Answer:
[tex]\cos(2\, x) = 1 - 2\, (\sin(x))^2[/tex].
Step-by-step explanation:
Angle sum identity for cosine: [tex]\cos(a + b) = \cos(a) \, \cos(b) - \sin(a) \, \sin(b)[/tex].
Pythagorean identity: [tex](\cos(a))^{2} + (\sin(a))^{2} = 1[/tex] for all real [tex]a[/tex].
Subtract [tex](\cos(x))^{2}[/tex] from both sides of the Pythagorean identity to obtain: [tex](\sin(a))^{2} = 1 - (\cos(a))^{2}[/tex].
Apply angle sum identity to rewrite [tex]\cos(2\, x)[/tex].
[tex]\begin{aligned}&\cos(2\, x)\\ &= \cos(x + x) \\ &= \cos(x) \, \cos(x) - \sin(x)\, \sin(x) \\ &= (\cos(x))^{2} + (\sin(x))^{2}\end{aligned}[/tex].
[tex](\sin(a))^{2} = 1 - (\cos(a))^{2}[/tex] follows from the Pythagorean identity. Hence, it would be possible to replace the [tex](\cos(x))^{2}[/tex] in the previous expression with [tex](1 - (\sin(x))^{2})[/tex].
[tex]\begin{aligned}&(\cos(x))^{2} - (\sin(x))^{2}\\ &= \left[1 - (\sin(x))^{2}\right] - (\sin(x))^{2} \\ &= 1 - 2\, (\sin(x))^{2} \end{aligned}[/tex].
Conclusion:
[tex]\begin{aligned}&\cos(2\, x) \\ &= (\cos(x))^{2} + (\sin(x))^{2} \\ &=1 - 2\, (\sin(x))^{2}\end{aligned}[/tex]
90 dollars ratio in 1:2:3
Answer:
3:6:9
Step-by-step explanation:
1/1+2+3 × 90 = 15
2/1+2+3 × 90 = 30
3/1+2+3 × 90 = 45
HELP PLEASE IM STUCK!
1. Which of these describes the relation for this set of coordinate pairs?
{(-1, 5), (12, 18), (0, 6), (-3, 3), (4, ?), (?, 11)}
a. x - y = 6 b. f(x) = x +6 c. f(x) = 6 d. y = 6x e. None of these
Answer:
b) f(x) = x + 6
Step-by-step explanation:
The coordinate (0, 6) makes the y-intercept = 6. Only one of these functions has that intercept: f(x) = x + 6. If you plug in each coordinate the outputted y-value matches up, making this the right answer.
Suppose the weights of apples are normally distributed with a mean of 85 grams and a standard deviation of 8 grams. The weights of oranges are also normally distributed with a mean of 131 grams and a standard deviation of 20 grams. Amy has an apple that weighs 90 grams and an orange that weighs 155 grams.
Required:
a. Find the probability a randomly chosen apple exceeds 100 g in weight.
b. What weight do 80% of the apples exceed?
Answer:
a) 0.0304 = 3.04% probability a randomly chosen apple exceeds 100 g in weight.
b) The weight that 80% of the apples exceed is of 78.28g.
Step-by-step explanation:
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Weights of apples are normally distributed with a mean of 85 grams and a standard deviation of 8 grams.
This means that [tex]\mu = 85, \sigma = 8[/tex]
a. Find the probability a randomly chosen apple exceeds 100 g in weight.
This is 1 subtracted by the p-value of Z when X = 100. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{100 - 85}{8}[/tex]
[tex]Z = 1.875[/tex]
[tex]Z = 1.875[/tex] has a p-value of 0.9697
1 - 0.9696 = 0.0304
0.0304 = 3.04% probability a randomly chosen apple exceeds 100 g in weight.
b. What weight do 80% of the apples exceed?
This is the 100 - 80 = 20th percentile, which is X when Z has a p-value of 0.2, so X when Z = -0.84.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.84 = \frac{X- 85}{8}[/tex]
[tex]X - 85 = -0.84*8[/tex]
[tex]X = 78.28[/tex]
The weight that 80% of the apples exceed is of 78.28g.
Are these triangles congruent?
Answer:
yes...
Step-by-step explanation:
its a congrate triangle
what percentage of the appies are yellow?
Answer:
20%
Step-by-step explanation:
6 out of 30. = 1/5 = multiply 5*20= 100 and 1*20= 100 so it is 20% of 100.
A researcher has obtained the number of hours worked per week during the summer for a sample of 15 students. 40 25 35 30 20 40 30 20 40 10 30 20 10 5 20 Using this data set, compute the following: a. Median b. Mean c. Mode d. 40th percentile e. Range f. Sample variance g. Standard deviation
Answer:
Mean = 25
Median = 25
Mode = 6
Step-by-step explanation:
Given the data :
Using calculator :
Mode = highest occurring data point
Range = max - min = 40 - 5 = 35
Sample variance = 128.57 (calculator)
Sample standard deviation = sqrt(variance) = 11.34
ANSWER ASAP IM TIMED 30 POINTS
Which shape below will form a tessellation?
A. regular heptagons
B. regular pentagons
C. regular octagons
D. regular hexagons
Answer:
Step-by-step explanation:
D ANSWER D
Answer:
d
Step-by-step explanation:
17.
What is the value of the expression
2a + 5b + 3c for a = 12, b = 6, and
C=3?
A 10
B 21
C49
D 63
D. 63
2a+5b+3c
2(12)+5(6)+3(3)
24+30+9=63
Hope this helps! :)
Select the correct answer.
At a high school there are 53 players on the football team, 15 players on the baseball team, and 12 players on
the basketball team. How many ways can a committee be formed with 1 representative from each team?
Math
Answer:
53*15*12=9540
Step-by-step explanation:
its just 53 times 15 times 12 for all the possibilities.
Say the first football player was picked, same with the first baseball player, and the first basketball player were all picked, then another possiblity would be the first football player, the second baseball player, and the first basketball player, here is a numerical example.
Football Baseball Basketball
1 1 1
1 2 1
1 2 2
1 2 3
1 2 4
and so on including all the patterns it would be 9540 possibilities
The total number of ways of forming a committee by selecting one representative from each team is 9540.
What is combination?
A combination is a mathematical technique that determines the number of possible arrangements or the number of ways in a collection of items where the order of the selection does not matter.
Combination Formula[tex]nC_{r}= \frac{n!}{r!(n-r)!}[/tex]
Where,
[tex]nC_{r}[/tex] is a number of combination.
n is total number of objects in the set.
r is the number of choosing objects from the set.
Multiplication rule in combination?According to the multiplication rule in combination if there are a ways of doing something and b ways of doing another thing, then there are a · b ways of performing both actions.
According to the given question
Total number of football players = 53
Total number of baseball players = 15
Total number of basket ball players = 12
Therefore,
Number of ways of selecting one representative from football team is given by
[tex]53C_{1} =\frac{53!}{1!52!} = 53[/tex]
Number of ways of selecting one representative from baseball team is given by
[tex]15C_{1} =\frac{15!}{1!14!}=15[/tex]
Number of ways of selecting one representative from basketball team is given by
[tex]12C_{1} =\frac{12!}{1!11!} =12[/tex]
So, the total number of ways of forming a committee by selecting one representative from each team = 53 × 15 × 12 =9540.
Hence, total number of ways of forming a committee by selecting one representative from each team 9540.
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How do you solve x[tex]x^{2} +4x+3=0[/tex]?
Answer:
[tex]{ \tt{ {x}^{2} + 4x + 3 = 0}} \\ { \tt{(x + 1)(x + 3) = 0}} \\ \\ { \tt{x = - 1 \: \: and \: \: - 3}}[/tex]
What is the value of 6 / x + 2x squared when x = 3
Answer:
8
Step-by-step explanation:
The equation will become 6/3+2(3)
-->6/3=2
-->Because of Order of Operation we will multiply the 2 and 3 before adding so 2(3) = 6
--> 6+2=8
Answer:
x=6
Step-by-step explanation:
when its squared you multiply by 2
CAN SOMEONE PLEASE HELP ME GOOD, I need this to graduate ): 5. Given that AABC - ADEC, find the
value of x.
Answer:
ans: 4
Step-by-step explanation:
corresponding sides are proportional since given triangle are similar triangle, I.e
(4/5.5) = { (2x+8)/(6x-2)}
8/11 = ( x+ 4 ) / ( 3x - 1 )
8( 3x - 1 )= 11( x + 4 )
24x - 8 = 11x + 44
13x = 52
x = 4
Find the value of y and show work
Answer:
75
Step-by-step explanation:
∠K and ∠ R are congruent (equal)
Triangle Sum Theory - angles of all triangles add to 180
180 - 79 - 26 = 75
An expression is shown below:
3(m + 5 + 9m)
Part A: Write two expressions that are equivalent to the given expression. (3 points)
Part B: Show that one of your expressions in Part A is equivalent to the given expression using algebraic properties. Explain which properties you used. (4 points)
Part C: Show that your other expression from Part A is equivalent to the given expression by substituting a number for m. (3 points)
The answers are as follows part A = 30m+15 part B =3m+15+27m
and partC = 30m+15
What is an expression?Expression in maths is defined as the collection of the numbers variables and functions by using signs like addition,substraction, multiplication and division.
Part A:- Two expressions that are equivalent to the given expressions are:-
3m + 15 + 27m
30m + 15
Part B: Show that one of your expressions in Part A is equivalent to the given expression using algebraic properties.
3 ( m + 5 + 9m )
Open the bracket by multiplying 3 by what is in the bracket
3m + 15 + 27m
Part C: Show that your other expression from Part A is equivalent to the given expression by substituting a number for m.
3 ( m + 5 + 9m )
Open the bracket by multiplying 3 by what is in the bracket
3m + 15 + 27m
Collect like terms together
3m + 27m + 15
= 30m + 15
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Instructions: Find the value of x
I’ll mark brainliest please help
The little lines theu each section are telling you all those sections are identical, which mean they are the same length.
You are told one section is 10, which means x is also 10
X = 10
A square-based tent covers an area of 64 square feet. What is the length of one side of the tent?
Answer:
8 feet
Step-by-step explanation:
A square's area can be expressed as l², with l representing one side of the tent. Therefore, as our area is 64, we can say that 64 = l². Taking the square root of both sides, we get that √64 = l, and l = 8 feet
When comparing two box-plots that show the same type of information, what determines agreement within the data?
A.the range of the quartiles in each data set
B.the median of each data set
C.the mean of each data set
D.the number of values in each data set
Answer:
c.the mean of each data set
Answer:
A
Step-by-step explanation:
What is the probability of rolling a number less than or equal to 8 with the
sum of two dice, given that at least one of the dice must show a 6?
Answer:
I hope this helps
the outcomes are the compulsory 6, and 1 or 2
Step-by-step explanation:
[tex] \frac{3}{6} \\ \frac{1}{2} or \: 0.5[/tex]
Need help on this question been stuck on it
Answer:
Exponential Function
Step-by-step explanation:
y values increase by x4
Answer:
exponential function
the ans is b
Which of the following sets of data does not contain an outlier?
A.16, 17, 20, 19.48
B.59. 60. 61, 67.65
C.95.99.97.94.60
D.-1.2.1.0.5.16
Answer:
it is a letter b
Step-by-step explanation:
that does not contain an outlet
[3⋅5]7=
help pls anyone pls help me
You buy a family-size box of laundry detergent that contains 40 cups. , how many loads of wash can you do?
SEE IMAGE BELOW
Answer:
30 loads
Step-by-step explanation:
You simply divide 40 by 1 1/3 which gives you 30 meaning you can wash 30 loads.
Answer:
30
Step-by-step explanation:
To solve this, you want to do 40÷[tex]1\frac{1}{3}[/tex], or [tex]\frac{40}{1}[/tex]÷[tex]\frac{4}{3}[/tex], which is the same as [tex]\frac{40}{1}[/tex]×[tex]\frac{3}{4}[/tex].
This can be solved to [tex]\frac{120}{4}[/tex] and simplifies to 30.
**This content involves writing expressions from worded questions and operations with fractions, which you may wish to revise. I'm always happy to help!
Question 11 of 45
Which similarity postulate or theorem can be used to verify that the two
triangles shown below are similar?
12
6 R
X 2 z
O A. Similarity cannot be determined,
B. SSS theorem
O C. AA postulate
D. SAS theorem
Answer:
You are only given a Side, an Angle, and then a side. So that is what I would choose. It can't be AA because you weren't given two angles. It can't be SSS because you weren't given 3 sides.
Step-by-step explanation:
ΔPQR and ΔXYZ are similar due to the SSS theorem.
Option B is the correct answer.
What is triangle congruency?There are ways to prove that two triangles are congruent.
- Side-Side-Side (SSS) Congruence.
The three sides of one triangle are equal to the corresponding three sides of another triangle.
- Side-Angle-Side (SAS) Congruence.
The two sides and the included angle of one triangle are equal to the corresponding two sides and included angle of another triangle.
- Angle-Side-Angle (ASA) Congruence.
The two angles and the included side of one triangle are equal to the corresponding two angles and included side of another triangle.
- Angle-Angle-Side (AAS) Congruence.
We have,
Similar triangles are triangles that have the same shape but may have different sizes.
Two triangles are similar if their corresponding angles are equal, and their corresponding sides are proportional.
When two triangles are similar, they have the same shape, but one may be larger or smaller than the other.
Now,
ΔPQR and ΔXYZ
PQ/XY = 12/4 = 3
PR/XZ = 6/2 = 3
Since the two sides are proportional,
QR/YZ will be proportional.
Now,
PQ/XY = PR/XZ = QR/YZ
ΔPQR and ΔXYZ are similar due to the SSS theorem.
Thus,
ΔPQR and ΔXYZ are similar due to the SSS theorem.
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Which expression are greater than 1/2? Choose all the apply
Answer:
25/30
5/8
Step-by-step explanation:
Which fraction is it out of all of these 6/14,5/8,25/30,or 3/6?
to determine which fractions are greater than 1/2, convert the fractions to decimals
to convert to decimals, divide the numerator by the denominator
1/2 = 0.5 less than half
6/14 = 0.43 less than half
5/8 = 0.625 greater than half
25 / 30 = 0.83 greater than half
3 / 6 = 0.5 equal to half
anyone help me, let's prove
Answer:
In my opinion the limit is equal to 1 not 0, sorry.
Step-by-step explanation: 6 25 13 43
lim n ⇒∞ ((2n - 1)/2n)
lim n ⇒∞ (2n/2n) - 1)/2n) 2n/2n = 1 1/∞ = 0
= 1 - 0
= 1
when I graphed the function I also got 1
At Western University the historical mean of scholarship examination scores for freshman applications is 900. Ahistorical population standard deviation \sigmaσ= 180 is assumed known.
Each year, the assistant dean uses a sample of applications to determine whether the mean examination score for the new freshman applications has changed.
a. State the hypotheses.
b. What is the 95% confidence interval estimate of the population mean examination
score if a sample of 200 applications provided a sample mean \overline x
x
= 935?
c. Use the confidence interval to conduct a hypothesis test. Using \alphaα= .05, what is your
conclusion?
d. What is the p-value?
Answer:
(910.053 ; 959.947)
Pvalue = 0.00596
Step-by-step explanation:
Given :
Population mean, μ = 900
Sample size, n = 200
Population standard deviation, σ = 180
The hypothesis :
H0 : μ = 900
H0 : μ ≠ 900
The 95% confidence interval:
Xbar ± Margin of error
Margin of Error = Zcritical * σ/√n
Since the σ is known, we use the z- distribution
Zcritical at 95% confidence = 1.96
Hence,
Margin of Error = 1.96 * 180/√200
Margin of Error = 24.947
95% confidence interval is :
935 ± 24.947
Lower boundary = 935 - 24.947 = 910.053
Upper boundary = 935 + 24.947 = 959.947
(910.053 ; 959.947)
Hypothesis test :
Test statistic
(935- 900) ÷ (180/√(200))
Test statistic = 2.750
Pvalue from Test statistic ;
Pvalue = 0.00596
Pvalue < α ; Reject H0 and conclude that score has changed
Hence, we can conclude that the score has changed
Use the substitution method to solve the system of equations. Choose the correct ordered pair. 4(x + 4) = 8(y + 2); 18y - 22 = 3x + 2
x = 30
y = 2
Get the explanation from the image I have shared.
Hope it helps you
Segment [tex]$s_1$[/tex] has endpoints at [tex]$(3+\sqrt{2},5)$[/tex] and[tex]$(4,7)$[/tex]. Segment [tex]$s_2$[/tex] has endpoints at [tex]$(6-\sqrt{2},3)$[/tex] and[tex]$(3,5)$[/tex]. Find the midpoint of the segment with endpoints at the midpoints of [tex]$s_1$[/tex] and [tex]$s_2$[/tex]. Express your answer as [tex]$(a,b)$[/tex].
Answer:
The midpoint of the segment with endpoints at the midpoints of s1 and s2 is (4,5).
Step-by-step explanation:
Midpoint of a segment:
The coordinates of the midpoint of a segment are the mean of the coordinates of the endpoints of the segment.
Midpoint of s1:
Using the endpoints given in the exercise.
[tex]x = \frac{3 + \sqrt{2} + 4}{2} = \frac{7 + \sqrt{2}}{2}[/tex]
[tex]y = \frac{5 + 7}{2} = \frac{12}{2} = 6[/tex]
Thus:
[tex]M_{s1} = (\frac{7 + \sqrt{2}}{2},6)[/tex]
Midpoint of s2:
[tex]x = \frac{6 - \sqrt{2} + 3}{2} = \frac{9 - \sqrt{2}}{2}[/tex]
[tex]y = \frac{3 + 5}{2} = \frac{8}{2} = 4[/tex]
Thus:
[tex]M_{s2} = (\frac{9 - \sqrt{2}}{2}, 4)[/tex]
Find the midpoint of the segment with endpoints at the midpoints of s1 and s2.
Now the midpoint of the segment with endpoints [tex]M_{s1}[/tex] and [tex]M_{s2}[/tex]. So
[tex]x = \frac{\frac{7 + \sqrt{2}}{2} + \frac{9 - \sqrt{2}}{2}}{2} = \frac{16}{4} = 4[/tex]
[tex]y = \frac{6 + 4}{2} = \frac{10}{2} = 5[/tex]
The midpoint of the segment with endpoints at the midpoints of s1 and s2 is (4,5).
In a regression analysis involving 30 observations, the following estimated regressionequation was obtained.y^ =17.6+3.8x 1 −2.3x 2 +7.6x 3 +2.7x 4For this estimated regression equation SST = 1805 and SSR = 1760. a. At \alpha =α= .05, test the significance of the relationship among the variables.Suppose variables x 1 and x 4 are dropped from the model and the following estimatedregression equation is obtained.y^ =11.1−3.6x 2 +8.1x 3For this model SST = 1805 and SSR = 1705.b. Compute SSE(x 1 ,x 2 ,x 3 ,x 4 )c. Compute SSE (x2 ,x3 ) d. Use an F test and a .05 level of significance to determine whether x1 and x4 contribute significantly to the model.
Answer:
(a) There is a significant relationship between y and [tex]x_1, x_2, x_3, x_4[/tex]
(b) [tex]SSE_{(x_1 ,x_2 ,x_3 ,x_4) }= 45[/tex]
(c) [tex]SSE_{(x_2,x_3)} = 100[/tex]
(d) [tex]x_1[/tex] and [tex]x_4[/tex] are significant
Step-by-step explanation:
Given
[tex]y = 17.6+3.8x_1 - 2.3x_2 +7.6x_3 +2.7x_4[/tex] --- estimated regression equation
[tex]n = 30[/tex]
[tex]p = 4[/tex] --- independent variables i.e. x1 to x4
[tex]SSR = 1760[/tex]
[tex]SST = 1805[/tex]
[tex]\alpha = 0.05[/tex]
Solving (a): Test of significance
We have:
[tex]H_o :[/tex] There is no significant relationship between y and [tex]x_1, x_2, x_3, x_4[/tex]
[tex]H_a :[/tex] There is a significant relationship between y and [tex]x_1, x_2, x_3, x_4[/tex]
First, we calculate the t-score using:
[tex]t = \frac{SSR}{p} \div \frac{SST - SSR}{n - p - 1}[/tex]
[tex]t = \frac{1760}{4} \div \frac{1805- 1760}{30 - 4 - 1}[/tex]
[tex]t = 440 \div \frac{45}{25}[/tex]
[tex]t = 440 \div 1.8[/tex]
[tex]t = 244.44[/tex]
Next, we calculate the p value from the t score
Where:
[tex]df = n - p - 1[/tex]
[tex]df = 30 -4 - 1=25[/tex]
The p value when [tex]t = 244.44[/tex] and [tex]df = 25[/tex] is:
[tex]p =0[/tex]
So:
[tex]p < \alpha[/tex] i.e. [tex]0 < 0.05[/tex]
Solving (b): [tex]SSE(x_1 ,x_2 ,x_3 ,x_4)[/tex]
To calculate SSE, we use:
[tex]SSE = SST - SSR[/tex]
Given that:
[tex]SSR = 1760[/tex] ----------- [tex](x_1 ,x_2 ,x_3 ,x_4)[/tex]
[tex]SST = 1805[/tex]
So:
[tex]SSE_{(x_1 ,x_2 ,x_3 ,x_4)} = 1805 - 1760[/tex]
[tex]SSE_{(x_1 ,x_2 ,x_3 ,x_4) }= 45[/tex]
Solving (c): [tex]SSE(x_2 ,x_3)[/tex]
To calculate SSE, we use:
[tex]SSE = SST - SSR[/tex]
Given that:
[tex]SSR = 1705[/tex] ----------- [tex](x_2 ,x_3)[/tex]
[tex]SST = 1805[/tex]
So:
[tex]SSE_{(x_2,x_3)} = 1805 - 1705[/tex]
[tex]SSE_{(x_2,x_3)} = 100[/tex]
Solving (d): F test of significance
The null and alternate hypothesis are:
We have:
[tex]H_o :[/tex] [tex]x_1[/tex] and [tex]x_4[/tex] are not significant
[tex]H_a :[/tex] [tex]x_1[/tex] and [tex]x_4[/tex] are significant
For this model:
[tex]y =11.1 -3.6x_2+8.1x_3[/tex]
[tex]SSE_{(x_2,x_3)} = 100[/tex]
[tex]SST = 1805[/tex]
[tex]SSR_{(x_2 ,x_3)} = 1705[/tex]
[tex]SSE_{(x_1 ,x_2 ,x_3 ,x_4) }= 45[/tex]
[tex]p_{(x_2,x_3)} = 2[/tex]
[tex]\alpha = 0.05[/tex]
Calculate the t-score
[tex]t = \frac{SSE_{(x_2,x_3)}-SSE_{(x_1,x_2,x_3,x_4)}}{p_{(x_2,x_3)}} \div \frac{SSE_{(x_1,x_2,x_3,x_4)}}{n - p - 1}[/tex]
[tex]t = \frac{100-45}{2} \div \frac{45}{30 - 4 - 1}[/tex]
[tex]t = \frac{55}{2} \div \frac{45}{25}[/tex]
[tex]t = 27.5 \div 1.8[/tex]
[tex]t = 15.28[/tex]
Next, we calculate the p value from the t score
Where:
[tex]df = n - p - 1[/tex]
[tex]df = 30 -4 - 1=25[/tex]
The p value when [tex]t = 15.28[/tex] and [tex]df = 25[/tex] is:
[tex]p =0[/tex]
So:
[tex]p < \alpha[/tex] i.e. [tex]0 < 0.05[/tex]
Hence, we reject the null hypothesis
Assume that x and y are both differentiable functions of t and find the required values of dy/dt and dx/dt.
xy = 2
a. Find dy/dt, when x = 4, given that dx/dt = 13.
b. Find dx/dt, when x = 1, given that dy/dt = -9.
Answer:
a. [tex]\frac{dy}{dt} = -\frac{13}{8}[/tex]
b. [tex]\frac{dx}{dt} = \frac{9}{2}[/tex]
Step-by-step explanation:
To solve this question, we apply implicit differentiation.
xy = 2
Applying the implicit differentiation:
[tex]y\frac{dx}{dt} + x\frac{dy}{dt} = \frac{d}{dt}(2)[/tex]
[tex]y\frac{dx}{dt} + x\frac{dy}{dt} = 0[/tex]
a. Find dy/dt, when x = 4, given that dx/dt = 13.
x = 4
So
[tex]xy = 2[/tex]
[tex]4y = 2[/tex]
[tex]y = \frac{2}{4} = \frac{1}{2}[/tex]
Then
[tex]y\frac{dx}{dt} + x\frac{dy}{dt} = 0[/tex]
[tex]\frac{1}{2}(13) + 4\frac{dy}{dt} = 0[/tex]
[tex]4\frac{dy}{dt} = -\frac{13}{2}[/tex]
[tex]\frac{dy}{dt} = -\frac{13}{8}[/tex]
b. Find dx/dt, when x = 1, given that dy/dt = -9.
x = 1
So
[tex]xy = 2[/tex]
[tex]y = 2[/tex]
Then
[tex]y\frac{dx}{dt} + x\frac{dy}{dt} = 0[/tex]
[tex]2\frac{dx}{dt} - 9 = 0[/tex]
[tex]2\frac{dx}{dt} = 9[/tex]
[tex]\frac{dx}{dt} = \frac{9}{2}[/tex]
