Answers
For the differential equation y = [(x² - 1) sin x] / (sin x + 1), the derivative is dy/dx = [(x² + 2x) sin² x] / (sin x + 1)².
What is differential equation?
Any equation with one or more terms and one or more derivatives of the dependent variable with respect to the independent variable is referred to as a differential equation.
To find dy/dx, we need to use the quotient rule of differentiation, which states that -
d/dx [f(x)/g(x)] = [g(x) × f'(x) - f(x) × g'(x)] / [g(x)]²
Applying this rule to the given function, we get -
y = [(x² - 1) sin x] / (sin x + 1)
y' = [(sin x + 1) d/dx[(x² - 1) sin x] - (x² - 1) sin x d/dx[sin x + 1]] / (sin x + 1)²
Now we need to find the derivatives of the two terms in the numerator separately -
d/dx[(x² - 1) sin x] = (2x sin x + (x² - 1) cos x) sin x
d/dx[sin x + 1] = cos x
Substituting these values in the formula for y', we get -
y' = [(sin x + 1) * (2x sin x + (x² - 1) cos x) - (x² - 1) sin x * cos x] / (sin x + 1)^2
Simplifying this expression, we get -
y' = [(x² + 1) sin² x + 2x sin x cos x - (x² - 1) sin x cos x - (x² - 1) sin x cos x] / (sin x + 1)²
y' = [(x² + 2x) sin² x] / (sin x + 1)²
Therefore, the derivative of y with respect to x (dy/dx) is -
dy/dx = [(x² + 2x) sin² x] / (sin x + 1)²
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Related Questions
Write the equation of the circle in standard form
X^2+y^2-8x-6y+16=0
Show work
Answers
The equation of the circle x² + y² - 8x - 6y + 16 = 0 in standard form is (x - 4)² + (y - 3)² = 3².
What is the equation of the circle in standard form?Given the equation of the circle ;
x² + y² - 8x - 6y + 16 = 0
To write the equation of a circle in standard form, we need to express it as:
(x - h)² + (y - k)² = r²
where (h, k) is the center of the circle and r is the radius.
To do this, we need to complete the square for both x and y terms in the given equation.
Starting with the x terms:
x² - 8x = x² - 8x + 16 - 16
= (x - 4)² - 16
Similarly, for the y terms:
y² - 6y = y² - 6y + 9 - 9
= (y - 3)² - 9
Substituting these results into the original equation, we get:
x² + y² - 8x - 6y + 16 = 0
(x - 4)² - 16 + (y - 3)² - 9 + 16 = 0
Simplifying, we get:
(x - 4)² + (y - 3)² - 9 = 0
(x - 4)² + (y - 3)² = 9
So the equation of the circle in standard form is:
(x - 4)² + (y - 3)² = 3²
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A line passes through the points (-1, 3) and (1, -5). Which points lie on the same line?
Select all that apply.
(0, -2)
(-3, 12)
(-5, 1)
(-2,7)
(4, 15)
(2, -9)
Answers
( 2,-9) points lie on the same line.
Line point intercept is defined.
Quantifying soil cover, including vegetation, litter, rocks, and biotic crusts, quickly and accurately can be done using the line-point intercept approach.
These metrics deal with the erosion caused by wind and water, water infiltration, and the site's capacity to withstand damage and recover from it.
k = -5 - 3/1 - (-1) = -8/2 = -4
the formula of the line is
y - 3 = -4(x+ 1)
when x = -3 , y = 11 ≠ 12 the points aren't on line.
when x = -5, y = 19 ≠ 1
when x = 4, y = -7 ≠ 15
when x = 2, y = -9 ≠ 7
when x = 0, y = -1 ≠ -2
( 2,-9) are on the same line.
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In a triangle ABC angle A is twice as large as angle B and B is 20 more than angle c. what are the measure of each angles
Answers
Answer:
Step-by-step explanation:
catss
There are 1 3/4 cookies. If there is 1/8 of a cookie in each bag, how many bags are there
Answers
Answer: 8
Step-by-step explanation:
Since 1 batch uses 1/8 of a bag of flour, I would use equivalent fractions to find how many 8ths are in 3/4. 3/4=?/8
4 times 2 equals 8, so I would multiply 3 by 2 also. 3/4=6/8. If 1 batch takes 1/8, then 6 batches would use 6/8 or 3/4 of a bag of flour.
solve the value of n 20 = 2 (n - 3)
Answers
Answer:
13
Step-by-step explanation:
1. 20/2 = n - 3
2. 10 = n - 3
3. 10 + 3 = n
4. 13 = n
A farmer sells 6.7 kilograms of pears and apples at the farmer's market. 2 5 of this weight is pears, and the rest is apples. How many kilograms of apples did she sell at the farmer's market?
Answers
By performing subtraction we know that the farmer sold 4.2 kilograms of apples in the market.
What is subtraction?One of the four operations used in mathematics, along with addition, multiplication, and division, is subtraction.
Removal of items from a collection is represented by the operation of subtraction.
For instance, in the following image, there are 5 2 peaches, which means that 5 peaches have had 2 removed, leaving a total of 3 peaches.
Subtraction in mathematics is the process of subtracting one integer from another. In other words, the result of subtracting two from five is three. After addition, subtraction is usually the second operation you learn in math class.
So, to find the weight of apples, perform subtraction as follows:
= 6.7 - 2.5
= 4.2 kilograms
Therefore, by performing subtraction we know that the farmer sold 4.2 kilograms of apples in the market.
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You have a portfolio consisting solely of Stock A and Stock B. The portfolio has an expected return of 10.2 percent. Stock A has an expected return of 11.7 percent while Stock B is expected to return 8.3 percent. What is the portfolio weight of Stock A?
Answers
The portfolio weight of Stock A is 0.625 for the given situation.
What use do portfolio weights serve?The percentage of a portfolio's investments that are assigned to each asset or security is shown by the portfolio weights. Investors can meet their chosen risk and return objectives by varying the weights of various assets in their portfolios. Portfolio weights are a key component of asset allocation and portfolio optimization methodologies since they are used to determine the expected return and risk of a portfolio.
Let w be the weight (proportion) of Stock A in the portfolio.
The weight of Stock B would be (1-w).
Given that, the portfolio has an expected return of 10.2 percent.
0.102 = w(0.117) + (1-w)(0.083)
w = (0.102 - 0.083)/(0.117 - 0.083) ≈ 0.625
Hence, the portfolio weight of Stock A is 0.625 for the given situation.
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Evaluate the expression 10a - 2b when
a = 2 and b = - 3?
Answers
Answer:
26
Step-by-step explanation:
10(2) -2(-3)=20+6=26
Answer:
[tex]\bf 26[/tex]Step-by-step explanation:
[tex]\tt 10a-2b[/tex]
[tex]\tt a=2[/tex]
[tex]\tt b=-3[/tex]
______
[tex]\tt 10\times \:2-2\left(-3\right)[/tex]Simplify: 10×2 = 20
[tex]\tt 20-2x-3[/tex]Simplify: 2×-3= -6
[tex]\tt 20-(-6)[/tex][tex]\tt 20+6[/tex]Simplify:-
[tex]\tt 26[/tex]_____________________
Hope this helps! :)
A self-serve car wash charges $5.25 to use its facilities, plus an additional $0.69 for each minute the customer is at the self-serve car wash. If Tiffany was at the car wash for 8 minutes, how much was she charged?
Answers
Using linear equations, Tiffany was charged $10.77 for 8 minutes at the self-serve car wash.
What Does It Mean to Solve Linear Equations?Solving a linear equation involves determining the answer to a linear equation with one, two, three, or more variables. The answer to a linear equation is the value or values of the variable(s) that are part of the equation. In the elimination procedure, any coefficient is first made equal, then deleted. After elimination, the equations are solved to provide the other equation.
Given that, the car wash charges $5.25 to use its facilities, plus an additional $0.69 for each minute.
C = 5.25 + 0.69x
For 8 minutes:
C = 5.25 + 0.69(8)
C = $10.77
Hence, Tiffany was charged $10.77 for 8 minutes at the self-serve car wash.
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Jason will receive a series of payments at the beginning of each year for 20 years. The first payment is 100. The subsequent 9 payments increase by 10% from the previous payment. After the 10th payment, each payment decreases by 10% from the previous payment. At an annual effective interest rate of 4%, calculate the present value of these payments at the time of the first payment.
A) 2,078.99
B) 2,118.01
C) 2,164.98
D) 2,193.24
E) 2,516.16
Answers
Answer:
2,118.01
Step-by-step explanation:
The distance between points A and B is 35 feet, the distance between points B and C is 35 feet, and the distance between points C and D is 80 feet. How wide is the
canyon? Explain your reasoning.
Answers
The answer of the question based on distance and width of the canyon is width of the trapezoid is approximately 67.43 feet.
What is Trapezoid?A trapezoid is quadrilateral ( four-sided polygon) that has two parallel sides and two non-parallel sides. The parallel sides are referred to as bases of the trapezoid, while the non-parallel sides are called legs. The distance between two parallel sides is called height or altitude of trapezoid.
The figure in the provided link is a trapezoid with parallel sides AB and CD, and non-parallel sides BC and AD.
Since AB and CD are parallel, we know that the length of the altitude is the same as the distance between them.
Let's call the width of the trapezoid "x". Then, we can use the following equation using the Pythagorean Theorem:
(AC)^2 = (BC)^2 + (AB - CD + 2x)^2
where AC is the diagonal connecting points A and C.
Substituting the given values, we get:
(AC)^2 = (35)^2 + (35 - 80 + 2x)^2
(AC)^2 = 1225 + (2x - 45)^2
(AC)^2 = 1225 + 4x^2 - 180x + 2025
(AC)^2 = 4x^2 - 180x + 3250
Now, we can use the distance formula to find the length of AC:
AC = sqrt[(A - C)^2 + (B - D)^2]
AC = sqrt[(35 + 35)^2 + 80^2]
AC = sqrt[4900 + 6400]
AC = sqrt(11300)
AC = 106.23 feet (rounded to two decimal places)
Substituting the value in equation above, we will get:
(106.23)^2 = 4x^2 - 180x + 3250
11299.77 = 4x^2 - 180x + 3250
4x^2 - 180x + 8049.77 = 0
Solving the quadratic equation using quadratic formula, we will get:
x = [180 ± sqrt(180^2 - 4(4)(8049.77))] / (2(4))
x = [180 ± sqrt(129322.08)] / 8
x = [180 ± 359.44] / 8
We take the positive root, since x represents a length, and we get:
x = (180 + 359.44) / 8
x = 67.43 feet (rounded to two decimal places)
Therefore, the width of the trapezoid is approximately 67.43 feet.
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Robin was given a $40 monthly allowance. She wants to go to the movies as many times as possible and have at least $12.50 left at the end
of the month to go to a concert. A movie ticket costs $5. Write and solve an inequality to determine how many times Robin can go to the
movies this month. Then interpret the solution.
PLEASE WRITE THE INEQUALITY WITH THE ANSWER AND A SUMMARY OF HOW TO SOLVE IT
Answers
Answer:
If Robin was given a $40 monthly allowance. She wants to go to the movies as many times as possible and have at least $12.50 left at the end of the month to go to a concert. A movie ticket costs $5. Then she can go six times to the movies.
Step-by-step explanation:
Two or more expressions with an Equal sign is called as Equation.
Given,
Robin was given a $40 monthly allowance.
She wants to go to the movies as many times as possible and have at least $12.50 left at the end of the month to go to a concert.
The amount spent on movies
40-12.5
$27.5
A movie ticket costs $5.
We need to find how many times can she go to the movies.
To find this we need to divide $27.5 by 5
$27.5/5=5.5
Robin can go 6 times to the movies.
please help.
a helicopter hovers 1400 feet above a small island. the figure shows that the angle of the depression from the to the point P is 30 degrees. how far off the coast, to the nearest foot, is the island?
Answers
Step-by-step explanation:
this (outside) angle of depression is equal to the inner angle at P.
because the line of sight from the helicopter to P is like the diagonal in a rectangle spring the rectangle into 2 equal right-angled triangles (one of just rotated by 180° of the other).
so, the angle at P is 30°.
1400 ft = sin(30)×radius
radius being the line of sight from the helicopter to P.
we need to calculate the radius, because we need it for d :
d = cos(30)×radius
radius = 1400/sin(30) = 2800 ft
d = cos(30)×2800 = 2,424.871131... ft ≈ 2,425 ft
A meteorologist reports that the chance of snow is less than 30%. The correct inequality to represent this comparison is s < 30. The variable s represents the likelihood of snow.
Which numbers are solutions of the inequality?
Choose all that apply.
20%
35%
17%
30%
29 and one-half percent%
1.5%
Answers
The solutions to the inequality s < 30 are 20%, 17%, 29 and one-half percent (which is equivalent to 29.5%), and 1.5%.
The disparity that best illustrates the phrase "the possibility of snow is less than 30%" is:
s < 30
This implies that the inequality can be solved for any value of s that is less than 30.
We must compare the numbers to 30 and see if they are less than it in order to identify which ones are the answers to the inequality.
The answer is yes because 20% = 0.2 is less than 30%.
It is not a solution because 35% = 0.35 is not less than 30%.
The answer is yes because 17% = 0.17 is less than 30%.
It is not a solution because 30% = 0.3 is not less than 30%.
That is a solution since 29 and a half percent = 0.295 is less than 30%.
The answer is yes because 1.5% = 0.015 is less than 30%.
As a result, the answers to the inequality s 30 are 20%, 17%, 29 and a half percent (or 29.5%), and 1.5 percent.
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What is the total surface area of the prism?
82 sq. cm
12 sq. cm
94 sq. cm
HEPLP WLL MAR BRANLEST
Answers
Answer:
12 as. cm that's the answer
What is the area of the figure?
Answers
8 x 2= 16
I think that’s the answer! Please correct me if
I’m wrong :)
I’ll give BRAINLIEST if correct as well as 5star ratings
Given the polynomial 9x^2y^6 - 25x^4y^8 rewrite as a product of polynomials
Answers
Answer:
The given polynomial already has two terms that can be factored out:
9x^2y^6 - 25x^4y^8 = (3xy^3)^2 - (5x^2y^4)^2
Now we have a difference of squares:
= (3xy^3 - 5x^2y^4)(3xy^3 + 5x^2y^4)
Therefore, 9x^2y^6 - 25x^4y^8 can be rewritten as the product of (3xy^3 - 5x^2y^4) and (3xy^3 + 5x^2y^4).
Answer: The answer would be (3xy^3-5x^2y^4)(3xy^3+5^2y^4)
(C)
Step-by-step explanation: Cant explain how to but for a quick answer it is C, and you can check the image for proof.
Dividing integers. Dan's scores for speed this season are -1, -3, 1, -1, -2, 0. What is his average speed score for the season? (The average is the sum of the points divided by the number of scores)
Answers
Dan's average speed score for the season is -1.
What is the arithmetic operations?
Arithmetic operations are mathematical operations that involve manipulating numbers to perform calculations. The four basic arithmetic operations are addition, subtraction, multiplication, and division.
To find Dan's average speed score for the season, we need to add up all his scores and divide by the total number of scores. We can do this as follows:
-1 + (-3) + 1 + (-1) + (-2) + 0 = -6
Dan's total score for the season is -6. To find his average score, we need to divide this total by the number of scores, which is 6 in this case:
-6 / 6 = -1
Hence, Dan's average speed score for the season is -1.
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HELP I NEED IT ASAP PLS ILL GIVE BRAINLIEST TOO!! 100 POINTS!!!
A race car drove around a circular track that was 0.4 mile. If 1 mile = 5,280 feet, what is the radius of the track, in feet? Use π = 3.14 and round to the nearest hundredth.
107.11 feet
214.21 feet
336.31 feet
672.61 feet
Answers
Step-by-step explanation:
The circumference of a circle is given by the formula C = 2πr, where C is the circumference and r is the radius. In this case, the circumference of the circular track is 0.4 mile, which is equal to 2112 feet (0.4 x 5280 = 2112).
Therefore, we can write the equation:
2112 = 2πr
Solving for r, we get:
r = 2112 / (2π)
r ≈ 336.31 feet (rounded to the nearest hundredth)
Therefore, the radius of the track, in feet, is approximately 336.31 feet. The closest option to this answer is 336.31 feet.
Answer:
334.31
Step-by-step explanation:
please give brainliest
If a seed is planted, it has a 95% chance of growing into a healthy plant.
If 8 seeds are planted, what is the probability that exactly 3 don't grow?
Answers
The probability that exactly three plants don't grow is given as follows:
0.0054 = 0.54%.
What is the binomial distribution formula?The mass probability formula, giving the probability of x successes, is of:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters are given by:
n is the number of trials of the experiment.p is the probability of a success on a single trial of the experiment.The parameter values for this problem are given as follows:
n = 8, as there are 8 plants.p = 0.05, as 95% of the plants grow, hence 5% don't grow.The probability that exactly three don't grow is P(X = 3), hence:
P(X = 3) = C(8,3) x (0.05)^3 x (0.95)^5 = 0.0054.
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How many terms are in this expression? 7p + 6 + r
Answers
Please help this too
Answers
b.) 144° (360-36-72-108)
c.) b= 20, 10% c= 80, 40% d=60, 30%
Question content area top
Part 1
To the right are the outcomes that are possible when a couple has three children. Assume that boys and girls are equally likely, so that the eight simple events are equally likely. Find the probability that when a couple has three children, there are exactly 0 girls.
1st
2nd
3rd
boy
−
boy
−
boy
boy
−
boy
−
girl
boy
−
girl
−
boy
boy
−
girl
−
girl
girl
−
boy
−
boy
girl
−
boy
−
girl
girl
−
girl
−
boy
girl
−
girl
−
girl
Question content area bottom
Part 1
What is the probability of exactly 0 girls out of three children?
enter your response here (Type an integer or a simplified fraction.)
Answers
The probability that when a couple has three children, there are exactly 0 girls is 12.5%
What is mean by Probability?The term probability refers to the likelihood of an event occurring. Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is expressed from zero to one.
Calculation of the probability:
Here we assume b for boy, g for girl
Now, the probability conditions are;
g - g - g
g - g - b
g - b - g
g - b - b
b - g - g
b - g - b
b - b - g
b - b - b
Thus, There are 8 outcomes, one of which (b - b - b) with exactly 0 girls.
So, We get;
⇒ 1 ÷ 8
= 0.125
= 12.5%
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Fancy Flowers Florist has just received a shipment of 250 red roses. They use 200 of the roses
to decorate for the grand opening of the Historical Museum. What percentage of the red roses
did Fancy Flowers Florist use to decorate for the grand opening at the museum?
A. 60%
B. 70%
C. 80%
D. 90%
Answers
The percentage of the red roses did Fancy Flowers Florist use to decorate for the grand opening at the museum is 80%
What is percentage?A percentage is a portion of a whole expressed as a number between 0 and 100 rather than as a fraction.
Given that, a flower shop received a shipment of 250 red roses, they will use 200 out of this for decorating for the grand opening of the Historical Museum,
We need to find the percentage of the red roses used there,
So, the percentage of the red roses used in decoration =
200 / 250 x 100
= 0.8 x 100
= 80%
Hence. the percentage of the red roses did Fancy Flowers Florist use to decorate for the grand opening at the museum is 80%
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it is 4/10 of a mile to john's house from frank's house. How much less than a mile is the distance from frank's house to john's house and then back to frank's house
Answers
The distance from frank's house to john's house and then back to frank's house is less than a mile by 1/5 units.
How do you figure out the overall distance?While travelling from one location to another and back again just multiplying the distance between the two places will give you the total distance travelled when travelling from one spot to another and returning to the beginning position.
The distance from frank's house to john's house = 4/10.
The distance back to Frank's house = 4/10.
Total distance = 4/10 + 4/10
Total distance = 8/10
The distance is thus,
1 - 8/10
Take the LCM:
(10 - 8) / 10
2/10 = 1/5
Hence, the distance from frank's house to john's house and then back to frank's house is less than a mile by 1/5 units.
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The diameter of a circle is 10 3/4 inches.
What is the radius, r, of the circle?
Enter your answer as a mixed number in simplest form by filling in the boxes.
Answers
Answer: The radius of the circle is 5 3/8 inches.
Step-by-step explanation:
The radius of the circle is half of the diameter. We can start by converting the mixed number diameter to an improper fraction:
10 3/4 = (10 × 4 + 3)/4 = 43/4
So, the diameter of the circle is 43/4 inches. The radius is half of this, which we can find by dividing by 2:
r = (43/4) ÷ 2 = 43/8
To simplify the fraction, we can divide the numerator and denominator by their greatest common factor (GCF), which is 1:
r = 43/8 = 5 3/8
Therefore, the radius of the circle is 5 3/8 inches.
Answer:5 3/4
Step-by-step explanation:
10 3/4 * 1/2
1/2 * 43/4 = 43/8
8/43=5 3/4
find the axis of symmetry for f(x)=x^{2}+12x+31
Answers
so since the squared variable is the "x", we're looking at a vertical parabola, and its axis of symmetry will be at the x-coordinate for its vertex, let's find its vertex then
[tex]\textit{vertex of a vertical parabola, using coefficients} \\\\ f(x)=\stackrel{\stackrel{a}{\downarrow }}{1}x^2\stackrel{\stackrel{b}{\downarrow }}{+12}x\stackrel{\stackrel{c}{\downarrow }}{+31} \qquad \qquad \left(-\cfrac{ b}{2 a}~~~~ ,~~~~ c-\cfrac{ b^2}{4 a}\right)[/tex]
[tex]\left(-\cfrac{ 12}{2(1)}~~~~ ,~~~~ 31-\cfrac{ (12)^2}{4(1)}\right) \implies \left( - \cfrac{ 12 }{ 2 }~~,~~31 - \cfrac{ 144 }{ 4 } \right) \\\\\\ \left( -6 ~~~~ ,~~~~ 31 -36 \right)\implies (\stackrel{ x }{-6}~~,~-5)\hspace{5em}\stackrel{ \textit{axis of symmetry} }{x=-6}[/tex]
Use the tables of values of f and g to answer the problem.
(g ∘ f)(2)
Answers
Answer:
giacsybsc
Step-by-step explanation:
vdgsivetshdvx
Need help with number 80
Answers
Answer:
a no
b. no
c.no
d. yes
Step-by-step explanation:
Volume of 5in, 9in , 10in Triangular Prism
Answers
Answer:
[tex]\mathrm{225\:in\:\right)^3}[/tex]
Step-by-step explanation:
Given:
base = b = 10 in
height = h = 9 in
length = l = 5 in
[tex]\mathrm{Use\:formula}\:\mathrm{V= \dfrac{1}{2}\times\:b\times\:h\times\:l:}[/tex]
[tex]\mathrm{V= \dfrac{1}{2}\times\:10\times\:9\times\:5}[/tex]
[tex]\mathrm{= 225\:in\:\right)^3}[/tex]
Final answer is rounded to 3 decimal places.