Answers
Answer:
Option A i the right option.
First blank is 110-[tex]10\sqrt{61}[/tex] or 10(11-[tex]\sqrt{61}[/tex])
Second blank is 31.898
Let me know if anything didn't make sense.
Step-by-step explanation:
So a diagonal through a rectangle makes two triangles. The question wants to know how much walking is saved walking down the diagonal vs walking along two sides that make the diagonal. in this case the two non diagonal sides walked are 60 paces and 50 paces.
A diagonal through a rectangle specifically makes a right triangle, so to find the diagonal we can use the pythagorean theorem.
c^2 = 60^2 + 50^2
c = [tex]\sqrt{60^2 + 50^2}[/tex]
c = [tex]\sqrt{6100} = 10\sqrt{61}[/tex]
if you don't get how to simplify a radical like that let me know.
Anyway, looking at the answers you can see right away the second option says no approximation is necessary. Well, you need to approximate square root of 61, so we can say the second answer is not right. So now we need to know what to fill in for option 1.
it wants the distance saved, well we know the distance of the diagonal is [tex]10\sqrt{61}[/tex] Hopefully you can see the disctance walking the two other sides is just adding them up so 50+60=110.
Now, to find the difference, that is subtraction. So subtract the smaller number from the larger number. You do need to remember with a right triangle, the sum of the to non diagonal (hypotenuse) sides are always longer than said hypotenuse. so that's 110-[tex]10\sqrt{61}[/tex]. That is the exact form. Or you could use 10(11-[tex]\sqrt{61}[/tex]) They are the same.
Then just plug that into a calculator for a decimal approximation.
Related Questions
consider the differential equation x3y ''' + 8x2y '' + 9xy ' − 9y = 0; x, x−3, x−3 ln(x), (0, [infinity]). Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. The functions satisfy the differential equation and are linearly independent since W(x, x−3, x−3 ln(x)) = ≠ 0 for 0 < x < [infinity].
Answers
Verifying that a given expression is a solution to the equation is just a matter of plugging in the expression and its derivatives, and making sure that the given expressions are indeed linearly independent.
For example, if y = x, then y' = 1 and the other derivatives vanish. So the DE after substitution reduces to
9x - 9x = 0
which is true for all 0 < x < ∞.
To check for linear independence, you compute the Wronskian, which, judging by what you wrote, you've already done...
commission received
Answers
Answer:
Commission Received refers to a percentage amount received by the company (or) an individual on the total sales incurred. It is an indirect income/revenue recorded on the credit side of profit and loss account.
Step-by-step explanation:
mark me brainliest if my answer is correct
Use the following information to answer the next six exercises. There are 23 countries in North America, 12 countries in South America, 47 countries in Europe, 44 countries in Asia, 54 countries in Africa, and 14 in Oceania (Pacific Ocean region).
Let A = the event that a country is in Asia.
Let E = the event that a country is in Europe.
Let F = the event that a country is in Africa.
Let N = the event that a country is in North America. Let O = the event that a country is in Oceania.
Let S = the event that a country is in South America.
18. What is the probability of drawing a red card in a standard deck of 52 cards?
19. What is the probability of drawing a club in a standard deck of 52 cards?
Answers
The probabilities we found in this exercise are.
0.2268 = 22.68% probability that a country is in Asia.0.2423 = 24.23% probability that a country is in Europe.0.2784 = 27.84% probability that a country is in Africa.0.1186 = 11.86% probability that a country is in North America.0.0722 = 7.22% probability that a country is in Oceania.0.0619 = 6.19% probability that a country is in South America.0.5 = 50% probability of drawing a red card in a standard deck of 52 cards.0.25 = 25% probability of drawing a club in a standard deck of 52 cards.In this exercise, probability concepts are used.
A probability is the number of desired outcomes divided by the number of total outcomes.
Total number of countries:
23 + 12 + 47 + 44 + 54 + 14 = 194
Let A = the event that a country is in Asia.
44 of the 194 countries are in Asia, thus:
[tex]P(A) = \frac{44}{194} = 0.2268[/tex]
0.2268 = 22.68% probability that a country is in Asia.
Let E = the event that a country is in Europe.
47 out of 194 countries are in Europe, thus:
[tex]P(E) = \frac{47}{194} = 0.2423[/tex]
0.2423 = 24.23% probability that a country is in Europe.
Let F = the event that a country is in Africa.
54 out of 194 countries are in Africa, thus:
[tex]P(F) = \frac{54}{194} = 0.2784[/tex]
0.2784 = 27.84% probability that a country is in Africa.
Let N = the event that a country is in North America.
23 out of 194 countries are in North America, thus:
[tex]P(N) = \frac{23}{194} = 0.1186[/tex]
0.1186 = 11.86% probability that a country is in North America.
Let O = the event that a country is in Oceania.
14 out of 194 countries are in Oceania, thus:
[tex]P(O) = \frac{14}{194} = 0.0722[/tex]
0.0722 = 7.22% probability that a country is in Oceania.
Let S = the event that a country is in South America.
12 out of 194 countries are in South America, thus:
[tex]P(S) = \frac{12}{194} = 0.0619[/tex]
0.0619 = 6.19% probability that a country is in South America.
18. What is the probability of drawing a red card in a standard deck of 52 cards?
In a standard deck of 52 cards, 26 are red, and thus:
[tex]p = \frac{26}{52} = 0.5[/tex]
0.5 = 50% probability of drawing a red card in a standard deck of 52 cards.
19. What is the probability of drawing a club in a standard deck of 52 cards?
In a standard deck of 52 cards, 13 are clubs, and thus:
[tex]p = \frac{13}{52} = 0.25[/tex]
0.25 = 25% probability of drawing a club in a standard deck of 52 cards.
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What is the sum of 2 and 3 subtracted from the product of 2 and the difference of 7 and 4? The answer is 1, but how is it solved?
Answers
Answer: -1
Step-by-step explanation:
sum of 2 and 3 subtracted from the product of 2 difference of 7 and 4
(2+3) - 2 ( 7 - 4 ) = -1
(2+3)-2(7-4) =
5 - 2(3) =
5-6 = -1
The sum of 2 and 3 subtracted from the product of 2 and the difference of 7 and 4 is equivalent to 1.
What is Equation Modelling?Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.
We have the sum of 2 and 3 subtracted from the product of 2 and the difference of 7 and 4
From the question, we can model the equation as -
x = 2 × (7 - 4) - (2 + 3)
x = 2(3) - 5
x = 6 - 5
x = 1
Therefore, the sum of 2 and 3 subtracted from the product of 2 and the difference of 7 and 4 is equivalent to 1.
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Twenty students randomly assigned to an experimental group receive an instructional program; 30 in a control group do not. After 6 months, both groups are tested on their knowledge. The experimental group has a mean of 38 on the test (with an estimated population standard deviation of 3); the control group has a mean of 35 (with an estimated population standard deviation of 5). Using the .05 level, evaluate the researcher's hypothesis that the instructional program affects students' knowledge. What is the correct cutoff score(s)
Answers
Answer:
The solution according to the problem given is provided below in the explanation segment.
Step-by-step explanation:
According to the question,
[tex]H_o: \mu_1=\mu_2[/tex]
[tex]H_a: \mu_1 \neq \mu_2[/tex]
Level of significance,
[tex]\alpha = .05[/tex]
The test statistics will be:
⇒ [tex]Z = \frac{(\bar x_1 - \bar x_2)}{\sqrt{\frac{\sigma_1^2}{n_1} +\frac{\sigma_2^2}{n_2} } }[/tex]
[tex]=\frac{(38-35)}{\sqrt{\frac{(3)^2}{30} +\frac{(5)^2}{30} } }[/tex]
[tex]=2.82[/tex]
The p-value will be:
= [tex]0.0024[/tex]
If the slope of a wheelchair ramp is 1/11 then what is the angle of inclination to the nearest tenth of a degree?
Answers
Answer
4.8 degrees to the nearest tenth.
Step-by-step explanation:
The slope = rise / run = opposite side / adjacent side.
So the angle of inclination is the angle whose tangent is 1/12.
To the nearest tenth of a degree it is 4.8 degrees.
Which expressions are equivalent to -6(b+2)+8
Choose all answers that apply:
A. -6b+2+8
B. -6b-4
C. None of the above
Answers
Answer:
B. -6b-4
Step-by-step explanation:
-6(b+2)+8
Distribute
-6b-12+8
Combine like terms
-6b-4
What type of line is PQ?
A. altitude
B. angle bisector
C. side bisector
D. median
Answers
The line PQ of the triangle is an altitude. The correct option is A.
What is the altitude of the triangle?
A line segment passing through a triangle's vertex and running perpendicular to the line containing the base is the triangle's height in geometry.
The extended base of the altitude is the name given to this line that contains the opposing side. The foot of the altitude is the point at where, the extended base and the height converge.
In the given triangle the line segment PQ is passing through a triangle's vertex and running perpendicular to the line containing the base is the triangle's height in geometry.
Therefore, the line PQ of the triangle is an altitude. The correct option is A.
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Use the information below to complete the problem: p(x)=1/x+1 and q(x)=1/x-1 Perform the operation and show that it results in another rational expression. p(x) + q(x)
Answers
Answer:
hope u will understand...if u like this answer plz mark as brainlist
Answer:
[tex]\displaystyle p(x) + q(x) = \frac{2x}{(x+1)(x-1)}[/tex]
The result is indeed another rational expression.
Step-by-step explanation:
We are given the two functions:
[tex]\displaystyle p(x) = \frac{1}{x+1}\text{ and } q(x) = \frac{1}{x-1}[/tex]
And we want to perform the operation:
[tex]\displaystyle p(x) + q(x)[/tex]
And show that the result is another rational expression.
Add:
[tex]\displaystyle = \frac{1}{x+1} + \frac{1}{x-1}[/tex]
To combine the fractions, we will need a common denominator. So, we can multiply the first fraction by (x - 1) and the second by (x + 1):
[tex]\displaystyle = \frac{1}{x+1}\left(\frac{x-1}{x-1}\right) + \frac{1}{x-1}\left(\frac{x+1}{x+1}\right)[/tex]
Simplify:
[tex]=\displaystyle \frac{x-1}{(x+1)(x-1)} + \frac{x+1}{(x+1)(x-1)}[/tex]
Add:
[tex]\displaystyle = \frac{(x-1)+(x+1)}{(x+1)(x-1)}[/tex]
Simplify. Hence:
[tex]\displaystyle p(x) + q(x) = \frac{2x}{(x+1)(x-1)}[/tex]
The result is indeed another rational expression.
write your answer as an integer or as a decimal rounded to the nearest tenth
Answers
Answer:
Step-by-step explanation:
CE and are the sides making up the sine of an angle.
CE is the side opposite the angle
DE is the side hypotenuse.
<D = 61 degrees
Sin(D) = opposite / hypotenuse
hypotenuse = 8
Sin(61) = 0.8746
CE = ?
sin(61) = CE / 8 multiply both sides by 8
8 sin(61) = CE
CE = 8 * 0.8746
CE = 6.9969
CE = 7.0
That 0 should be included in the answer, but I think it is safe to say that if you enter 7, you will get it right.
Answer:
7.0
Step-by-step explanation:
The scatterplot shows the number of bedrooms in a house and the selling price for that house.
Calculate the residual for the house with 6 bedrooms, to the nearest thousand.
The residual for the house with 6 bedrooms is ___
-30k
-26k
26k
30k
Answers
Answer:
-26k, -26,000 ; (B)
ED2021
The residual for the house with 6 bedrooms is -26k.
What is Scatterplots?Scatterplots are the plotting of set of points on a vertical axis and an horizontal axis. The shows the correlation extent between the numbers of the observed quantities.
Given is a scatterplot showing the selling price of houses with the number of bedrooms.
Equation of the line of best fit is given as,
y = 6.3x + 4.8
Residual = Actual value - Predicted value
From the scatterplot,
Actual selling price of house with 6 bedrooms = 40 × 10,000 = 400,000
Predicted selling price for 6 bedrooms = (6.3 × 6) + 4.8
= 42.6 × 10,000
= 426,000
Residual = 400,000 - 426,000
= -26,000
Hence the residual is -26k.
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What is the surface area of a cube with a side length of 6 m?
156 m2
300 m2
216 m2
360 m2
Answers
Answer:
216 m²
Step-by-step explanation:
Surface area of a cube = 6a², when a = length of one side
so,
6a²
= 6×6²
= 6×36
= 216 m²
Answered by GAUTHMATH
Answer:
216 m²
Step-by-step explanation:
Write the equation of the line in fully simplified slope-intercept form.
Answers
Answer:
y = 6/5x-1
Step-by-step explanation:
We have two points so we can find the slope
(-5,-7) and (5,5)
The slope is
m = ( y2-y1)/(x2-x1)
= ( 5- -7)/( 5 - -5)
= (5+7)/(5+5)
= 12/10
= 6/5
The slope intercept form of a line is
y = mx+b
y = 6/5x+b
Using the point (5,5)
5 = 6/5(5)+b
5=6+b
b=-1
y = 6/5x-1
Find the area of the shaded regions.
Answers
Answer:
around 22, or 21.98
Step-by-step explanation:
[tex]s1 = {r}^{2} \times \pi = 9 \times 3.14 = 28.26[/tex]
[tex]s2 = 1 \times 3.14 = 3.14[/tex]
[tex]s3 = 28.26 - (3.14 \times 2) = 28.26 - 6.28 = 21.98[/tex]
somebody help me please
Answers
Answer:
115
Step-by-step explanation:
Since the lines are parallel, PWX+WXR=180, WXR=65. So PWX=(180-65)=115
A 40-foot tree casts a shadow 60 feet long. How long would the shadow of a 6-foot man be at that time?
Answers
Answer:
26 ft
Step-by-step explanation:
I'm guessing this is how it's done
60-40= 20
there for at this time any shadow would be 20x it's original height/length
so 6+20=26 ft
lmk if I'm correct
Taking ratios
Let the shadow length=x ft
[tex]\\ \sf\longmapsto 40:60=6:x[/tex]
[tex]\\ \sf\longmapsto \dfrac{40}{60}=\dfrac{6}{x}[/tex]
[tex]\\ \sf\longmapsto \dfrac{4}{6}=\dfrac{6}{x}[/tex]
[tex]\\ \sf\longmapsto 4x=6(6)[/tex]
[tex]\\ \sf\longmapsto 4x=36[/tex]
[tex]\\ \sf\longmapsto x=\dfrac{36}{4}[/tex]
[tex]\\ \sf\longmapsto x=9[/tex]
2/5 + 1/10= In simplest form
A. 3/15
B. 5/10
C. 1/4
D. 1/2
Answers
Answer:
1/2
Step-by-step explanation:
2/5 + 1/10
Get a common denominator of 10
2/5 * 2/2 + 1/10
4/10 + 1/10
5/10
simplify
Divide the top and bottom by 5
1/2
Answer:
[tex] \frac{1}{2} [/tex]
Answer D is correctStep-by-step explanation:
[tex] \frac{2}{5} + \frac{1}{10} \\ \frac{2 \times 2}{5 \times 2} + \frac{1}{10} \\ \frac{4}{10} + \frac{1}{10} \\ \frac{5}{10} \\ \frac{5 \div 5}{10 \div 5} \\ = \frac{1}{2} [/tex]
find the LCM of 210, 280, 360 by prime factorisation
Answers
Answer:
Step-by-step explanation:
210=2x3x5x7
280=2x2x2x5x7
360=2x2x2x3x3x5
Answer:
210= 2×3×5×7
280=2×2×2×5×7
360=2×2×2×3×3×5
common factors=2×2×2×3×5×7=840
uncommon factors=3
L.C.M=Common factors× uncommon factors
L.C.M=840×3
L.C.M=2520
Step-by-step explanation:
i hope it will be helpful
plzz mark as brainliest
The number of defective circuit boards coming off a soldering machine follows a Poisson distribution. During a specific ten-hour period, one defective circuit board was found. (a) Find the probability that it was produced during the first hour of operation during that period. (Round your answer to four decimal places.) (b) Find the probability that it was produced during the last hour of operation during that period. (Round your answer to four decimal places.) (c) Given that no defective circuit boards were produced during the first five hours of operation, find the probability that the defective board was manufactured during the sixth hour. (Round your answer to four decimal places.)
Answers
Answer:
a) the probability that the defective board was produced during the first hour of operation is [tex]\frac{1}{10}[/tex] or 0.1000
b) the probability that the defective board was produced during the last hour of operation is [tex]\frac{1}{10}[/tex] or 0.1000
c) the required probability is 0.2000
Step-by-step explanation:
Given the data in the question;
During a specific ten-hour period, one defective circuit board was found.
Lets X represent the number of defective circuit boards coming out of the machine , following Poisson distribution on a particular 10-hours workday which one defective board was found.
Also let Y represent the event of producing one defective circuit board, Y is uniformly distributed over ( 0, 10 ) intervals.
f(y) = [tex]\left \{ {{\frac{1}{b-a} }\\\ }} \right _0[/tex]; ( a ≤ y ≤ b )[tex]_{elsewhere[/tex]
= [tex]\left \{ {{\frac{1}{10-0} }\\\ }} \right _0[/tex]; ( 0 ≤ y ≤ 10 )[tex]_{elsewhere[/tex]
f(y) = [tex]\left \{ {{\frac{1}{10} }\\\ }} \right _0[/tex]; ( 0 ≤ y ≤ 10 )[tex]_{elsewhere[/tex]
Now,
a) the probability that it was produced during the first hour of operation during that period;
P( Y < 1 ) = [tex]\int\limits^1_0 {f(y)} \, dy[/tex]
we substitute
= [tex]\int\limits^1_0 {\frac{1}{10} } \, dy[/tex]
= [tex]\frac{1}{10} [y]^1_0[/tex]
= [tex]\frac{1}{10} [ 1 - 0 ][/tex]
= [tex]\frac{1}{10}[/tex] or 0.1000
Therefore, the probability that the defective board was produced during the first hour of operation is [tex]\frac{1}{10}[/tex] or 0.1000
b) The probability that it was produced during the last hour of operation during that period.
P( Y > 9 ) = [tex]\int\limits^{10}_9 {f(y)} \, dy[/tex]
we substitute
= [tex]\int\limits^{10}_9 {\frac{1}{10} } \, dy[/tex]
= [tex]\frac{1}{10} [y]^{10}_9[/tex]
= [tex]\frac{1}{10} [ 10 - 9 ][/tex]
= [tex]\frac{1}{10}[/tex] or 0.1000
Therefore, the probability that the defective board was produced during the last hour of operation is [tex]\frac{1}{10}[/tex] or 0.1000
c)
no defective circuit boards were produced during the first five hours of operation.
probability that the defective board was manufactured during the sixth hour will be;
P( 5 < Y < 6 | Y > 5 ) = P[ ( 5 < Y < 6 ) ∩ ( Y > 5 ) ] / P( Y > 5 )
= P( 5 < Y < 6 ) / P( Y > 5 )
we substitute
[tex]= (\int\limits^{6}_5 {\frac{1}{10} } \, dy) / (\int\limits^{10}_5 {\frac{1}{10} } \, dy)[/tex]
[tex]= (\frac{1}{10} [y]^{6}_5) / (\frac{1}{10} [y]^{10}_5)[/tex]
= ( 6-5 ) / ( 10 - 5 )
= 0.2000
Therefore, the required probability is 0.2000
ko dung may tinh hay so sanh
3√7 vs 7√3
Answers
Answer:
what this makes bi sense haha
Step-by-step explanation:
but ok
Question Which of the following is a benefit of using email to communicate at work ? a) You can express yourself in a limited number of characters b) You don't have to worry about using proper grammar. c) You always get a response right away. d ) You can reach a large audience with one communication .
Answers
Answer:
d) you can reach a large audience with one communication
Step-by-step explanation:
common sense
I need help ASAP please and thank you
Answers
Answer:
"a" is the answer because 3x+2 can not be equal to zero
Step-by-step explanation:
the perimeter of a square is less than or equal to 50 find the range of the value of the length of the square
Answers
Answer:
12.5 ≥ s >0
Step-by-step explanation:
The perimeter of a square is given by
P = 4s where s is the side length
50 ≥ 4s
Divide each side by 4
50/4 ≥ 4s/4
25/2 ≥ s
12.5 ≥ s
A company pays $20 per hour for up to 8 hours of work, and $30 per hour for overtime hours (hours beyond 8 hours). For up to 8 hours worked, the equation for total pay (y) for hours worked (x) is y = 20x. For over 8 hours worked, what is the equation for total pay (y) as a function of total hours worked (x)?
Answers
Answer: y = 30x
Step-by-step explanation:
Because we are talking about over 8 hours. The question states that you get 30$ per hour for overtime hours. That means if you work over 8 hours your dollars per hour increases to 30. So because the amount of dollars increases to 30 you can infer that all you have to do is make the same equation as the 20 dollar's per hour equation. Except you put 30 making it y = 30x.
Land surveyors outlined a park as shown. What is the area of the park?
Answers
la cuadra se llama 6minutos
Calculus II Question
Identify the function represented by the following power series.
[tex]\sum_{k = 0}^\infty (-1)^k \frac{x^{k + 2}}{4^k}[/tex]
Answers
With some rewriting, you get
[tex]\displaystyle \sum_{k=0}^\infty (-1)^k\frac{x^{k+2}}{4^k} = x^2 \sum_{k=0}^\infty \left(-\frac x4\right)^k[/tex]
Recall that for |x| < 1, you have
[tex]\displaystyle \frac1{1-x} = \sum_{k=0}^\infty x^k[/tex]
So as long as |-x/4| = |x/4| < 1, or |x| < 4, your series converges to
[tex]\displaystyle x^2 \sum_{k=0}^\infty \left(-\frac x4\right)^k = \frac{x^2}{1-\left(-\frac x4\right)} = \frac{x^2}{1+\frac x4} = \boxed{\frac{4x^2}{4+x}}[/tex]
Based on known expressions from Taylor series, the power series [tex]\sum \limits_{k = 0}^{\infty} (-1)^{k}\cdot \frac{x^{k+2}}{4^{k}}[/tex]Taylor series-derived formula of the rational function [tex]\frac{4\cdot x^{2}}{4+x}[/tex].
How to derive a function behind the approximated formula by Taylor seriesTaylor series are polynomic approximations used to estimate values both from trascendental and non-trascendental functions. It is commonly used in trigonometric, potential, logarithmic and even rational functions.
In this question we must use series properties and common Taylor series-derived formulas to infer the expression behind the given series. Now we proceed to find the expression:
[tex]\sum \limits_{k = 0}^{\infty} (-1)^{k}\cdot \frac{x^{k+2}}{4^{k}}[/tex]
[tex]x^{2}\cdot \sum\limits_{k = 0}^{\infty} \left(-\frac{x}{4} \right)^{k}[/tex]
[tex]x^{2}\cdot \left(\frac{1}{1+\frac{x}{4} } \right)[/tex]
[tex]\frac{4\cdot x^{2}}{4+x}[/tex]
Based on power and series properties and most common Taylor series- derived formulas, the power series [tex]\sum \limits_{k = 0}^{\infty} (-1)^{k}\cdot \frac{x^{k+2}}{4^{k}}[/tex] represents a Taylor series-derived formula of the rational function [tex]\frac{4\cdot x^{2}}{4+x}[/tex]. [tex]\blacksquare[/tex]
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Estimate the average rate of change from x 1 to x = 4. Enter your estimate as a decimal number (not as a fraction), rounded to one decimal place. Average rate of change = Number
Answers
Answer:Mark brainliest please
Answer is - 0.5
Step-by-step explanation:
The average rate of change is -0.5 which is the average rate of change from x 1 to x = 4 the answer is -0.5.
What is the rate of change?It is defined as the change in values of a dependent variable with respect to the independent variables.
As we know an average is a single number that represents the mean value for the given set of data or the closed value for each entry given in the set of data.
We have a graph of functions shown in the picture.
Estimate the average rate of change from x 1 to x = 4.
At x = 1,
y = 5
At x = 4
y = 3.5(approx)
The average rate of change = (3.5 - 5)/(4 - 1)
The average rate of change = -1.5/3
The average rate of change = -0.5
Thus, the average rate of change is -0.5 which is the average rate of change from x 1 to x = 4 the answer is -0.5.
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5
13
The probabilities that three men win their respective races are 1/3,3/5and 3/4.what is theprobability that
a) all of them win their races)
b) only one of them win his race?
Answers
Answer:
a
Step-by-step explanation:
1/3 x 3/5 x 3/4 =7/12 so therefore that's what the answer isn't
In a class of students, the following data table summarizes how many students have a cat or a dog. What is the probability that a student who has a cat also has a dog?
Has a cat Does not have a cat
Has a dog 7 6
Does not have a dog 8 2
Answers
Differentiate the following Functions
5x^2-2xy + 4y^3= 5
Answers
Answer:
[tex]\displaystyle y' = \frac{y - 5x}{x + 6y^2}[/tex]
General Formulas and Concepts:
Algebra I
Terms/CoefficientsFactoringCalculus
Differentiation
DerivativesDerivative NotationImplicit DifferentiationDerivative Property [Multiplied Constant]: [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]
Derivative Property [Addition/Subtraction]: [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]
Basic Power Rule:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Derivative Rule [Product Rule]: [tex]\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)[/tex]
Derivative Rule [Chain Rule]: [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle 5x^2 - 2xy + 4y^3 = 5[/tex]
Step 2: Differentiate
Implicit Differentiation: [tex]\displaystyle \frac{dy}{dx}[5x^2 - 2xy + 4y^3] = \frac{dy}{dx}[5][/tex]Rewrite [Derivative Property - Addition/Subtraction]: [tex]\displaystyle \frac{dy}{dx}[5x^2] - \frac{dy}{dx}[2xy] + \frac{dy}{dx}[4y^3] = \frac{dy}{dx}[5][/tex]Rewrite [Derivative Property - Multiplied Constant]: [tex]\displaystyle 5\frac{dy}{dx}[x^2] - 2\frac{dy}{dx}[xy] + 4\frac{dy}{dx}[y^3] = \frac{dy}{dx}[5][/tex]Basic Power Rule [Chain Rule]: [tex]\displaystyle 10x - 2\frac{dy}{dx}[xy] + 12y^2y' = 0[/tex]Product Rule: [tex]\displaystyle 10x - 2\bigg[ \frac{dy}{dx}[x]y + x\frac{dy}{dx}[y] \bigg] + 12y^2y' = 0[/tex]Basic Power Rule [Chain Rule]: [tex]\displaystyle 10x - 2\bigg[ y + xy' \bigg] + 12y^2y' = 0[/tex]Simplify: [tex]\displaystyle 10x - 2y + 2xy' + 12y^2y' = 0[/tex]Isolate y' terms: [tex]\displaystyle 2xy' + 12y^2y' = 2y - 10x[/tex]Factor: [tex]\displaystyle y'(2x + 12y^2) = 2y - 10x[/tex]Isolate y': [tex]\displaystyle y' = \frac{2y - 10x}{2x + 12y^2}[/tex]Factor: [tex]\displaystyle y' = \frac{2(y - 5x)}{2(x + 6y^2)}[/tex]Simplify: [tex]\displaystyle y' = \frac{y - 5x}{x + 6y^2}[/tex]Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
Book: College Calculus 10e
