2 Consider the ellipsoid x² 4 4 ² 2²=18 Find an egregethe tangent plane to ellipsoid at the point (1,2,1) 6 Find parametric equations of the time that is normal to the ellipsoid at the point (421) Find the aut acute angle that the tangent plan at the point (1, 2, 1)
Answers
The equation of the tangent plane to the ellipsoid x²/18 + 4y²/18 + z²/2 = 1 at the point (1, 2, 1) is x/3 + 2y/3 + z/√2 = 5/3.
To find the equation of the tangent plane, we first need to find the gradient vector of the ellipsoid at the point (1, 2, 1). The gradient vector is given by:
grad(f) = (2x/18, 8y/18, 2z/2) = (x/9, 4y/9, z)
Substituting the coordinates of the point (1, 2, 1), we get:
grad(f)(1, 2, 1) = (1/9, 8/9, 1)
This vector is perpendicular to the tangent plane. Therefore, the equation of the tangent plane can be written as:
(x - 1)/9 + 4(y - 2)/9 + (z - 1)/√2 = 0
Simplifying, we get:
x/9 + 4y/9 + z/√2 = 5/3
Thus, the equation of the tangent plane to the ellipsoid at the point (1, 2, 1) is x/3 + 2y/3 + z/√2 = 5/3.
To find the normal line to the ellipsoid at the point (4, 2, 1), we can use the gradient vector found earlier. The parametric equations of the line are:
x = 1 + t/9
y = 2 + 4t/9
z = 1 + t/√2
where t is a parameter.
To find the acute angle between the tangent plane at the point (1, 2, 1) and the x-axis, we need to find the angle between the normal vector to the plane and the x-axis. The normal vector is (1/9, 8/9, 1/√2), and the x-axis vector is (1, 0, 0). Using the dot product formula, we get:
cosθ = (1/9) / √(1/81 + 64/81 + 1/2)
Simplifying, we get:
cosθ = 1/√234
Therefore, the acute angle between the tangent plane at the point (1, 2, 1) and the x-axis is cos⁻¹(1/√234).
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Related Questions
find the partial derivatives of the function (8y-8x)/(9x 8y)
Answers
The partial derivative of the function with respect to y is: ∂/∂y [(8y-8x)/(9x+8y)] = 8/(9x+8y) - (64x)/(9x+8y)^2To find the partial derivatives of the function (8y-8x)/(9x+8y), we need to take the derivative with respect to each variable separately.
First, let's find the partial derivative with respect to x. To do this, we treat y as a constant and differentiate the function with respect to x:
(8y-8x)/(9x+8y)
= (8y)/(9x+8y) - (8x)/(9x+8y)
Using the quotient rule, we can simplify this expression:
= (-8y(9))/((9x+8y)^2) - 8/(9x+8y)
Simplifying further, we get:
= (-72y)/(9x+8y)^2 - 8/(9x+8y)
Therefore, the partial derivative of the function with respect to x is:
∂/∂x [(8y-8x)/(9x+8y)] = (-72y)/(9x+8y)^2 - 8/(9x+8y)
Now, let's find the partial derivative with respect to y. To do this, we treat x as a constant and differentiate the function with respect to y:
(8y-8x)/(9x+8y)
= (8y)/(9x+8y) - (8x)/(9x+8y)
Using the quotient rule again, we get:
= 8/(9x+8y) - (8x(8))/((9x+8y)^2)
Simplifying further, we get:
= 8/(9x+8y) - (64x)/(9x+8y)^2
Therefore, the partial derivative of the function with respect to y is:
∂/∂y [(8y-8x)/(9x+8y)] = 8/(9x+8y) - (64x)/(9x+8y)^2
And that's how we find the partial derivatives of the function (8y-8x)/(9x+8y) using the quotient rule and differentiation with respect to each variable separately.
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if two sides of a square field were increased by five feet, as seen in the diagram, the area of the field would increase by 245 ft2 . find the area of the original square
Answers
If increasing the sides of a square field by five feet will increase the area by 245 ft², then the area of the original square is 484 ft².
To find the area of the original square, we can use the following formula:
Area of the original square = x²
where x is the original length of the square field.
Given that the increase in the length and width of the square field is 5 ft, the side length of the new square is (x + 5) ft. Therefore, the area of the new square is (x + 5)² ft².
Given that the area of the new square is 245 ft² more than the area of the smaller square, we can write:
(x + 5)² = 245 + x²
Expanding the left-hand side of the equation and simplifying, we get:
x² + 10x + 25 = 245 + x²
Solving for x, we get:
10x + 25 = 245
x = 22
Plugging x = 22 into the formula, we can find the area of the original square:
Area of the original square = x² = 22² = 484 ft²
Therefore, the area of the original square is 484 ft².
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Function f is a linear function and function g is defined by g(x) = f(x) + k. If the value of k is 7, how does the graph of g compare with the graph of f ?..
The graph of g is the graph of f
translated up 7 units
translated left 7 units
translated right 7 units
translated down 7 units
stretched vertically by a scale factor of 7
stretched horizontally by a scale factor of 7
Answers
The graph of g is Transformation up 7 units.
what is Transformation?
A graph transformation is a method for altering an existing graph or graphed equation to create a different version of the graph that follows. The modification of algebraic equations is a typical form of algebraic issue.
solution:
From the question, we have the following parameters that can be used in our computation:
g(x) = f(x) + k
Also, we have the value of k to be
k = 7
From the question, we can interpret g(x) = f(x) + k as f(x) is shifted up by k units
This means that
g(x) = f(x) + k
So, we have
g(x) = f(x) + 7
Hence, the transformation is (a) translated up 7 units
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Mimi wants to spend a $25 gift card on two kinds of in -app purchases for her favorite game: premium skins that cost $2.99, and tools that cost $4.99.
Answers
The equation that shows the way that Mimi can spend a $25 is 2.99p + 4.99t = 25.
What is an equation?A mathematical equation is the statement that illustrates that the variables given. In this case, two or more components are taken into consideration to describe the scenario.
Let the premium skins be p.
Let the number of tools be t.
This will be illustrated as:
(2.99 × p) + (4.99 × t) = 25
2.99p + 4.99t = 25
This illustrates the equation.
The complete question is given below.
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Mimi wants to spend a $25 gift card on two kinds of in -app purchases for her favorite game: premium skins that cost $2.99, and tools that cost $4.99. What is the equation to depict this?
Two coworkers picked up some writing instruments at the office supply store. Luther selected 8 boxes of pencils and 3 boxes of ballpoint pens, paying $17. Next, Cora spent $20 on 8 boxes of pencils and 4 boxes of ballpoint pens. How much does a box of each cost?
Answers
A box of ballpoint pens costs $5, but a box of pencils costs $4.
What is equation?An equation is a mathematical statement that shows the equality of two expressions. It consists of variables, numbers, and mathematical operations, and it states that the value of one expression is equal to the value of another expression. Equations can be used to model real-world problems and to solve mathematical problems.
Here,
Let x be the cost of a box of pencils and y be the cost of a box of ballpoint pens.
From Luther's purchases, we have the equation:
8x + 3y = 17
And from Cora's purchases, we have the equation:
8x + 4y = 20
We now have two equations with two unknowns, which we can solve using substitution or elimination. Let's use substitution.
We'll solve the first equation for y and substitute the result into the second equation:
y = 17 - 8x / 3
Substituting into the second equation:
8x + 4 (17 - 8x / 3) = 20
Expanding and simplifying the right side:
8x + 68 - 32x / 3 = 20
Combining like terms:
-24x / 3 + 68 = 20
Subtracting 68 from both sides:
-24x / 3 = -48
Dividing both sides by -24/3:
x = 4
So a box of pencils costs $4. To find the cost of a box of ballpoint pens, we can substitute x = 4 into either of the original equations:
8 * 4 + 3y = 17
Expanding and solving for y:
32 + 3y = 17
Subtracting 32 from both sides:
3y = -15
Dividing both sides by 3:
y = -5
So a box of ballpoint pens costs $5 and a box of pencils costs $4.
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HELPPPPPPPPPPP ME!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answers
Answer:76,904,685
Step-by-step explanation:
Need help figuring out this assignment
Answers
Answer:
Parallelogram. A parralelogram looks like a rectangle, but the end it tilted a bit. a parallelogram is a simple quadrilateral with two pairs of parallel sides.
Step-by-step explanation:
So, connect the arrows with one another.
arrow 1 --- connect with arrow 2
arrow 3 --- connect with arrow 4
then you get a parralelogram!
what is the biconditional statement of the following conditional statement? "if a polygon has four sides, then it is a quadrilateral." if a polygon does not have four sides, then it is a not quadrilateral. if a polygon is not a quadrilateral, then it is does not have four sides. a polygon has four sides if and only if it is a quadrilateral. if a polygon is a quadrilateral, then it has four sides.
Answers
The biconditional statement is a polygon has four sides if and only if it is a quadrilateral. Option C
What is a biconditional statement?A biconditional statement is defined as a statement that combines both an converse and conditional statement.
It is written in the "if and only if" from.
It is a statement that states a condition depends on another being and vice versa.
From the statement given;
"if a polygon has four sides, then it is a quadrilateral
The biconditional statement is ;
Polygon has four sides if and only if it is a quadrilateral.
Thus, the biconditional statement is a polygon has four sides if and only if it is a quadrilateral. Option C
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Based on current dividend yields and expected capital gains, the expected rates of return on portfolios A and B are 12% and 16%, respectively. The beta of A is .7, while that of B is 1.4. The T-bill rate is currently 5%, whereas the expected rate of return of the S&P 500 index is 13%. The standard deviation of portfolio A is 12% annually, that of B is 31%, and that of the S&P 500 index is 18%. a. Calculate the alphas for the two portfolios. (Negative amount should be indicated by a minus sign. Round your answers to 1 decimal place.)
Answers
the T-bill rate is 5%, and the expected rate of return of the S&P 500 index is 13%.
The alpha of a portfolio measures its risk-adjusted performance compared to the market. It is calculated by subtracting the risk-free rate from the portfolio's expected rate of return and adjusting for the portfolio's beta.
For portfolio A, the alpha can be calculated as follows:
Alpha_A = Expected return_A - Risk-free rate - Beta_A × (Expected return_market - Risk-free rate)
Alpha_A = 12% - 5% - 0.7 × (13% - 5%)
For portfolio B, the alpha can be calculated as follows:
Alpha_B = Expected return_B - Risk-free rate - Beta_B × (Expected return_market - Risk-free rate)
Alpha_B = 16% - 5% - 1.4 × (13% - 5%)
Calculating these expressions will give the alpha values for portfolios A and B.
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PLS ANSWER)
Side measures are preserved in which of the following types of transformations?
I. rotations
II. Reflections
III. Translations
A. I and III only
B. I and II only
C. I, II, and III
D. I only
Answers
I’m sure it’s the answer not positive though, I’m sorry.
1
Mrs. Wang has a bag that contains 15 pounds of sugar. For each cake she makes, she uses pound of sugar. How
5
many cakes can she make from the 15 pound bag of sugar?
F 20
75
H
5
15
1
20
Sharon Wells Mathematics Cumulum. Inc 2100-
Grade 5 So Wecks Week 1 Assessment
Answers
Answer: she can make 3 cakes with 15 pounds of sugar
Step-by-step explanation: 15÷5=3
how can you determine the coordinates of S(-7,1) after a reflection in the line y=3
Answers
The coordinates of S(-7,1) after a reflection over the line y = 3 is given as follows:
(-7,5).
How to obtain the coordinates after the reflection?The coordinates of the original point in this problem are given as follows:
S(-7,1).
The reflection line is given as follows:
y = 3.
The reflection line is an horizontal line, meaning that:
The x-coordinate will remain constant.The y-coordinate will be the same distance from the reflection line, but in an opposite direction.The y-coordinate of S is 2 units below of the reflection line, hence after the reflection it will be two units above, as follows:
3 + 2 = 5.
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Taylor read 32 pages on Monday. On Tuesday, she read twice as much as she read on Monday. On Wednesday, she read 14 fewer pages than she did on Tuesday. How many total pages did Taylor read during the three days?
Answers
Answer:
Taylor read 146 pages during these three days.
Step-by-step explanation:
32 pages on Monday.
On Tuesday twice, as much as Monday. So 2*32 = 64 on Tuesday.
On Wednesday, 14 less pages than Tuesday. 64 - 14 = 50. So he read 50 pages on Wednesday.
How many total pages did Taylor read during these three days?
32 + 64 + 50 = 146
Taylor read 146 pages during these three days.
The length of a rectangle is the same as its width. If the perimeter is 32 M what is its area?
Answers
The perimeter of a rectangle is calculated using the formula:
\(P=2(l+w)\)The question gives that the length and width of the rectangle are equal. This means that:
\(l=w\)Therefore, the perimeter will be calculated to be:
\(\begin{gathered} P=2(l+l)=2(2l) \\ P=4l \end{gathered}\)The perimeter is given to be 32 m. Therefore, the length of the rectangle will be:
\(\begin{gathered} 32=4l \\ l=\frac{32}{4} \\ l=8 \end{gathered}\)Using the formula for the area:
\(A=l^2\)Therefore, the area will be:
\(\begin{gathered} A=8^2 \\ A=64\text{ m}^2 \end{gathered}\)The area is 64 square meters.
Evaluate the expression for n=5. No spaces please.
4(n+4)+10
Answers
4(5+4)+10
20+16+10
=46
4(5+4)+10.... use PEMDAS
4(9)+10... multiply
36+10... add
46
have a nice day and if i helped in any way please consider giving me brainliest!
Select the correct answer from each drop-down menu. Brian makes an investment in his company's stock. What does the stock chart say about the stock price today? The down arrow indicates the stock price has decreased from yesterday's price of $ at the of the market day.
Answers
Answer: $66.20; close
Step-by-step explanation:
The down arrow indicates the stock price has decreased from yesterday’s price of $ 66.20 at the close of the market day.
Use the given information to find the new price. Round to the nearest cent Original Price: $28 Discount Rate: 25%
Answers
Answer:
To find the new price after a discount, you can use the formula: new price = original price * (1 - discount rate). In this case, the new price would be $28 * (1 - 0.25) = $28 * 0.75 = $<<28*0.75=21>>21. So the new price, rounded to the nearest cent, would be $21.
Step-by-step explanation:
At the tri-county track meet Jada won the 100 meter race in 12 9/10 seconds. The runner up finished in 13 1/10 seconds how many seconds faster was Jada then the funner up?
Answers
We have
\(\begin{gathered} 12\text{ }\frac{9}{10}\text{ = 12+}\frac{9}{10}\text{ = }\frac{120}{10}+\frac{9}{10}=\frac{129}{10} \\ 13\text{ }\frac{1}{10}=13+\frac{1}{10}=\frac{130}{10}+\frac{1}{10}=\frac{131}{10} \end{gathered}\)Then, the diference is
\(\frac{131}{10}-\frac{129}{10}=\frac{2}{10}=\frac{1}{5}\)So, Jada was 1/5 seconds faster than the second place.
The rectangular model is made up of squares. Each square is of equal size. What percentage of the model is shaded?
Answers
Answer:
44.5 percent is shaded.
from whole model, 44.5% is shaded of 100%
12r2s + 21rs2 – 3rs=
Answers
63rs
explanation:
collect like terms
12*2=24rs
21*2=42rs
then add 24+42=66rs
then subtract 3rs
then just add rs
Select the correct answer. rational functions v and w both have a point of discontinuity at x = 7. which equation could represent function w? a. w(x) = v(x − 7) b. w(x) = v(x 7) c. w(x) = v(x − 7) 7 d. w(x) = v(x) 7
Answers
The following equation could be used to represent a function w:
= w(x)=v(x-7)+7
According to the information provided,
The point of discontinuity of rational functions is at x=7.
When a rational function has a point of discontinuity, it generally occurs when,
q(x) = r(x-a), where x = a
In this case, we must pay attention to the following relationship, which is a combination of a parent rational function and a vertical translation:, (2)
If we know that a=7 and k=7.
The equation which can represent w is as follows,
w(x) = v ( x-7 ) + 7
A rational function can be represented as a polynomial split by another polynomial. Because polynomials are defined everywhere, the domain of a rational function is the set of all numbers except the zeros in the denominator.
Example: x = f(x) (x - 3). The denominator, x = 3, has only one zero. Rational functions are no longer defined when the denominator is zero.
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One number is 13 less than another number. Let x represent the greater number. What is the sum of these two numbers?
Answers
Answer:
2x - 13
Step-by-step explanation:
If x represents the greater number, then the other number is x - 13. The sum of these two numbers is:
x + (x - 13) = 2x - 13
Jon wanted to solve this problem: Given that sin 0=1/4, what is the value of cos 0? What did he do wrong?
Answers
Answer:
Step 2 is wrong
Step-by-step explanation:
\((Sin Ф )^{2}\)= \((1/4)^{2}\) = 1/16
I need help with finding 12x+3y=-3 on a graph.
Answers
Let's graph the equation
\(12x+3y=-3\)Notice that this equation corresponds to a line.
1. First, let's clear y, so we can get the equation of the line in the slope-intercept form:
\(\begin{gathered} 12x+3y=-3 \\ \rightarrow3y=-12x-3 \\ \rightarrow y=-4x-1 \end{gathered}\)This way, he equation of the line in the slope-intercept form is
\(y=-4x-1\)2. Now, we have to expose two points contained in the line. To do so, we give x any value we want and we calculate the corresponding y value.
Let's do this with x = 0 and x = 1
\(\begin{gathered} y=-4x-1 \\ \\ x=0\Rightarrow y=-4(0)-1\rightarrow y=-1 \\ \text{This way, the point} \\ (0,-1) \\ \text{Belongs to the line} \\ \\ x=1\Rightarrow y=-4(1)-1\rightarrow y=-5 \\ \text{This way, the point } \\ (1,-5) \\ \text{belongs to the line} \end{gathered}\)3. Put those points in the cartesian plane:
4. Draw a straight line that passes through both points. This will be the graph of the line
a consumer group is investigating the number of flights at a certain airline that are overbooked. they conducted a simulation to estimate the probability of overbooked flights in the next 5 flights. the results of 1,000 trials are shown in the following histogram. based on the histogram, what is the probability that at least 4 of the next 5 flights at the airline will be overbooked?
Answers
The probability of at least 4 of the next 5 flights being overbooked is 1.0%.
The probability of an event occurring is determined by analyzing the data from a sample.
A histogram is often used to visualize the distribution of data, which can be used to calculate the probability of an event occurring.
By examining the histogram, you can determine the probability of a certain event occurring based on the height of the corresponding bar in the graph.
From the histogram, it is clear that the probability of at least 4 of the next 5 flights being overbooked is 1.0%.
This is because there is only one bar in the histogram that corresponds to the probability of 4 out of 5 flights being overbooked.
This bar has a height of 1.0%, which indicates a 1.0% probability of at least 4 out of 5 flights being overbooked.
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Evan's new poetry collection will contain 55 poems. Evan can write poetry at a rate of two words a minute for a maximum of six hours each day. If each poem in Evan's collection contains an average of 213 words, how many days will it take Evan to finish the collection? (Round to the nearest whole number.)
Answers
Answer:
4 days
Step-by-step explanation:
If there are 213 words in each poem, the the total number of words in all 55 poems will be 213 * 55 = 11715 words. He can also type two words per minute so using proportion,
If 2 words = 1 minute
11715 words = ?minutes
= \(\frac{11715}{2} * 1minute\)
= 5857.5 minutes
Then we convert this into hours and then into days
60 mins = 1 hr
5857.5 mins = ?
= \(\frac{5875.5}{60} * 1hr\)
= 97.625 hrs
24 hrs = 1 day
97.625 hrs = ?
= \(\frac{97.625}{24} * 1 day\)
= 4.07 days approximately 4 days
help me plzzzzzzzzzzzzzzzzzzzzz
Answers
Connect points W and Y. We form the triangle WXY on top and triangle WZY on the bottom. They are congruent triangles. We can prove this using the SSS (side side side) congruence theorem. The tickmarks tells us which pair of sides are congruent. For instance, XY= ZY due to the double tickmarks.
given this information, in order to use her 4 hours of time spent studying to get the highest possible test score, how many hours should she have spent solving multiple choice problems, and how many hours should she have spent reviewing lecture notes? 0 hours working on problems, 4 hours reading 2 hours working on problems, 2 hours reading 3 hours working on problems, 1 hour reading 4 hours working on problems, 0 hours reading
Answers
The most effective way to use her 4 hours of study time to get the highest possibility or probability of test scores would be to spend 4 hours working on multiple-choice problems and 0 hours reviewing lecture notes.
From the information given, it is clear that working on multiple-choice problems is the most effective way to improve her test score. Therefore, the highest possibility or probability of test scores would be achieved by spending the most time working on multiple-choice problems.
The available time for studying is 4 hours, so the maximum time that can be spent working on multiple-choice problems is 4 hours. If she spends 4 hours working on multiple-choice problems, she will not have any time left to review lecture notes.
So, the most effective way to use her 4 hours of study time to get the highest possible test score would be to spend 4 hours working on multiple-choice problems and 0 hours reviewing lecture notes.
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6 is added to the
product of 7 and a number.
Answers
The equation model of the given statement in question is 6 + 7× x.
What are equation models and arithmetic?The model of equation is defined as the model of the given situation in the form of an equation using constants and variables.
In math, it deals with numbers of operations given according to the statements. The four major arithmetic operators are, addition, subtraction, multiplication, and division.
Given that;
6 is added to product of a number and 7
Now,
Let, the number multiplied by 7 be x,
=7*x
6 is added to the product;
= 6 + 7 × x
Therefore, the equation of model will be given as 6 + 7 × x;
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a group of n people bought a ticket for a vacation rm 2520. suddenly 4 people canceled and the remaining needs to pay rm 224 each. show than n^2 -4n - 45=0
Answers
Answer:
The remaining 9 people each had to pay RM224.
Step-by-step explanation:
The equation n^2 - 4n - 45 = 0 represents the relationship between the number of people who bought a ticket and the cost per person after 4 people cancel. To find the value of n, we can use the quadratic formula:
n = (-b ± √(b^2 - 4ac)) / 2a
where a = 1, b = -4, and c = -45. Plugging these values into the formula, we get:
n = (-(-4) ± √((-4)^2 - 4(1)(-45))) / 2(1)
n = (4 ± √(16 + 180)) / 2
n = (4 ± √(196)) / 2
n = (4 ± 14) / 2
n = 9 or n = -5
However, the number of people cannot be negative, so the solution is n = 9. This means that there were originally 9 people who bought the ticket, and after 4 people cancelled, the remaining 9 people each had to pay RM224.