[x^2-x^{12}]dx[/tex]
[tex]V=49\pi [\frac{x^3}{3}-\frac{x^{13}}{13}]_0^1[/tex]
[tex]V=49\pi [\frac{1^3}{3}-\frac{1^{13}}{13}-(0-0)][/tex]
[tex]V=49\pi [\frac{1}{3}-\frac{1}{13}][/tex]
[tex]V=49\pi (\frac{13-3}{39})[/tex]
[tex]V=49\pi (\frac{10}{39})[/tex]
[tex]V=\frac{490\pi}{39}[/tex]
Therefore the volume is [tex]\frac{490\pi}{39}[/tex] cubic units.
The volume of the solid obtained by rotating the region bounded by the curves y = 7x6 and y = 7x, for x ≥ 0, around the x-axis is determined by setting up and evaluating a volume integral using the method of cylindrical shells. The intersection points of the curves, that serve as the limits of the integral, are found by equating the two functions, giving x = 0 and x = 1.
Explanation:To find the volume of the solid obtained by rotating the area between the curves y = 7x6 and y = 7x around the x-axis, we use the method of cylindrical shells.
Firstly, we find the intersection points of the two curves by setting them equal to each other: 7x6 = 7x, which gives us x = 0 and x = 1.
Next, we set the volume integral and evaluate it over the range [0,1]. The general formula is V = 2π ∫ from a to b [x * (f(x) - g(x)) dx]. Here, f(x) = 7x6 and g(x) = 7x, therefore,
V = 2π ∫ from 0 to 1 [x * (7x6 - 7x) dx] = 2π ∫ from 0 to 1 [7x7 - 7x2 dx],
which can be integrated term-by-term. The antiderivatives of x7 and x2 are (1/8)x8 and (1/3)x3, respectively. By implementing these with the Fundamental Theorem of Calculus, we then substitute the limits of integration, subtract, and simplify to reach the final volume of the rotated solid.
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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).
Ms. Smith drove a total of 700 miles on a business to
If her car averaged 35 miles per gallon of gasoline a
gasoline cost $1.25 per gallon, what was the cost in
dollars of the gasoline for the trip?
Answer:
The total amount of gas Ms. Smith used on the trip is $25.
Step-by-step explanation:
To solve this problem, we first must figure out how many gallons of gas Ms. Smith used on her trip. If she drove 700 miles and averaged 35 miles per gallon, if we divide 700 by 35 we can figure out how many gallons she used.
700/35 = 20
Thus, Ms. Smith used 20 gallons on her trip. Next, to figure out what the cost of the gasoline was, we must multiply the number of gallons (20) by the cost of gasoline per gallon ($1.25).
20 gallons * $1.25/gal = $25
Therefore, the cost of gasoline for the trip was $25.
Hope this helps!
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%.
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.
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).
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could someone please help and explain .
Answer:
WXYZ = Ro(180°, (2, -3))(ABCD)
Step-by-step explanation:
A reflection across two perpendicular lines (y=-3, x=2) is equivalent to reflection across their point of intersection. That, in turn, is equivalent to rotation 180° about that point of intersection.
Your double reflection is equivalent to rotation 180° about (2, -3).
Suppose you're taking an Honors Algebra 1 multiple choice test. The test consists of 40 questions, each having 5 options. If you guess at all 40 questions, what is the mean of the number of correct answers?
Answer:8
Step-by-step explanation:
We have given test consists of 40 multiple choice questions having five options for each question.
Suppose there is one correct answer to each question
therefore probabilty of getting a correct answer on making a guess is [tex]\frac{1}{5}[/tex]
i.e. 1 out of 5 questions is correct
Using binomial distribution
where n=40 p=[tex]\frac{1}{5}[/tex]
mean of binomial distribution is np
therefore mean of no of correct answers=[tex]40\times \frac{1}{5}[/tex]
=8
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.
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.
In the year 1985, a house was valued at $113,000. By the year 2005, the value had appreciated exponentially to $155,000. What was the annual growth rate between 1985 and 2005 (Rourid your answer to two decimal places.) Assume that the value continued to grow by the same percentage. What was the value of the house in the year 2010? (Round your answer to the nearest dolar.)
Answer:
The growth rate is 1.02; the value of the house in 2010 is $185,388
Step-by-step explanation:
This is an exponential growth equation, therefore, it follow the standard form:
[tex]y=a(b)^x[/tex]
where y is the value of the house after a certain number of years,
a is the initial value of the house,
b is the growth rate, and
x is the year number.
We are going to make this easy on ourselves and call year 1985 year 0. Therefore, is year 1985 is year 0, then year 2005 is year 20, and year 2010 is year 25. We will make these the x coordinates in our coordinate pairs.
(0, 113000) and (20, 155000)
Filling into our standard form using the first coordinate pair will give us the initial value of the house at the start of our problem:
[tex]113000=a(b)^0[/tex]
Anything raised to the 0 power is equal to 1, so
113000 = a(1) and
a = 113000
Now we will use that value of a along with the second pair of coordinates and solve for b, the growth rate you're looking for:
[tex]155000=113000(b)^{20}[/tex]
Start by dividing both sides by 113000 to get a decimal:
[tex]1.371681416=b^{20}[/tex]
To solve for b, we have to undo that power of 20 by taking the 20th root of b. Because this is an equation, we have to take the 20th root of both sides:
[tex]\sqrt[20]{1.371681416}=\sqrt[20]{b^{20}}[/tex]
The 20th root and the power of 20 undo each other so all we have left on the right is a b, and taking the 20th root on your calculator of the decimal on the left gives you:
b = 1.0159 which rounds to
b = 1.02 This is our growth rate.
Now we can use this growth rate and the value of a we found to write the model for our situation:
[tex]y=113000(1.02)^x[/tex]
If we want to find the value of the house in the year 2010 (year 25 to us), we sub in a 25 for x and do the math:
[tex]y=113000(1.02)^{25}[/tex]
Raise 1.02 to the 25th power and get:
y = 113000(1.640605994) and multiply to get a final value of
y = $185,388
The annual exponential growth rate from 1985 to 2005 is 1.40%. Using this rate, the estimated value of the house in 2010 would be approximately $176,927.
Explanation:In order to calculate the exponential growth rate, we use the formula: R = (final value/initial value)^(1/n) - 1, where R is the annual rate, n is the number of years. So, R = (155,000/113,000)^(1/20) - 1 = 0.0140 or 1.40%.
To find the value of the house in 2010, we use the exponential growth formula: future value = present value * (1 + annual rate)^n. The value in 2010 would be $155,000 * (1 + 0.0140)^5 = $176,927 (rounded to the nearest dollar).
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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
checking congruence
Check the picture below.
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!
Stackable polystyrene cups have a height h1=12.5 cm. Two stacked cups have a height of h2=14 cm. Three stacked cups have a height of h3=15.5 cm. Find the equation for hx= 1.5 x+ Your friend is 200 cm tall. Find out how many cups you will need to reach the height of your friend. cups
Answer:
Approximately 59 stacked cups.
Step-by-step explanation:
Given,
Height of a cup = 12.5 cm,
Two stacked cups = 14 cm,
Three stacked cups = 15.5 cm,
........, so on,....
Thus, there is an AP that represents the given situation,
12.5, 14, 15.5,....
First term is, a = 12.5,
Common difference, d = 1.5 cm,
Thus, the height of x cups is,
[tex]h(x) = a+(x-1)d = 12.5 + (x-1)1.5 = 1.5x + 11[/tex]
According to the question,
h(x) = 200
⇒ 1.5x + 11 = 200
⇒ 1.5x = 189
⇒ x = 59.3333333333 ≈ 59,
Hence, approximately 59 stacked cups will need.
Answer:
hx = 1.5cm . x + 11 cm
126 cups
Step-by-step explanation:
We have the following ordered pairs (x, hx).
(1, 12.5 cm)(2, 14 cm)(3, 15.5 cm)We are looking for a linear equation of the form:
hx = a.x + b
where,
a is the slope
b is the y-intercept
To find the slope, we take any pair of ordered values and replace their values in the following expression.
[tex]a=\frac{\Delta hx }{\Delta x} =\frac{h2-h1}{2-1} =\frac{14cm-12.5cm}{2-1} =1.5cm[/tex]
Now, the general form is:
hx = 1.5cm . x + b
We can take any ordered pair and replace it in this expression to find b. Let's use h1.
h1 = 1.5cm . x1 + b
12.5 cm = 1.5 cm . 1 + b
b = 11 cm
The final equation is:
hx = 1.5cm . x + 11 cm
If hx = 200 cm,
200 cm = 1.5cm . x + 11 cm
189 cm = 1.5cm . x
x = 126
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
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.
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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, ...
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
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.
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].
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.
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 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]
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.
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} .
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
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.
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.
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
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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.
Find a particular solution to y" - y' + 9y = 3 sin 3x
Answer:
cos3x
Step-by-step explanation:
y" - y' + 9y = 3 sin 3x
[tex]D^{2}y-Dy+9y=3 sin3x[/tex]
[tex]y=\frac{3 sin 3x}{(D^{2} -D+9}=3 sin 3x[/tex]
here [tex]D^2[/tex] will be replaced by [tex]\alpha^2[/tex] where [tex]\alpha[/tex] is coefficient of x
[tex]y=\frac{3 sin 3x}{-3^{2} -D+9}[/tex]
[tex]y=-3\frac{sin 3x}{D}[/tex]
[tex]y=-3\int\ {sin 3x} \, dx[/tex]
[tex]y=-3\frac{cos3x}{-3}[/tex]
y=cos3x
hence Particular solution is cos3x