Answers
(i) Given that
[tex]\tan^{-1}(x) + \tan^{-1}(y) + \tan^{-1}(xy) = \dfrac{7\pi}{12}[/tex]
when [tex]x=1[/tex] this reduces to
[tex]\tan^{-1}(1) + 2 \tan^{-1}(y) = \dfrac{7\pi}{12}[/tex]
[tex]\dfrac\pi4 + 2 \tan^{-1}(y) = \dfrac{7\pi}{12}[/tex]
[tex]2 \tan^{-1}(y) = \dfrac\pi3[/tex]
[tex]\tan^{-1}(y) = \dfrac\pi6[/tex]
[tex]\tan\left(\tan^{-1}(y)\right) = \tan\left(\dfrac\pi6\right)[/tex]
[tex]\implies \boxed{y = \dfrac1{\sqrt3}}[/tex]
(ii) Differentiate [tex]\tan^{-1}(xy)[/tex] implicitly with respect to [tex]x[/tex]. By the chain and product rules,
[tex]\dfrac d{dx} \tan^{-1}(xy) = \dfrac1{1+(xy)^2} \times \dfrac d{dx}xy = \boxed{\dfrac{y + x\frac{dy}{dx}}{1 + x^2y^2}}[/tex]
(iii) Differentiating both sides of the given equation leads to
[tex]\dfrac1{1+x^2} + \dfrac1{1+y^2} \dfrac{dy}{dx} + \dfrac{y + x\frac{dy}{dx}}{1+x^2y^2} = 0[/tex]
where we use the result from (ii) for the derivative of [tex]\tan^{-1}(xy)[/tex].
Solve for [tex]\frac{dy}{dx}[/tex] :
[tex]\dfrac1{1+x^2} + \left(\dfrac1{1+y^2} + \dfrac x{1+x^2y^2}\right) \dfrac{dy}{dx} + \dfrac y{1+x^2y^2} = 0[/tex]
[tex]\left(\dfrac1{1+y^2} + \dfrac x{1+x^2y^2}\right) \dfrac{dy}{dx} = -\left(\dfrac1{1+x^2} + \dfrac y{1+x^2y^2}\right)[/tex]
[tex]\dfrac{1+x^2y^2 + x(1+y^2)}{(1+y^2)(1+x^2y^2)} \dfrac{dy}{dx} = - \dfrac{1+x^2y^2 + y(1+x^2)}{(1+x^2)(1+x^2y^2)}[/tex]
[tex]\implies \dfrac{dy}{dx} = - \dfrac{(1 + x^2y^2 + y + x^2y) (1 + y^2) (1 + x^2y^2)}{(1 + x^2y^2 + x + xy^2) (1+x^2) (1+x^2y^2)}[/tex]
[tex]\implies \dfrac{dy}{dx} = -\dfrac{(1 + x^2y^2 + y + x^2y) (1 + y^2)}{(1 + x^2y^2 + x + xy^2) (1+x^2)}[/tex]
From part (i), we have [tex]x=1[/tex] and [tex]y=\frac1{\sqrt3}[/tex], and substituting these leads to
[tex]\dfrac{dy}{dx} = -\dfrac{\left(1 + \frac13 + \frac1{\sqrt3} + \frac1{\sqrt3}\right) \left(1 + \frac13\right)}{\left(1 + \frac13 + 1 + \frac13\right) \left(1 + 1\right)}[/tex]
[tex]\dfrac{dy}{dx} = -\dfrac{\left(\frac43 + \frac2{\sqrt3}\right) \times \frac43}{\frac83 \times 2}[/tex]
[tex]\dfrac{dy}{dx} = -\dfrac13 - \dfrac1{2\sqrt3}[/tex]
as required.
Related Questions
Tim Hortons is hiring and offers $200 every week plus $5 per hour. McDonalds offers $300 every week plus $2 per hour. State the conditions under which Tim Hortons is the better employer.
Answers
Answer: After working for 33 [tex]\frac{1}{3}[/tex] hours a week
Step-by-step explanation:
Let us write equations to represent these two places. Let x be hours worked and y be money earned.
Tim Hortons:
$200 + $5x = y
McDonalds:
$300 + $2x = y
Now, to find the conditions of which Tim Hortons is the better employer (on the basis of money earned) we must find the interval that Tim Hortons pays more. This can be found by setting up another equation, or by graphing. I have shown both. See attached for the graph.
$200 + $5x > $300 + $2x
$5x > $100 + $2x
$3x > $100
x > [tex]\frac{100}{3}[/tex]
x > 33.3334
Tim Hortons is the better employer after an employee has worked for 33 [tex]\frac{1}{3}[/tex] hours a week.
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Solve each triangle – find any missing side and angle measures. Round answers to the nearest tenth.
Answers
Answer:
A = 55 degrees
B = 35 degrees
C = 90 degrees
side BA = 4.8 units
side AC = 2.7 units
side BC = 3.9 units
Step-by-step explanation:
Hello!
The things we know:
A = 55 degrees
C = 90 degrees
side BA = 4.8 units
AC = adjacent side
BC = opposite side
AB = hypothenuse
B = 180 - 55 - 90 degrees
B = 35 degrees
Let's find side AC
we can use the cos of theta.
cos(55) = AC/4.8
multiply 4.8 on both sides.
4.8cos(55) = AC
AC = 2.7 units
And now we just have to find BC
to calculate BC we can either use tangent or sin.
I will just use sin
sin(55) = BC/4.8
multiply 4.8 on both sides.
4.8sin(55) = BC
BC = 3.9
And now we know everything about the triangle
Hope I Helped!
To get from his house to the lecture hall at school, Lin walked west 651 feet. After class, he walked northeast 910 feet to the gym. Finally, he walked 615 feet back to his house from the gym.
A triangle has points gym, Lin apostrophe s house, and lecture hall. The distance between the lecture hall and gym is 910 feet. The distance between the gym and Lin apostrophe s house is 615 feet. The distance between the lecture hall and Lin apostrophe s house is 651 feet.
What general direction did Lin walk from the gym to his house, and what type of triangle did his walking path form?
Lin walked south, creating a right triangle.
Lin walked southwest, creating an obtuse triangle.
Lin walked southeast, creating an acute triangle.
Lin walked directly east, creating a right triangle.
Answers
The general direction that Lin walked from the gym to his house is; B: Lin walked southwest, creating an obtuse triangle.
How to interpret distance bearing?
We are given;
Distance between the lecture hall and gym = 910 feet.
Distance between the gym and Lin apostrophe's house = 615 feet.
Distance between the lecture hall and Lin apostrophe's house = 651 feet.
Now, since this 3 distances form a triangle and the 3 distances are unequal, then we can call it an obtuse triangle since he walked west and then walked northwest.
Now, since he walked back to his house from the gym, we can say that he walked southwest if we picture the bearing of his first two directions.
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Which graph shows the solution to y2-2x+10?
Answers
PLEASE HELP
This table contains an Arithmetic Sequence. Find the missing terms in the table.
Answers
help me out plsss I need to turn this in now
Answers
Answer:
Complementary angles are:
∠ORS and ∠SRQ
Supplementary angles are:
∠ORP and ∠PRQ
Maria bought a stock for x dollars, and she sold it at a profit of 20 %. Write a
mathematical expression for the selling price.
Answers
Answer: 1.2x
Step-by-step explanation:
Profit = Selling price - Cost
She bought the stock at a cost of x dollars
Her profit was 20% of that, which is 0.2x dollars
Selling price = Profit + Cost = x + 0.2x = 1.2x
help asap grade 5 math help fast help quick !!!
Answers
Answer:
B and D
Step-by-step explanation:
A
[tex]\frac{5}{6}[/tex] × 13 = [tex]\frac{5(13)}{6}[/tex] = [tex]\frac{65}{6}[/tex] = 10 [tex]\frac{5}{6}[/tex] < 13
B
4 × 13 = 52 > 13
C
[tex]\frac{3}{4}[/tex] × 13 = [tex]\frac{3(13)}{4}[/tex] = [tex]\frac{39}{4}[/tex] = 4 [tex]\frac{3}{4}[/tex] < 13
D
[tex]\frac{10}{10}[/tex] × 13 = 1 × 13 = 13
Band D are equal to or greater than 13
please help meeee ………..
Answers
Answer:
x = 5
Step-by-step explanation:
The equality of bases property says powers of the same base will be equal if and only if the powers are equal. This property is used to solve exponential equations.
Application[tex]\dfrac{(2)^x}{2}=16\qquad\text{copy of the original equation}\\\\2\times\dfrac{(2)^x}{2}=2\times16\qquad\text{multiply by 2 to isolate the base}\\\\2^x=32\qquad\text{simplify}\\\\2^x=2^5\qquad\text{rewrite the constant as a power of 2}\\\\\boxed{x=5}\qquad\text{use the Equality of Bases Property}[/tex]
__
Additional comment
Equating the exponents is fully equivalent to taking the logarithm of both sides of the equation, to that base.
[tex]\log_2(2^x)=\log_2(2^5)\ \Longrightarrow\ x=5[/tex]
the ratio equivalent to tan P and tan Q
Answers
Answer:
Tan p is 20/21
Tan q is 21/20
Step-by-step explanation:
Tan is short for tangeant.
tangeant means opposite/adjacent
the opposite of vertices p is 20.
the adjacent of vertices p is 21.
so tan p= 20/21
The opposite of vertices q is 21
the adjacent of vertices q is 20
so tan q is 21/20
Find the volume of the prism.
Answers
Answer:
216kmStep-by-step explanation:
The formula for finding the volume of a triangular prims is:
0.5 * b * h * length, where b is the length of the base of the triangle, h is the height of the triangle and length is prism length.
So, let's substitute those variables with the base length, height length, and prism length
The base length is 6km, the height length is 8km, and the prism length is 9km
So, we get:
0.5 * 6 * 8 * 9
= 1/2 * 6 * 8 * 9
= 1/2 * 48 * 9
= 1/2 * 432
= 216
The volume of this prism is 216km.
Which of the lines is the graph of x=1−y?
Answers
Answer:
y = -x + 1
(Image of what graph looks like is attached)
Step-by-step explanation:
x = 1 - y can be rewritten into y = mx + b format.
x = 1 - y (subtract 1 from both sides)
x - 1 = -y (multiply by -1 to both sides)
y = -x + 1
Brainliest, please :)
Watch help video
Mason is a songwriter who collects royalties on his songs whenever they are played in
a commercial or a movie. Mason will earn $30 every time one of his songs is played in
a commercial and he will earn $80 every time one of his songs is played in a movie.
Mason earned a total of $470 in royalties on 9 commercials and movies. Write a
system of equations that could be used to determine the number of commercials and
the number of movies on which Mason's songs were played. Define the variables that
you use to write the system.
Answers
Answer:479
Step-by-step explanation:470+9=479
We can write the system of equation that determines the number of commercials and the number of movies on which Mason's songs were played as → 30x + 80y = 470 and x + y = 9. The variables used in the system of equations are -
x - Number of times his song played in a commercial
y - Number of times his song played in a movie.
What is equation modelling?
Equation modelling is a method of writing a mathematical relation from a mathematical statement for mathematical analyses of the problem given.
Given is Mason who collects royalties on his songs whenever they are played in commercials or movies.
Assume that his song played in 'x' commercials and 'y' movies.
Cost per song when played in commercial = $30
Then for 'x' commercials, total cost will be = $30x
Cost per song when played in movie = $80
Then for 'y' movies, total cost will be = $80y
Total money earned through both = $470
We can write the system of equation that determines the number of commercials and the number of movies on which Mason's songs were played as -
30x + 80y = 470
The total number of commercials and movies combined -
x + y = 9
The variables used in the system of equations are -
x - Number of times his song played in a commercial
y - Number of times his song played in a movie.
Therefore, We can write the system of equation that determines the number of commercials and the number of movies on which Mason's songs were played as → 30x + 80y = 470 and x + y = 9. The variables used in the system of equations are -
x - Number of times his song played in a commercial
y - Number of times his song played in a movie.
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What is the solution to the equation 2^x+4 -12=20
Answers
Answer:
2^x = 28Step-by-step explanation:
What is the solution to the equation?
[tex]2^{x}[/tex] + 4 - 12 = 20
[tex]2^{x}[/tex] = 20 - 4 + 12
[tex]2^{x}[/tex] = 16 + 12
[tex]2^{x}[/tex] = 28
Answer:
x-1
Step-by-step explanation:
because i said so
x^2/0.35-x=1.3*10^-5
Solve for x
Answers
The solution to the value of x in the given equation is x = 0.00212658 or x = -0.00213958
Solving algebraic equations:The process of solving the unknown value of an algebraic equation involves equating the values of the unknown to the constant value.
Given that:
[tex]\mathbf{\dfrac{x^2}{0.35-x}= 1.3 \times 10^{-5} }[/tex]
[tex]\mathbf{x^2 = 1.3 \times 10^{-5} (0.35-x) }[/tex]
x² = 4.725 × 10⁻⁶ - 1.3 × 10⁻⁵x
= [tex]x^2 + 1.3\times10^{-5}x - 4.725 \times 10^{-6}[/tex]
= x²+0.000013x-0.000004725
Solving for (x) by using the quadratic equation:x = 0.00212658 OR x = -0.00213958
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Fiona divided 3x2 5x-3 by 3x 2. the expression represents the remainder over the divisor. what is the value of a? -5 -1 1 5
Answers
The value of a is -5 if the provided polynomial is divided by (3x + 2) option first -5 is correct.
What is polynomial?Polynomial is the combination of variables and constants systematically with "n" number of power in ascending or descending order.
[tex]\rm a_1x+a_2x^2+a_3x^3+a_4x^4..........a_nx^n[/tex]
The question is incomplete.
The complete question is in the picture, please refer to the attached picture.
We have a polynomial:
3x² + 5x - 3
Which is divided by (3x + 2)
= (3x² + 5x - 3) ÷ (3x + 2)
[tex]\rm = \dfrac{\left(3x^2+5x-3\right)}{\left(3x+2\right)}[/tex]
[tex]\rm =x+\dfrac{3x-3}{3x+2}[/tex]
[tex]\rm =x+1+\dfrac{-5}{3x+2}[/tex]
[tex]\rm =x+1-\dfrac{5}{3x+2}[/tex]
By comparing with the given remainder.
a = -5
Thus, the value of a is -5 if the provided polynomial is divided by (3x + 2) option first -5 is correct.
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3. Find the point of intersection of the graphs for each system. a.2x + 2y = 3 and 6x-6y = -1
Answers
Given :
2x + 2y = 3 (Equation #1)6x - 6y = -1 (Equation #2)Multiply Equation #1 by 3 :
3(2x + 2y) = 3(3)6x + 6y = 9 (Equation #3)Add Equations 2 and 3 :
6x - 6y + 6x + 6y = -1 + 912x = 8x = 2/3Now, input x in Equation #3 to find y :
6(2/3) + 6y = 94 + 6y = 96y = 5y = 5/6∴ Hence, the point of intersection is (2/3, 5/6).
help me please!!! 30 points
Answers
Answer:
17 square centimeters.
Step-by-step explanation:
Divide the rectangle into a 3x5 rectangle and a 2x1 rectangle.
(3x5)+(2x1)=17
17 square centimeters.
What is the range of the function shown in this table?
OA. (3, 4), (4, 4), (5, 2), (6, 5)
OB. (3,4)
O C. (2, 4, 5)
O D. (3, 4, 5, 6}
Answers
Answer: A
Step-by-step explanation:
Answer:
C
Step-by-step explanation:
The range is just the y-values. You don't have to write any of the repeated values. You do need to write them in numerical order.
4, 4, 2, 5
becomes
2,4,5
PLEASE HELP!!!!!!!!!!!!!!!!!!!!!!!!! GEOMTREY
Answers
Answer:
[tex]m\angle{ABC}=144^{\circ}[/tex]
Step-by-step explanation:
By using Angle Addition Postulate, it holds:
[tex]m\angle{ABD}+m\angle{DBC}=m\angle{ABC}[/tex]
Substitute [tex]m\angle{ABD}=76^{\circ}[/tex] and [tex]m\angle{DBC}=68^{\circ}[/tex] into the equation:
[tex]76^{\circ}+68^{\circ}=m\angle{ABC}[/tex]
Add the measures:
[tex]m\angle{ABC}=144^{\circ}[/tex]
Please need help ASAP
Answers
Answer:
Can u please upload more clearer pictures ?
Select the correct answer. What is one root of this equation? 2x^2-4x+9
Answers
Answer:
1±i sqrt(7/2)
Step-by-step explanation:
For this equation you would need to use quadratic formula: x=(-b+-square root b^2-4ac)/2a
Answer:the answer is option E (see pic)
Step-by-step explanation:
One root would be: x =(4-√88)/-4=1+1/2√ 22 = 1.345
Which explains why the graph is not a function? It is not a function because the points are not connected to each other. It is not a function because the points are not related by a single equation. It is not a function because there are two different x-values for a single y-value. It is not a function because there are two different y-values for a single x-value.On a coordinate plane, solid circles appear at the following points: (negative 2, negative 5), (negative 1, 3), (1, negative 2), (3, 0), (4, negative 2), (4, 4).
Answers
The given points of the expression is not a function because there are two different y-values for a single x-value.
Given The points are : (-2,-5), (-1,3), (1,-2) , (3,0) ,(4,-2) (4,4)
Function refers to a relation or expression involving one or more variables. In a function each value of x corresponds to each value of y. A relation which is having two or more values of y for x values is not a function.
We have been given points (-2,-5), (-1,3), (1,-2) , (3,0) ,(4,-2) (4,4).
These are not points of a function because of a simple reason that it has two values of y for each one value of x.
Value of relation at x=-2 is -5.
Value of relation at x=-1 is 3.
Value of relation at x=1 is -2.
Value of relation at x=3 is 0.
Value of relation at x=4 is -2.
Value of relation at x=4 also equal to 4.
It has two values at x=4 one is -2 and other one is 4.
Hence the relation is not a function because there are two different y-values for a single x-value.
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The given points of the expression is not a function because there are two different y-values for a single x-value.
Given The points are : (-2,-5), (-1,3), (1,-2) , (3,0) ,(4,-2) (4,4)
Function refers to a relation or expression involving one or more variables. In a function each value of x corresponds to each value of y. A relation which is having two or more values of y for x values is not a function.
We have been given points (-2,-5), (-1,3), (1,-2) , (3,0) ,(4,-2) (4,4).
These are not points of a function because of a simple reason that it has two values of y for each one value of x.
Value of relation at x=-2 is -5.
Value of relation at x=-1 is 3.
Value of relation at x=1 is -2.
Value of relation at x=3 is 0.
Value of relation at x=4 is -2.
Value of relation at x=4 also equal to 4.
It has two values at x=4 one is -2 and other one is 4.
Hence the relation is not a function because there are two different y-values for a single x-value.
The radius of a sphere is 3 inches. Which represents the volume of the sphere?
• 12t cubic inches
• 367 cubic inches
• 647T cubic inches
• 817 cubic inches
Answers
Answer:
Step-by-step explanation:
The formula for the volume of a sphere of radius r is V = (4/3)(pi)r^3.
Here, with r = 3 in, the volume is
V = (4/3)(3.14)(3 in)^2, or V = 36(pi)(27 in)^3 = 36pi in^3
What is the difference of the rational expressions below?
4x 2x
9
9
O A.
6x
O B.
2x
O C.
O D. Ex
Answers
Answer:
B
Step-by-step explanation:
[tex]\frac{4x}{9}[/tex] - [tex]\frac{2x}{9}[/tex]
since the denominators are common , then subtract the numerators, leaving the common denominator.
= [tex]\frac{4x-2x}{9}[/tex]
= [tex]\frac{2x}{9}[/tex]
Suppose A and B are 2x2 matrices and that AB = BA =
which of the following statements is correct?
Answers
The correct statement about the matrix is B. B^-1.
How to illustrate the matrix?From the information, it can be seen that the matrix is 2 by 2. Based on the information about the matrix, 1/3AB = 1/3BA.
From this, it can be depicted that the inverse of B illustrates that:
A(1/3B) = (1/3B)
This illustrates that the correct is B.
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i need help right now
Answers
here u go hope this helps
write each polynomial in standard form. Then give the leading coefficient. You may need to combine like terms before writing the polynomial in standard form.
18y^5-3y^8+10y
Answers
Answer: y(-3y^7+18y^4+10)
Step-by-step explanation:
For what values of k will the function f(x) = 9x² + 4x + k have 1 real roots.
Answers
Hello,
Answer:
for k = 4/9
Step-by-step explanation:
a = 9 ; b = 4 ; c = k
we search Δ = 0 (because 1 real root)
Δ = b² - 4ac = 4² - 4 × 9 × k = 16 - 36k
Δ = 0 ⇔ 16 - 36k = 0 ⇔ 36k = 16 ⇔ k = 16/36 = (4 × 4)/(4 × 9) = 4/9
⇒ k = 4/9
Answer:
[tex]\{\frac{4}{9} \}[/tex]
Step-by-step explanation:
First off, we are given a parabola by definition:
[tex]f(x) = 9x^2 + 4x + k \\ a = 9, b = 4, c = k;[/tex]
Since [tex]a > 0[/tex], our parabola has an upward opening.
Should we think about it graphically, it will be no wonder that we should pay our attention to the points of the parabola intersecting the abscissa axis. In other words, we need our vertex to intersect the x axis only once per se.
The vertex:
[tex]V_{x} = \frac{-b}{2a} = \frac{-4}{2 * 9} = - \frac{2}{9} \\ V_{y} = 9V_{x}^2 + 4V_{x} + k = 9(- \frac{2}{9})^2 + 4(- \frac{2}{9}) + k = 9 * \frac{4}{9^2} - \frac{8}{9} + k = \frac{4}{9} - \frac{8}{9} + k = - \frac{4}{9} + k[/tex]
As a result, our [tex]V_{y}[/tex] is parametric. [tex]k = \frac{4}{9}[/tex] suits us because [tex]V_{y} = - \frac{4}{9} + \frac{4}{9} = 0[/tex]. If [tex]k > \frac{4}{9}[/tex], we do not have any roots at all. We have two roots if [tex]k < \frac{4}{9}[/tex].
if m²+1/m = 4, find m²+ 1/m²
Answers
A) The solution to m² + (1/m²) is gotten as; 14
B) The solution to m³ + ¹/₃ is gotten as; 52.312
How to simplify Algebra?We are given the algebraic equation;
(m² + 1)/m = 4
Multiply through by m to get;
m² + 1 = 4m
m² - 4m + 1 = 0
From online quadratic equation calculator, we have;
m = 3.732 or 0.268
A) Thus;
m² + (1/m²) = 3.732² + (1/3.732²)
⇒ 14
B) m³ + ¹/₃ gives;
3.732³ + (1/3) = 52.312
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