Answers
a. The equation of the tangent plane is z = 1 + x.
b. The point (4/3, 8/3, 5) lies on both the plane T(0,1) and the line l.
c. The other points on the graph of f(x) whose tangent plane contains the line l are (-9/8, -1/2, -21/8) and (-25/32, -3/4, -47/32)
What is the function?The function is defined as a mathematical expression that defines a relationship between one variable and another variable.
(a) The partial derivatives of f(x) at this point:
∂f/∂x = 2x + y²
∂f/∂y = x² + 2xy
Evaluating these at (0,1), we get:
∂f/∂x(0,1) = 1
∂f/∂y(0,1) = 0
So the equation of the tangent plane is:
z = f(0,1) + ∂f/∂x(0,1)(x-0) + ∂f/∂y(0,1)(y-1)
z = 1 + x
(b)
The plane T(0,1) contains the line l:
2x = y = (z+3)/2
Substituting y = 2x and z = 2y-3, we get:
z = 2(2x) - 3 = 4x - 3
So any point on the line l can be written as (x, 2x, 4x-3).
Substituting these values into the equation of the plane, we get:
4x-3 = 1 + x
3x = 4
So x = 4/3, y = 8/3, and z = 5.
Therefore, the point (4/3, 8/3, 5) lies on both the plane T(0,1) and the line l.
(c) To find all other points on the graph of f whose tangent plane contains the line l, we need to find the intersection of the graph of f with the plane T(0,1) + λ(2,0,1), where λ is a scalar. This gives us the system of equations:
x² + y²xy = 1 + x + 2λ
2x + y² = 0
y - 1 = λ
Solving for x and y in terms of λ, we get:
y = λ + 1
x = -(y²)/2 = -(λ+1)²/2
Substituting these values into the first equation, we get:
(λ+1)² + (λ+1)²(-(λ+1)²/2) = 1 - (λ+1)²/2 + 2λ
Simplifying and solving for λ, we get:
λ = -1/2, -1/4
Therefore, the other points on the graph of f(x) whose tangent plane contains the line l are:
(-9/8, -1/2, -21/8) and (-25/32, -3/4, -47/32)
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Related Questions
Find the equation of the line shown.
Answers
Answer:
y=x+1
Step-by-step explanation:
y=mx+b
y=mx+1
2/2=1
y=1x+1
y=x+1
All of the following employee information is included on a W-2 form EXCEPT..
Answers
Overall, the W-2 form is an important document for employees to receive and keep for tax purposes, but it is not comprehensive in terms of all the financial details of their employment.
What is equation?An equation is a mathematical statement that indicates the equality of two expressions. It consists of two expressions separated by an equal sign (=). The expression on the left side of the equal sign is equivalent to the expression on the right side. Equations are used in many areas of mathematics and science to represent relationships between variables and to solve problems. They are also used in various fields such as engineering, physics, and economics to model real-world situations and make predictions based on mathematical analysis.
Here,
A W-2 form is an Internal Revenue Service (IRS) tax form that employers are required to provide to their employees every year. It reports the employee's annual wages and the amount of taxes withheld from their paychecks. While a W-2 form contains a lot of important information about an employee's earnings and tax withholding, it does not include every detail about their employment or financial situation. Some examples of information that are not included on a W-2 form include:
Information about non-wage income, such as interest or dividends
Information about benefits received, such as health insurance or retirement plan contributions
Information about deductions or credits, such as charitable contributions or education expenses
Information about stock options or other forms of equity compensation
Information about expenses related to employment, such as travel or equipment costs.
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Solve the equation by factoring y= x^2-8x+16=7
Answers
The solution of the equation by factoring is x = 6.645 and x = 1.355.
What is factoring?The breaking or breakdown of an entity (such as a number, a matrix, or a polynomial) into a product of another entity, or factors, whose multiplication results in the original number, matrix, etc., is known as factorization or factoring in mathematics. An algebraic expression is factorised when it is written as the product of its factors. Certain variables, factors, or algebraic expressions might be present.
The given equation is:
x² - 8x + 16 = 7
x² - 8x + 16 - 7 = 0
x² - 8x + 9 = 0
Using the quadratic formula we have:
x = 8 ±√(8)² - 4(1)(9) /2(1)
x = 8 ± 5.29/2
x = 8 + 5.29/2 and x = 8 - 5.29/2
x = 6.645 and x = 1.355
Hence, the solution of the equation by factoring is x = 6.645 and x = 1.355.
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I don’t understand how to do these problems and I have a test on them tomorrow. Pls help
Answers
Answer: There are no questions posted
Step-by-step explanation:
The graph of the rational function f left parenthesis x right parenthesis equals fraction numerator x minus 2 over denominator x squared plus 4 x minus 5 end fraction will have a vertical asymptote at _________.
Answers
As a result, the vertical asymptotes of the function's curve will be at x=-5 and x=1 of quadratic equation .
what is function ?A function in mathematics is a formula that associates each element from a set (referred to as the domain) with a single element from another set (called the range). A machine that generates an outcome from an input (the independent variable) can be compared to a function (the dependent variable). Algebraic expressions or equations are frequently used to describe functions, such as f(x) = x2 + 3x - 2. In a coordinate plane, where the x-axis denotes the input values and the y-axis denotes the output values, they can also be graphically depicted as a curve or line.
given
Since division by zero is undefined, we must search for the values of x that cause the denominator of the function to equal zero in order to determine the vertical asymptote of a rational function.
The function is undefined at x=-5 and x=1 because the denominator of f(x) = (x-2)/(x2+4x-5) is a quadratic equation that can be factored as (x+5)(x-1).
As a result, the vertical asymptotes of the function's curve will be at x=-5 and x=1 of quadratic equation .
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If - 3y = 1 and 11y - x = -5, then what does x - 10y = ?
Answers
The answer for x - 10y = 14/3.
Solving for x and y by substitution methodTo solve for x and y, we can use the first equation to solve for y in terms of x:
-3y = 1
y = -1/3
Now we can substitute this value of y into the second equation and solve for x:
11y - x = -5
11(-1/3) - x = -5
-x = -5 + 11/3
-x = -4/3
x = 4/3
Now that we have found the values of x and y, we can substitute them into the expression x - 10y:
x - 10y = (4/3) - 10(-1/3)
x - 10y = 4/3 + 10/3
x - 10y = 14/3
Therefore, x - 10y = 14/3.
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put the steps to prove the vertical angle theorem in the correct order
Answers
Create four angles by drawing a straight line that meets another straight line at an angle. To keep track of the angles, assign letters to each one. Make a note of the angles' measurements using a protractor.
What lines connect to create the four right angles?A right angle is created when two perpendicular lines cross at an exact 90-degree angle. Four 90-degree right angles are formed by intersecting perpendicular lines.
Which of the four angles that are created when two lines connect is 90 degrees?A right angle is a right angle if one of the four angles created by two intersecting lines is a right angle. Straight lines that are perpendicular to one another intersect at a 90° angle.
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Find the Value of the sphere
Answers
The volume of the sphere according to the question will be approximately 7232.752cm³.
What is a sphere?
A geometric shape with a sphere-like appearance. In three dimensions, the sphere is defined. The sphere is a three-dimensional solid with a volume and surface area.
What is the Volume of sphere?
A sphere is a perfectly round geometrical 3D object. The formula for its volume equals:
Formulae of Volume of sphere = (4/3) × π × r³
So according to the question:
r = 12cm(given)
Substitute in above formulae:
=(4/3)×3.14×12×12×12 cm³ [Here 4/3= 1.333, π=3.14,r=12]
= 1.333×3.14×12×12×12 cm³
Volume of sphere will be =7232.752 cm³
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Work out 40 +
5
Give your answer as a whole number or as
a fraction in its lowest terms.
please I beg help meeer
Answers
Using the addition operation, the sum of 40 + 5 is 45.
What is the addition operation?The addition operation is one of the basic mathematical operations.
Other mathematical operations include subtraction, division, multiplication, exponentiation, and functions.
The addition operation involves two addends added using the addition operand (+) and the equal symbol (=) we show the result of the algebraic operation called the sum or total.
Thus, when five is added to forty (40 + 5), the result or sum is forty-five (45).
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I need help! Please include an explanation.
Answers
Based on the information in the graph, the perimeter of the figure would be 48.
How to find the perimeter of the figure?To find the perimeter of the figure we must add the value of all the sides. In this case, this figure has 8 sides. However, in essence it is a rectangle because all the angles are 90°, so we could assume that the sides on the right add up to 6 as the side on the left. Also the bottom sides would add up to 18 like the top.
According to the above, we could infer that the perimeter of the figure is:
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I need the answer QUICKLY:
Answers
The experimental probability of choosing a nickel based on 13 trials would be 4 / 13
How to find the experimental probability ?The experimental probability can be found by the formula ;
= ( ( Probability of picking a Nickel x Number of trials ) + 1 ) / 13 trials
Probability of picking a Nickel - 1 / 4
Number of trials - 13 trials
The experimental probability for Landon picking a Nickel on the 13th trial is :
= ( ( 1 / 4 x 12 ) + 1 ) / 13
= ( 3 + 1 ) / 13
= 4 / 13
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If a student's rank in a class of 200 students is 144, find the student's percentile rank.
Answers
The student's percentile will be 28.
What is a Percentile?Calculating percentiles is a common practise. The majority of the time, we can define percentile as a score below which a predetermined percentage of other scores fall. We might be able to tell when someone received a test score of 65 out of 90. But without knowing what percentile he belongs to, this number is meaningless. When a score is in the 90th percentile, it signifies that it is higher than the scores of 90% of test-takers.
Given : Student's rank = 144 out of 200
So, there are 56 (200-144) students whose rank is below that student.
We know that, percentile
= no. of ranks below that rank / total no. of ranks * 100
So, his percentile = 56/200 * 100
= 28 percentile.
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How to convert a whole number into a mixed fraction
Answers
Answer: You always put the whole number over 1
Step-by-step explanation:
ex: 31= 31/1
Identify the inequality graphed on the number line.
A. x _> 14
B. x < 14
C. x > 14
D. x _< 14
Answers
D. x ≤ 14
Hope this helped :)
prove this geometry
given: trapezoid abcd where ba is congruent to cd angle abc is congruent to angle dcb
prove bd is congruent to ac
Answers
By the hypotenuse-leg congruence theorem (HL theorem), triangles ABD and CBD are congruent.
What is Trapezoid ?
A trapezoid is a four-sided polygon with at least one pair of parallel sides. It is also known as a trapezium in some countries.
To prove that BD is congruent to AC, we will use the fact that the opposite sides of a trapezoid are parallel.
Since AB and CD are parallel, and BC is a transversal, we know that angles ABC and DCB are supplementary (they add up to 180 degrees).
From the given information, we know that angle ABC is congruent to angle DCB. Therefore, both angles must be equal to 90 degrees (since their sum is 180 degrees).
Now consider the right triangles ABD and CBD.
In triangle ABD, we have a right angle at angle ABD, and we know that AB is congruent to CD (given).
In triangle CBD, we have a right angle at angle BCD, and we know that BC is congruent to BC (reflexive property).
We also know that angle ABD is congruent to angle BCD (since they are both equal to 90 degrees minus angle ABC/DCB).
Therefore, by the hypotenuse-leg congruence theorem (HL theorem), triangles ABD and CBD are congruent.
This means that BD is congruent to AC (the corresponding sides of congruent triangles are congruent).
Therefore, we have proven that BD is congruent to AC in trapezoid ABCD.
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y = 2x + 4
4y - 8x = 16
Answers
Answer: Infinitely many solutions
Step-by-step explanation:
Solve by Substitution :
Solve equation [2] for the variable y
y = 2x + 4
Plug this in for variable y in equation [1]
4•(2x+4) - 8x = 16
0 = 0 => Infinitely many solutions
Please make me brainliest!
If you 3 lobster raviolis and 5 steak salads about how much will you earn in commission (round to the nearest hundredth) If you get paid $50 for working today, how much will you earn in total, including commission ?
Answers
The tοtal amοunt earned, including cοmmissiοn and base pay, is $63.50, rοunded tο the nearest hundredth.
How to find the commission?Tο answer this questiοn, we need tο knοw the cοmmissiοn rate fοr the lοbster raviοli and k salads. Let's assume the cοmmissiοn rate is 10% fοr bοth items.
If the lοbster raviοli cοst $15 each and the steak salads cοst $12 each, then the tοtal cοmmissiοn earned wοuld be:
3 x $15 x 0.10 + 5 x $12 x 0.10 = $13.50
Therefοre, the cοmmissiοn earned is $13.50.
Tο find the tοtal amοunt earned, including cοmmissiοn and base pay, we simply add the cοmmissiοn earned tο the base pay:
$50 + $13.50 = $63.50
Therefοre, the tοtal amοunt earned, including cοmmissiοn and base pay, is $63.50, rοunded tο the nearest hundredth.
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15 POINTS
Identify the roots in the given
{-8}
{-2, 4}
{0,-8}
{-2, -8,4}
Answers
The set containing the roots of the graphed quadratic function is given as follows:
{-3, 1}
How to obtain the roots of a function?The roots of a function can be obtained from the graph of the function, as they also represent the x-intercepts of the function.
The x-intercepts of a function are given by the values of x at which the graph of the function crosses the x-axis, which is the horizontal axis.
The values of x for which the graphed function crosses the x-axis are given as follows:
x = -3 -> 3 units to the left of the y-axis.x = 1 -> 1 unit to the right of the y-axis.Hence the set containing the roots of the graphed quadratic function is given as follows:
{-3, 1}
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An aquarium can be modeled as a right rectangular prism and it can hold 1120 cubic inches of water when it is full. Its length is 20 in and width is 8 in. Find the height of the aquarium in inches. Round your answer to the nearest tenth if necessary.
Answers
The height of the right rectangular prism aquarium is 7 inches
How to determine the height of the aquariumWe can use the formula for the volume of a right rectangular prism to find the height of the aquarium.
The formula is V = l x w x h,
Where V is the volume, l is the length, w is the width, and h is the height.
In this case, we know that the volume of the aquarium is 1120 cubic inches, the length is 20 inches, and the width is 8 inches.
Therefore, we can write:
1120 = 20 x 8 x h
Simplifying the right side of the equation, we get:
1120 = 160h
Dividing both sides by 160, we obtain:
h = 1120/160 = 7
Therefore, the height of the aquarium is 7 inches.
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Answer:7
Step-by-step explanation:
Which is an equation of the line that is a perpendicular bisector of AB with A(-6, 8) and B(-1, 3)?
O A)y=x+9
OB) y = x + 7
O c)y=-x-5
OD) y = -x + 2
Answers
To find the equation of the perpendicular bisector of AB with A(-6, 8) and B(-1, 3), we need to first find the midpoint of AB, and then find the slope of the line perpendicular to AB that passes through the midpoint.
Midpoint of AB:
The midpoint of AB is given by the formula:
((x1 + x2)/2, (y1 + y2)/2)
Substituting the values, we get:
((-6 + (-1))/2, (8 + 3)/2) = (-3.5, 5.5)
So, the midpoint of AB is (-3.5, 5.5).
Slope of the line AB:
The slope of AB is given by the formula:
m = (y2 - y1)/(x2 - x1)
Substituting the values, we get:
m = (3 - 8)/(-1 - (-6)) = 5/5 = 1
So, the slope of AB is 1.
Slope of the perpendicular bisector:
Since the perpendicular bisector of AB is perpendicular to AB, its slope will be the negative reciprocal of the slope of AB. Therefore, the slope of the perpendicular bisector is -1.
Equation of the perpendicular bisector:
Now that we have the slope of the perpendicular bisector (-1) and the midpoint of AB (-3.5, 5.5), we can use the point-slope form of a line to find the equation of the perpendicular bisector:
y - y1 = m(x - x1)
Substituting the values, we get:
y - 5.5 = -1(x - (-3.5))
y - 5.5 = -x - 3.5
y = -x + 2
Therefore, the equation of the line that is the perpendicular bisector of AB with A(-6, 8) and B(-1, 3) is y = -x + 2, which is option (D).
- I Hope This Helps! :)
Bradenton Bakery is baking a cake for a customer's quinceañera. The cake mold is shaped like a cylinder with a diameter of 10 inches and height of 7 inches.
Which of the following shows a correct method to calculate the number of cubic units of cake batter needed to fill the mold? Approximate using pi equals 355 over 113.
V equals 355 over 113 times 5 squared times 7
V equals 355 over 113 times 7 squared times 5
V equals 355 over 113 times 7 squared times 10
V equals 355 over 113 times 10 squared times 7
Answers
The correct method to calculate the number of cubic units(volume) of cake batter needed to fill the mold is option (a) , i.e V equals 355 over 113 times 5 squared times 7.
What is mensuration?Mensuration is the branch of mathematics concerned with the study of various geometric shapes, their areas and volumes. In a broad sense, it is about the process of measurement. Measurement is a branch of mathematics. That is measurement.
Solution:
Given, Diameter = 10 inches
Radius = 5 inches
Height = 7 inches
Volume of cylinder = πr²h
= (355/113)×5²×7
= 3.14×25×7
= 549.77 inches³
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Solve 12=13+p
p = __
Answers
Answer:
[tex]12 = 13 + p \\ 12 - 13 = p \\ - 1 = p \\ [/tex]
please help me solve this
Answers
The speed of both car A and car B are the same which is 0.7 miles per minute
How to solve an equation?An equation is an expression containing numbers and variables linked together by mathematical operations such as addition, subtraction, division, multiplication and exponents.
Let x represent the distance travelled by car A and y represent the distance travelled by car B
Speed of car A = x /20
Car B reaches in 50 minutes, hence:
Speed of car B = y/50
Since both speed are same;
x/20 = y/50
20y = 50x (1)
Car B is 21 miles farther, then:
y - x = 21 (2)
x = 14, miles y = 35 miles
Speed of car A = speed of Car B = 14/20 = 35/50 = 0.7 miles per minute
The speed is 0.7 miles per minute
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An isosceles triangle, with equal sides labeled 20 feet. The length of base is labeled 32 feet. The altitude drawn on base is 12
Find the area of the isosceles triangle.
The area is?
Answers
this is because the altitude (12 ft) is your height and 32ft is your base
formula for area of a triangle is A = 1/2 bh
b: base
h: height
plug in your values to get that A= 192 ft ^2
hope that helps :)
Identify the inequality graphed on the number line.
Responses
Answers
Answer:
x is greater than or equal to 9.
Step-by-step explanation:
This is because the dot is going above 9 and it's a closed dot. The closed dot means the greater than or equal to and the x represents all real numbers at and after the 9
the scale factor between 0 and 1 ____ the size of an object, or makes it ___. A scale factor of 1 ______ the size of an object; that is, the size is ____. A scale factor greater than 1 ______ an object, or make it ___.
Answers
An object's size in real life and its size on a map or a blueprint are compared using a scale factor, which is a mathematical concept. The scale factor may exceed, fall short of, or be equal to 1.
What is scale factor and how is it calculated?The size of the object is decreased and becomes smaller if the scale factor is between 0 and 1. This is so because the scale factor only approximates the object's true size. The size of the thing on the map will be half that of the real object, for instance, if the scale factor is 0.5.
When the scale factor is 1, the object's size is said to remain constant. As a result, the map or blueprint is created using the same dimensions.
In other words, the map or blueprint is created at the exact same scale as the final product.
The size of the thing is enlarged and made larger if the scale factor is greater than 1. This is so because the scale factor is a multiple of the object's actual size. The size of an object on a map will appear twice as large as it is in reality, for instance, if the scale factor is 2.
In general, it's crucial to comprehend scale variables in order to appropriately depict an object's size on maps, blueprints, and other scaled representations.
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23/7 plus 76/14 as a MIXED NUMBER
Answers
Answer: 8 5/7
Step-by-step explanation:
Well, first you make the common denominator between the two, which would be 46/14 and 76/14. Now you add 46 and 76 and receive 122/14. You can simplify that and get 61/7, and now you turn it into a mixed number:
8 5/7
Because I’m pretty much fed up with college level algebra
Please explain to me (literally) how I might use it in everyday life
If you give me a good explanation, I might reconsider giving any CONSIDERATION to the subject
Answers
Answer:
A business person will use algebra to determine whether a piece of equipment does not lose its worth if it is in stock.
Step-by-step explanation:
A sandwich shop offers 4 types of bread, 5 meats, 7 cheeses, 4 condiment options, and you can pick veggies or no veggies. How many sandwiches are possible?
Answers
Answer:
Step-by-step explanation: OK so I don't know if I'm exactly right on this but what I would do is I would first read it more than 1 time and I would see that the sandwich shop offers 4 types of bread, that's 4 different types of bread. Then it says 5 meats, 7 cheeses, 4 condiment options, and you can pick veggies or no veggies. So what I did was I multiplied 4 by 5 and got 20, then I multiplied 7 by 4 and got 28, then I added 20 and 28 and got 48 sandwiches are possible. Then to make sure I was correct I did 4 times 5 times 7 times 4 by accident and I got 560. So I don't know which one is correct but I just thought that I could at least try to help you because even I have problems with problems like this one too. So I hope I helped you even a little bit.
The dimensions of poster board paper at a store are 55 cm by 70 cm, and it sells at
$5.49. A second poster board paper measures 27.5 cm by 35 cm. Given the price of
the first poster board paper, what would be an appropriate price for the second
poster board paper? Give reasons for your answer.
Answers
The appropriate price for the second poster is $1.37. The dimensions is not equal but less than that of the first poster
How to determine the amountFrom the information given, we have that the poster takes form of a rectangle.
The formula that is used to calculate the area of a rectangle is usually expressed with the equation;
A = lw
The parameters are enumerated as;
A is the area of the rectanglel is the length w is the widthFor the first poster, we have;
Area = 55(70)
Multiply the values
Area = 3850 cm^2
Then, we have that;
3850 cm^2 = $5.49
Then, 27.5(35) = x
cross multiply the values
x = $1. 37
The dimensions of the first poster is two times greater than that of the second.
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