To find the orthogonal trajectories of the given family of curves, we first need to understand what the term "orthogonal" means. In simple terms, two lines or curves are said to be orthogonal if they intersect at a right angle. Now, coming back to the problem, the given family of curves can be written as x^2 - 2y^2 = c, where c is a constant.
To find the orthogonal trajectories, we need to differentiate this equation with respect to y, treating x as a constant. This gives us:
-4xy = dy/dx
Now, we need to find the equation of the curves that intersect the given family of curves at a right angle, i.e., the slopes of the curves must be negative reciprocals of each other. Therefore, we can write:
dy/dx = 4xy/k
where k is a constant. To solve this differential equation, we can separate the variables and integrate:
∫dy/4xy = ∫dx/k
ln|y| - ln|x^2| = ln|c| + ln|k|
ln|y/x^2| = ln|ck|
y/x^2 = ±ck
Therefore, the orthogonal trajectories of the given family of curves are given by y = ±kx^2/c, where k is a constant. These curves intersect the original family of curves at right angles.
1. Identify the family of curves: The given equation is x^2 + 2y^2 = c, where c is a constant. This represents a family of ellipses with different sizes depending on the value of c.
2. Calculate the derivative: To find the orthogonal trajectories, we first need to find the derivative of the given equation with respect to x. Differentiate both sides with respect to x:
d/dx(x^2) + d/dx(2y^2) = d/dx(c)
2x + 4yy' = 0
3. Find the orthogonal slope: The slope of the orthogonal trajectory is the negative reciprocal of the original slope. Since the original slope is y', the orthogonal slope is -1/y':
Orthogonal slope = -1/y'
4. Replace the original slope with the orthogonal slope:
2x + 4y(-1/y') = 0
5. Solve for y':
y' = -2x/(4y)
6. Solve the differential equation: Now we have a first-order differential equation to find the equation of the orthogonal trajectories:
dy/dx = -2x/(4y)
Separate variables and integrate both sides:
∫(1/y) dy = ∫(-2x/4) dx
ln|y| = -x^2/4 + k
7. Solve for y:
y = e^(-x^2/4 + k) = C * e^(-x^2/4), where C is a new constant.
The orthogonal trajectories of the given family of curves are represented by the equation y = C * e^(-x^2/4).
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A woman wants to construct a box whose base length is twice the base width. The material to build the top and bottom is $9/m^2 and the material to build the sides is $6/m^2. If the woman wants the box to have a volume of 70 m3, determine the dimensions of the box (in metres) that will minimize the cost of production. What is the minimum cost?
The minimum cost of production is $278.46.
The base width of the box be x, then the base length is 2x. Let the height of the box be h.
The volume of the box is given by:
V = base area × height
[tex]70 = x[/tex] × [tex]2x[/tex] × [tex]h[/tex]
[tex]h = 35/x^2[/tex]
The cost of producing the box is given by:
[tex]C = 2[/tex] ×[tex](cost of top/bottom) + 4[/tex] × [tex](cost of side)[/tex]
[tex]C = 2[/tex]× [tex]9[/tex]× [tex](2x[/tex] × [tex]x) + 4[/tex] × [tex]6[/tex] × [tex](2x + 2h)[/tex]
[tex]C = 36x^2 + 48xh[/tex]
Substituting the expression for h obtained above:
[tex]C = 36x^2+ 48x(35/x^2)[/tex]
[tex]C = 36x^2+ 1680/x[/tex]
To minimize C, we take the derivative with respect to x and set it to zero:
[tex]dC/dx = 72x - 1680/x^2 = 0[/tex]
[tex]72x = 1680/x^2[/tex]
[tex]x^3 = 1680/72[/tex]
[tex]= 23.33[/tex]
x = 2.82 m (rounded to two decimal places)
Substituting this value of x in the expression for h, we get:
[tex]h = 35/(2.82)^2[/tex]
[tex]= 4.34 m[/tex] (rounded to two decimal places)
Therefore, the dimensions of the box that minimize the cost of production are:
[tex]Base width = x = 2.82 m[/tex]
[tex]Base length = 2x = 5.64 m[/tex]
[tex]Height = h = 4.34 m[/tex]
To find the minimum cost, we substitute these values of x and h in the expression for C:
[tex]C = 36(2.82)^2 + 1680/(2.82)[/tex]
[tex]C = $278.46[/tex] (rounded to two decimal places)
Therefore, the minimum cost of production is $278.46.
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if he sees a wolf, a boy will cry wolf with probability 0.8. if he does not see a wolf, the boy will cry wolf anyway with probability 0.4. if the probability that there is a wolf would otherwise be 0.25, what is the probability that there really is a wolf when the boy crys wolf?
The probability that there really is a wolf when the boy cries wolf is 0.54.
Let A be the event that the boy cries wolf and B be the event that there is a wolf. We are given the following probabilities:
P(A|B) = 0.8 (the probability that the boy cries wolf when there is a wolf)
P(A|B') = 0.4 (the probability that the boy cries wolf when there is no wolf)
P(B) = 0.25 (the probability that there is a wolf)
We want to find P(B|A), the probability that there really is a wolf given that the boy cries wolf.
We can use Bayes' theorem to calculate this:
P(B|A) = P(A|B) * P(B) / [P(A|B) * P(B) + P(A|B') * P(B')]
Substituting the given values, we get:
P(B|A) = 0.8 * 0.25 / [0.8 * 0.25 + 0.4 * 0.75] = 0.54
Therefore, the probability that there really is a wolf when the boy cries wolf is 0.54.
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Find the two following values
The value of angle TUW is 32⁰.
The value of angle UTV is 25⁰.
What is the value of angle TUW?
The value of angle TUW is calculated by applying the following formula.
angle TVW = angle TUW (vertical opposite angles are equal)
angle TVW = 32⁰
So, angle TUW = 32⁰
The value of angle UTV is calculated as;
angle UTV = VWU (vertical opposite angles are equal)
3x + 4 = 2x + 11
3x - 2x = 11 - 4
x = 7
angle UTV = 3x + 4
= 3(7) + 4
= 25⁰
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Which of the plotted points is in the second quadrant
Answer:
Point C is in the second quadrant. C is the correct answer.
Use the graph of the rational function to complete the following statement.
As , .
Question content area bottom left
Part 1
As ,
enter your response here.
.
.
.
Question content area right
Part 1
-10
-8
-6
-4
-2
2
4
6
8
10
-10
-8
-6
-4
-2
2
4
6
8
10
x
y
A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. A graph has three branches and asymptotes y= 1, x = negative 3 and x =3. The first branch is above y equals 1 and to the left of x equals negative 3 comma approaching both. The second branch opens downward between the vertical asymptotes comma reaching a maximum at left parenthesis 0 comma 0 right parenthesis . The third branch is above y equals 1 and to the right of x equals 3 comma approaching both.
Asymptotes are shown as dashed lines. The horizontal asymptote is y = 1 The vertical asymptotes are x = -3 and x=3
The end behavior of the rational function is described as follows:
As x -> ∞, f(x) -> 1.
What is the horizontal asymptote of a function?The horizontal asymptote is the value of f(x) as x goes to infinity, as long as this value is different of infinity.
For this problem, we have that both when x goes to negative infinity and when x goes to positive infinity, the graph of the function goes to y = 1, hence the end behavior of the function is defined by the horizontal asymptote as follows:
As x -> ∞, f(x) -> 1.
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a better estimate is obtained by assuming that each lake is a separate tank with only clean water flowing in. use this approach to determine how long ti would take the pol lution level ni each lake ot be reduced to 50% of its original level. how long would ti take ot reduce the pollution to %5 of its original level?
It would take around 8 hours to bring each lake's pollution level down to 50% of its starting point, and around 22 hours to bring it down to 5%.
Assuming each lake is a separate tank with only clean water flowing in, we can use the exponential decay model [tex]$A = A_0e^{-kt}$[/tex], where $A$ is the amount of pollutant at time t, A₀ is the initial amount of pollutant, and k is the decay constant.
To find the time it would take to reduce the pollution level in each lake to 50% of its original level, we need to solve the equation [tex]$0.5A_0 = A_0e^{-kt}$[/tex] for t:
[tex]0.5A_0 &= A_0e^{-kt} \\frac{0.5A_0}{A_0} &= e^{-kt} \\ln\left(\frac{0.5A_0}{A_0}\right) &= -kt \\ln(0.5) &= -kt \t &= \frac{\ln(0.5)}{-k}\end{align*}[/tex]
To find the time it would take to reduce the pollution level in each lake to 5% of its original level, we need to solve the equation[tex]$0.05A_0 = A_0e^{-kt}$[/tex] for t:
[tex]0.05A_0 &= A_0e^{-kt} \\frac{0.05A_0}{A_0} &= e^{-kt} \\ln\left(\frac{0.05A_0}{A_0}\right) &= -kt \\ln(0.05) &= -kt \t &= \frac{\ln(0.05)}{-k}\end{align*}[/tex]
The decay constant $k$ can be found by using the given information that each lake is replaced by clean water every 8 hours. This means that the half-life of the pollutant is 8 hours, which gives us:
[tex]0.5A_0 &= A_0e^{-k(8)} \\ln(0.5) &= -8k \k &= -\frac{\ln(0.5)}{8} \approx 0.08664\end{align*}[/tex]
Substituting this value of k into the equations we derived earlier, we get:
[tex]t_{50} = \frac{\ln(0.5)}{-k} \approx 8.006 \text{ hours}[/tex]
[tex]t_{5} &= \frac{\ln(0.05)}{-k} \approx 22.133 \text{ hours}[/tex]
Therefore, it would take approximately 8 hours to reduce the pollution level in each lake to 50% of its original level, and approximately 22 hours to reduce it to 5% of its original level.
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NEED ANSWERS ASAP!! PLS
The US government monitors the consumption of different products. The table shows y, the amount of ice cream consumed, in millions of pounds, for * years since 2010. The quadratic equation that models the amount of ice cream consumed, in millions of pounds, since 2010 is shown. y = 12(¢ - 6)2 + 3922 Determine when the amount of ice cream consumed in the United State would be 5,650 millions of pounds.
The amount of ice cream consumed in the United State would be 5,650 millions of pounds in 2028.
How to determine when the amount of ice cream is 5,650 millions of pounds?Based on the information provided about the mount of ice cream consumed in the United State, a quadratic equation that models the amount of ice cream consumed, in millions of pounds, since 2010 is given by;
y = 12(x - 6)² + 3922
Where:
y is the amount of ice cream consumed, in millions of pounds.x is the number of years since 2010.By substituting the value of y, the number of years can be calculated as follows;
5,650 = 12(x - 6)² + 3922
5,650 - 3922 = 12(x - 6)²
1728 = 12(x - 6)²
144 = (x - 6)²
12 = x - 6
x = 12 + 6
x = 18 years.
Since 2010; 2010 + 18 = 2028.
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How much greater is the value of the 6 inand 4786.53 Denton 3821.69
Find an ONB (orthonormal basis) for the following plane in R3 x + 5y + 4z = 0 First, solve the system, then assign parameters s and t to the free variables (in this order), and write the solution in vector form as su + tv. Now normalize u to have norm 1 and call it ū. Then find the component of v orthogonal to the line spanned by u and normalize it, call it ī. Below, enter the components of the vectors ū = [ū1, ū2, ū3]and ū = ū1, 72, 73)".
The ONB for the given plane in R3 is ū = [-5/√(26), 1/√(26), 0] and ī = [25/(√(26/13)), -5/(√(26/13)), 0].
To find an orthonormal basis for the plane x + 5y + 4z = 0, we first solve the system and get the parametric solution
x = -5t - 4s
y = t
z = s
Assigning parameters s and t to the free variables and writing the solution in vector form as su + tv, we get
[-5t - 4s, t, s] = t[-5, 1, 0] + s[-4, 0, 1]
Taking u = [-5, 1, 0] and v = [-4, 0, 1], we normalize u to have norm 1 by dividing it by its length
||u|| = √(26)
ū = [-5/√(26), 1/√(26), 0]
To find the component of v orthogonal to u, we take the dot product of v and u, and divide it by the dot product of u and u, and then multiply u by this scalar
v - ((v · u) / (u · u))u
v · u = -5
u · u = 26
v - (-5/26)[-5, 1, 0]
v - [25/26, -5/26, 0]
Finally, we normalize this vector to have norm 1
||v - proj_u v|| = √(26/13)
ī = [25/(2√(26/13)), -5/(2√(26/13)), 0]
Therefore, the orthonormal basis for the plane x + 5y + 4z = 0 is ū = [-5/√(26), 1/√(26), 0] and ī = [25/(√(26/13)), -5/(√(26/13)), 0].
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the weights of newborn baby boys born at a local hospital are believed to have a normal distribution with a mean weight of 3315 grams and a variance of 391,876 . if a newborn baby boy born at the local hospital is randomly selected, find the probability that the weight will be less than 4504 grams. round your answer to four decimal places.
The probability that the weight will be less than 4504 grams is 0.9713.
We need to standardize the value 4504 using the given mean and variance, and then use the standard normal distribution table to find the corresponding probability.
The standard deviation is the square root of the variance: √391876≈626.05
So, the z-score for a weight of 4504 grams is:
z=(4504−3315)/626.05≈1.8974
Using a standard normal distribution table, we find that the probability of a z-score being less than 1.8974 is approximately 0.9713.
Therefore, the probability that a newborn baby boy born at the local hospital will weigh less than 4504 grams is approximately 0.9713.
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For a four strokes engine, the camshaft that controls the valves a timing rotates at a. Double the engine rotational speed b. Half the engine rotational speed
c. Same engine rotational speed
d. none of the options
The correct answer is option (c) same engine rotational speed.
In a four-stroke engine, the camshaft rotates at half the speed of the crankshaft. The camshaft controls the opening and closing of the engine valves, which allows for the intake of fuel and air and the expulsion of exhaust gases.
When the engine rotational speed is doubled, the crankshaft will rotate twice as fast, but the camshaft will still rotate at half the speed of the crankshaft. This means that the camshaft will still rotate at the same speed as before, and the valve timing will remain the same.
Similarly, when the engine rotational speed is halved, the camshaft will still rotate at half the speed of the crankshaft. This means that the camshaft will still rotate at the same speed as before, and the valve timing will remain the same.
Therefore, the correct answer is option (c) same engine rotational speed.
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This question has two parts.
A wooden block is a prism, which is made up of two cuboids with the dimensions shown. The volume of the wooden block is 427 cubic inches.
Part A
What is the length of MN?
Write your answer and your work or explanation in the space below.
Part B
200 such wooden blocks are to be painted. What is the total surface area in square inches of the wooden blocks to be painted?
A) The length MN of the given wooden block is: 12
B) 80400 in²
How to find the surface area and volume of the prism?1) The formula for volume of a cuboid is:
Volume = Length * Width * Height
Thus:
427 = (MN * 7 * 3) + (5 * 5 * 7)
427 = 21MN + 175
21MN = 252
MN = 252/21
MN = 12
2) Surface area of entire object is:
TSA = 2(12 * 3) + 2(12 * 7) - (5 * 7) + 2(7 * 3) + 3(5 * 7) + 2(5 * 5)
= 402 in²
For 200 blocks:
TSA = 200 * 402 = 80400 in²
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In a large city, the average number of lawn mowings during summer is normally distributed with mean u and standard deviation 0-8.7. If I want the margin of error for a 90% confidence interval to be +3, I should select a simple random sample of size (4 decimal points)
To achieve a margin of error of +3 for a 90% confidence interval for the average number of lawn mowings during summer in a large city with a mean of u and a standard deviation of 0-8.7, a simple random sample of size 18 should be selected.
To determine the sample size needed to achieve a margin of error of +3 for a 90% confidence interval, we can use the formula:
n = (z * σ / E)^2
where n is the sample size, z is the z-score for the desired confidence level (in this case, 1.645 for 90% confidence), σ is the standard deviation, and E is the margin of error.
Substituting the given values into the formula, we get:
n = (1.645 * 0.8 / 3)^2 = 17.18
Rounding up to the nearest whole number, we get a sample size of 18.
Therefore, to achieve a margin of error of +3 for a 90% confidence interval for the average number of lawn mowings during summer in a large city with a mean of u and a standard deviation of 0-8.7, a simple random sample of size 18 should be selected. This sample size ensures that the estimate of the population means based on the sample mean is within +3 of the true population mean with 90% confidence.
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Please help I have to do this before state testing this I one out of 32 questions also if you so happen to be mrs Billie from Alhambra traditional school I hate you
Answer:turn right 45 degrees, then turn right another 45 degrees. flip the figure x-axis wise/horizontally
Step-by-step explanation:
The total cost for funding a trip for the senior class to go to the fall fair, C(x), is a function of the number of students that will make the trip, x. The trip will not be taken until at least 5 students sign up to go. This relationship can be modeled by the function shown.
C(x) = 350 + 7.50x
What is the domain and range for this situation?
The value of domain and range for this situation are,
Domain = (- ∞, ∞)
Range = (- ∞, ∞)
We have to given that;
The total cost for funding a trip for the senior class to go to the fall fair, C(x), is a function of the number of students that will make the trip, x.
Now, We have;
⇒ C (x) = 350 + 7.5x
Clearly, the function is a polynomial.
Hence, The value of domain and range for this situation are,
Domain = (- ∞, ∞)
Range = (- ∞, ∞)
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Brody has a jar with 1000 g of sugar in it. Each day, he empties out half
of the sugar that is in the jar. At the end of the first day, he is left with
500 g of sugar.
a) How much sugar will be left in the jar at the end of the 5th day? Give
your answer in grams (g).
b) Write a sentence to explain whether or not the jar will ever be empty
31.25 g on the 5th day
no it will never by empty because even when it gets down to one singular piece of sugar you would technically just cut it in half, then that in half, yes it would get impossible, that's why you wouldn't actually do it, but if you typed it into a calculator it would just keep getting a smaller and smaller decimal.
According to a poll of adults, about 46% work during their summer vacation. Assume that the true proportion of all adults that work during summer vacation is p=0.46. Now consider a random sample of 400 adults. Complete parts a and b below.
a. What is the probability that between 39% and 53% of the sampled adults work during summer vacation?
The probability is. (Round to three decimal places as needed.)
b. What is the probability that over 63% of the sampled adults work during summer vacation?
The probability is (Round to three decimal places as needed.)
To find the probability we do the following:
1/20 ÷ 5 could someone answer this
The value of the expression is 1/100.
We have,
To solve this expression, we can use the division property of fractions which states that dividing by a fraction is the same as multiplying by its reciprocal.
So, we have:
1/20 ÷ 5
= 1/20 x 1/5 (reciprocal of 5 is 1/5)
= 1/100 (multiply numerator with numerator and denominator with denominator)
Therefore,
The simplified result is 1/100.
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I need help on this 40 points I need to turn it in in like 5 min
Answer:
Step-by-step explanation:
13. B
14. A
15. D
Help me please and explain im so confused
The value of cos S to the nearest hundredth is 0.54
What is trigonometric ratio?Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle.
Sin(tetha) = opp/hyp
cos(tetha) = adj/hyp
tan(tetha) = opp/adj
Here in this triangle, the hypotenuse is 28
and the opposite to angle S is line TU
The adjascent is 15
therefore cos S = adj/hyp
= 15/28
= 0.54 ( nearest hundredth)
therefore the value of cos S is 0.54
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Which of the following statements is false concerning the hypothesis testing procedure for a regression model?The F-test statistic is used.An α level must be selected.The null hypothesis is that the true slope coefficient is equal to zero.The null hypothesis is rejected if the adjusted r2 is above the critical value.The alternative hypothesis is that the true slope coefficient is not equal to zero.
The statements that is false concerning the hypothesis testing procedure for a regression model is "The null hypothesis is rejected if the adjusted r2 is above the critical value".
The statement that the null hypothesis is rejected if the adjusted r2 is above the critical value is false concerning the hypothesis testing procedure for a regression model.
The F-test statistic is used to test the overall significance of the regression model, and an α level must be selected to determine the level of significance.
The null hypothesis is that the true slope coefficient is equal to zero, which means that there is no linear relationship between the dependent variable and the independent variable.
The alternative hypothesis is that the true slope coefficient is not equal to zero, which means that there is a linear relationship between the dependent variable and the independent variable.
The adjusted R-squared value is a measure of the goodness of fit of the regression model and represents the proportion of the variation in the dependent variable that is explained by the independent variable(s) in the model.
The null hypothesis is rejected if the F-test statistic is above the critical value, which indicates that the regression model is statistically significant and the independent variable(s) have a significant linear relationship with the dependent variable.
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Use the table of the probability distribution to find the variance
The variance of the given distribution is 0.9.
Calculating the anticipated value or distribution mean is the first step in determining a probability distribution's variance. The anticipated value is a weighted average of all potential outcomes, with each outcome's probability serving as the weight. The expected value can be expressed mathematically as follows:
[tex]E(X) =[/tex] Σ[tex][xi[/tex] × [tex]P(xi)][/tex]
where μ is the mean of the data.
Then, calculate μ:
μ [tex]= (1+2+3+4+5)/5[/tex]
[tex]=(15/5)[/tex]
[tex]= 3[/tex]
and replace this value and the values of xn and P(xn) into the formula for the variance, just as follow:
Hence, the variance of the given distribution is 0.9.
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Which of the following is not a valid probability
A. 1
B. 1.1
C. 0
D. 0.001
Answer:
B
Step-by-step explanation:
It is impossible for a possibility to be more than 1, or 100%.
Consider the the following series. [infinity] 1 n3 n = 1 (a) Use the sum of the first 10 terms to estimate the sum of the given series. (Round the answer to six decimal places. ) s10 = (b) Improve this estimate using the following inequalities with n = 10. (Round your answers to six decimal places. ) sn + [infinity] f(x) dx n + 1 ≤ s ≤ sn + [infinity] f(x) dx n ≤ s ≤ (c) Using the Remainder Estimate for the Integral Test, find a value of n that will ensure that the error in the approximation s ≈ sn is less than 10-5
The estimated sum of the given series using the sum of the first 10 terms is 302,500, the improved estimate for the sum of the given series is between 305,000 and 306,000, and the value of n is 8.
(a) Utilizing the equation for the entirety of the primary n terms of the arrangement, we have:
[tex]s10 = 1^3 + 2^3 + ... + 10^3[/tex]
= 1,000 + 8,000 + ... + 1,000,000
= 302,500
In this manner, the assessed whole of the given arrangement using the entirety of the primary 10 terms is 302,500.
(b) For n = 10, we have:
[tex]sn = 1^3 + 2^3 + ... + 10^3 ≈ 302,500[/tex]
Utilizing the disparities with[tex]f(x) = x^3[/tex], we have:
[tex]sn + ∫[10,∞] x^3 dx ≤ s ≤ sn + ∫[10,∞] x^3 dx + 10^3[/tex]
Utilizing calculus, ready to assess the integrand:
[tex]sn + ∫[10,∞] x^3 dx = sn + [1/4 x^4] [10,∞] = sn + 2500[/tex]
[tex]sn + ∫[10,∞] x^3 dx + 10^3 = sn + [1/4 x^4] [10,∞] + 10^3 = sn + 3500[/tex]
Substituting sn = 302,500, we get:
302,500 + 2500 ≤ s ≤ 302,500 + 3500
305,000 ≤ s ≤ 306,000
In this manner, the made strides assess for the sum of the given arrangement is between 305,000 and 306,000.
(c) The Leftover portion Gauge for the Necessarily Test states that the mistake E in approximating the whole s of an interminable arrangement by the nth halfway entirety sn is:
[tex]E ≤ ∫[n+1,∞] f(x) dx[/tex]
In this case, we need to discover mean of n such that E < 10 using the integral test
xss=removed xss=removed> [tex][(10^-5 x 4)^(1/4)] - 1[/tex]
n > 7.9378
Subsequently, we require n = 8 to guarantee that the blunder within the estimation s ≈ sn is less than[tex]10^-5.[/tex]
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A construction crew has just built a new road. They built the road at a rate of 6 kilometers per week. They built 7. 68 kilometers of road. How many weeks did it take them?
We can use the formula:
distance = rate × time
where distance is the total length of the road built, rate is the speed at which the road was built, and time is the number of weeks it took to build the road.
Substituting the given values, we get:
7.68 kilometers = 6 kilometers/week × time
To solve for time, we can divide both sides of the equation by 6 kilometers/week:
7.68 kilometers ÷ 6 kilometers/week = time
Simplifying the left-hand side, we get:
1.28 weeks = time
Therefore, it took the construction crew approximately 1.28 weeks to build the road.
Find the distance between the points given.
(3, 4) and (6, 8)
5
√22
√7
Answer:
To find the distance between two points, we can use the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) and (x2, y2) are the coordinates of the two points.
In this case, the two points are (3, 4) and (6, 8), so we have:
d = sqrt((6 - 3)^2 + (8 - 4)^2)
d = sqrt(3^2 + 4^2)
d = sqrt(9 + 16)
d = sqrt(25)
d = 5
Therefore, the distance between the points (3, 4) and (6, 8) is 5 units.
It's worth noting that the values 5√22 and √7 do not match the above
Sales tax is 7%. What is the tax on a book that costs $12?
Please help asap :( Find the exact length of arc ADC. In your final answer, include all of your calculations
Answer:
15 Pi m
Step-by-step explanation:
arc ADC = 360 Degrees - 60 Degrees divided by 360 Degrees Multiplied by 2 Pi Multiplied by 9
= 5/6 Times 18 Pi
=15 Pi m
Find a basis of the null space N(A) for the the matrix. Then find an orthogonal basis using Gram-Schmidt process. [1 2 1 3 2]
A= [4 1 0 6 1]
[1 1 2 4 5]
We apply the Gram-Schmidt process to these vectors to find an orthonormal basis:
v1 = x1 = [3, -4, 1
To find a basis of the null space N(A), we need to find all vectors x such that Ax = 0, where 0 is the zero vector.
To do this, we set up the augmented matrix [A | 0] and row reduce:
[ 1 2 1 3 2 | 0 ]
[ 4 1 0 6 1 | 0 ]
[ 1 1 2 4 5 | 0 ]
R2 - 4R1 -> R2:
[ 1 2 1 3 2 | 0 ]
[ 0 -7 -4 6 -7 | 0 ]
[ 1 1 2 4 5 | 0 ]
R3 - R1 -> R3:
[ 1 2 1 3 2 | 0 ]
[ 0 -7 -4 6 -7 | 0 ]
[ 0 -1 1 1 3 | 0 ]
R2 / -7 -> R2:
[ 1 2 1 3 2 | 0 ]
[ 0 1 4/7 -6/7 1 | 0 ]
[ 0 -1 1 1 3 | 0 ]
R1 - 2R2 - R3 -> R1:
[ 0 0 0 0 0 | 0 ]
[ 0 1 4/7 -6/7 1 | 0 ]
[ 0 0 11/7 -1/7 1 | 0 ]
We can write the system of equations corresponding to this row echelon form as:
x2 + (4/7)x3 - (6/7)x4 + x5 = 0
(11/7)x3 - (1/7)x4 + x5 = 0
Solving for the variables in terms of the free variables x3, x4, and x5, we get:
x1 = -[(4/7)x3 - (6/7)x4 - x5]/2
x2 = -(4/7)x3 + (6/7)x4 - x5
x3 = x3 (free variable)
x4 = x4 (free variable)
x5 = x5 (free variable)
So the null space N(A) is the set of all vectors of the form:
x = [ -[(4/7)x3 - (6/7)x4 - x5]/2, -(4/7)x3 + (6/7)x4 - x5, x3, x4, x5 ]
To find an orthogonal basis for N(A), we can use the Gram-Schmidt process. Let's call the columns of A a1, a2, a3, a4, and a5.
First, we need to find a basis for N(A) by setting the free variables to 1 and the others to 0:
x1 = [3, -4, 1, 0, 0]
x2 = [-2, 3, 0, 1, 0]
x3 = [-2, 1, 0, 0, 1]
Next, we apply the Gram-Schmidt process to these vectors to find an orthonormal basis:
v1 = x1 = [3, -4, 1
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supply of houses is determined by two variables (I and R) in the following way: h(1,R) = a log1 + b R + CR log1, where a, b, and c are all constants. How does housing supply respond to changes in I (a) and R (OR)? an an Select one: an a. ar an a+cR an I and an = b + clog1 2 an ī and a b+cR log1 = b + cR log1 O b. a1 an a+cR an C. ai and an an d. ar an + CR log 1 and aR = b + c log1
The housing supply function is given by h(I,R) = a log1 + bR + cR log1. The housing supply responds to changes in I with a rate of a, and to changes in R with a rate of b + c log1.
Based on the given equation, the housing supply (h) is determined by two variables: I and R. The equation shows that h is a function of R, with a log-linear relationship. The variable I only appears as a constant (a) in the equation, so changes in I do not directly affect the supply of houses.
On the other hand, changes in R (or OR, which is the same variable) do affect the supply of houses. Specifically, an increase in R leads to an increase in the supply of houses. The magnitude of this increase depends on the values of b and c in the equation.
To see this, we can take the partial derivative of h with respect to R:
dh/dR = b + cR/(ln(10))
This equation tells us how much the housing supply changes in response to a change in R. The derivative is positive (i.e. the supply increases) as long as c is positive. The larger c is, the greater the increase in supply for a given increase in R.
Therefore, the correct answer is:
b + cR/(ln(10))
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