Don’t get part b of the question. Very confusing any chance you may help me with this please.
Answers
To solve this problem, first, we will solve the given equation for y:
[tex]\begin{gathered} x=3\tan 2y, \\ \tan 2y=\frac{x}{3}, \\ 2y=\arctan (\frac{x}{3}), \\ y=\frac{\arctan(\frac{x}{3})}{2}=\frac{1}{2}\arctan (\frac{x}{3})\text{.} \end{gathered}[/tex]Once we have the above equation, now we compute the derivative. To compute the derivative we will use the following properties of derivatives:
[tex]\begin{gathered} \frac{d}{dx}\arctan (x)=\frac{1}{x^2+1}, \\ \frac{dkf(x)}{dx}=k\frac{df(x)}{dx}. \end{gathered}[/tex]Where k is a constant.
First, we use the second property above, and get that:
[tex]\frac{d\frac{\arctan(\frac{x}{3})}{2}}{dx}=\frac{d\arctan (\frac{x}{3})\times\frac{1}{2}}{dx}=\frac{1}{2}\frac{d\arctan (\frac{x}{3})}{dx}\text{.}[/tex]Now, from the chain rule, we get:
[tex]\frac{dy}{dx}=\frac{1}{2}\frac{d\text{ arctan(}\frac{x}{3})}{dx}=\frac{1}{2}\frac{d\arctan (\frac{x}{3})}{dx}|_{\frac{x}{3}}\frac{d\frac{x}{3}}{dx}\text{.}[/tex]Finally, computing the above derivatives (using the rule for the arctan), we get:
[tex]\frac{dy}{dx}=\frac{1}{2}\frac{\frac{1}{3}}{\frac{x^2}{9}+1}=\frac{1}{6}(\frac{1}{\frac{x^2}{9}+1})=\frac{3}{2(x^2+9)}.[/tex]Answer:
[tex]\frac{3}{2(x^2+9)}.[/tex]Related Questions
A function can have miltiple x intercepts A function can have multiple y intercepts To find the y intercept you must find the zeros The notation of the Zeros of the function is f(0)
Answers
The statements which are true regarding a function among the given answer choices are;
A function can have multiple x-intercepts.The notation of the zeroes of the function is; f(0).Which statements among the answer choices are true for functions?It follows from the complete task content that the statements which are true be identified from the given answer choices.
From the definition of a function; A function is a relation which assigns to every input value one single output value. Hence, it follows that no single input value has more than one output value assigned to it.
It therefore follows from the definition above that; a function can have multiple x-intercepts, but can only have one y-intercept.
Also, the zeroes of the function are represented by the function instance; f(0) at which point the input, x = 0.
Remarks;
The complete task content is such that; The statements which are correct about functions are to.be identified.
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The probability that John recieves junk mail is 11 percent. If he receives 94 pieces of mail in a week, about how many of them can he expect to be junk mail.a. 5 b. 15 c. 10 d.20
Answers
10 (option C)
Explanation:The probability of getting a junk mail = 11%
Number of mails received = 94
Amount that will be junk mail = The probability of getting a junk mail × Number of mails received
Amount that will be junk mail = 11% × 94
= 0.11 × 94 = 10.34
Since we can't have decimal number of mails, we would approximate to the nearest whole number
10.34 to the nearest whole number is 10
Hence, 10 junk mails are expected
The table gives a set of outcomes and their probabilities. Let A be the event the outcome is divisible by 3". Find P(A). 12 Outcome Probability Tim elaps 1 0.14 PAUS 2 0.02 3 0.19 Smart out of 4 0.01 5 0.04 7 6 0.17 7 0.15 8 0.28
Answers
Here, we want to get the probability that a selected outcome is divisible by 3
What we have to do here is ti select numbers that are multiples of 3 and add their probabilities
From the given table, the outcomes that are multiples of 3 are;
3 and 6 only
So, we proceed to add the probabilities of these outcomes
Mathematically, we have this as;
[tex]P(A)\text{ = 0.19 + 0.17 = 0.36}[/tex]11 gallons Blue Car 2 of gas 35.4 miles A gallons 27 miles Silver Car 5 14. You are running a fuel economy study. You want to find out which car can travel a greater distance on 1'gallon of gas. a. What is the gas mileage, in miles per gallon, for the blue car? b. What is the gas mileage, in miles per gallon, for the silver car? c. Which car could travel the greater distance on 1 gallon of gas?
Answers
Answer:
a) 23.67 miles per gallon
b) 34 miles per gallon
c) The silver car could travel a greater distance.
Step-by-step explanation:
a)
Conversion of the mixed numbers to fractions:
[tex]1\frac{1}{2}=\frac{1\ast2+1}{2}=\frac{2+1}{2}=\frac{3}{2}[/tex][tex]35\frac{1}{2}=\frac{35\ast2+1}{2}=\frac{70+1}{2}=\frac{71}{2}[/tex]Gas mileage:
3/2 gallons - 71/2 miles
1 gallons - x miles
Simplifying the top line by 2.
3 gallons - 71 miles
1 gallon - x miles
3x = 71
x = 71/3
x = 23.67 miles per gallon
b)
Conversion of the mixed number to fraction:
[tex]27\frac{1}{5}=\frac{27\ast5+1}{5}=\frac{135+1}{5}=\frac{136}{5}[/tex]Mileage:
4/5 gallons - 136/5 miles
1 gallon - x miles
Simplifying the top line by 5
4 gallons - 136 miles
1 gallon - x miles
4x = 136
x = 136/4
x = 34 miles per gallon
c)
Blue car: 23.67 miles per gallon
Silver car: 34 miles per gallon
Silver car could travel a greater distance.
Determine a third pair of congruent parts to establish congruence between the triangles. Give the congruence postulate involved
Answers
In this problem, we have that
mYO ≅ XO
The third pair of congruent parts is
m by vertical angles
therefore
triangle YOT ≅ triangle XOB ----> by ASA congruence postulate
Through (1,-2) parallel to y=-2x+5
Answers
Answer:
Step-by-step explanation:The line parallel to y = -2x + 5 that passes through the point(1,1)
Has the same slope, m but a different y intercept (0,b)
So lets start by using the given point (1, 1) and the slope intercept form of the line to calculate b
y = mx + b
m = -2
1 = -2(1) + b
1 = -2 + b
Add 2 to both sides of the equation to solve for b
1 + 2 = b
3 = b
The line is
y = -2x + 3
Please help. Find value of p
Answers
The value of p from the given diagram using the similarity theorem is -13.
Similarity theorem of triangleYou can use three triangle-specific theorems to quickly distinguish similar triangles. These three theorems, known as Angle - Angle (AA), Side - Angle - Side (SAS), and Side - Side - Side (SSS), are proof methods for determining similarity in triangles.
In order to determine the value of p from the given expression, we will use the expression below;
2p-5+3p/14+26 = 3p/26
5p+5/40 = 3p/26
Cross multiply
40 * 3p = 26(5p +5)
120p = 130p + 130
120-130p = 130
-10p = 130
Divide both sides by -10 to have:
-10p/-10 = 130/-10
p = -13
This gives the required value of p from the figure.
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Anna's room is a rectangle. Its length is 15 feet and its width is 4 yards. What is the perimeter of the room?
Answers
Answer:
38
Step-by-step explanation:
Perimeter is basically each side added together. 15 + 15 + 4 + 4 is 38. Therefore, it's 38.
The graph of f(a) = > has been transformed to create the graph of g(s) =
Answers
EXPLANATION
The graph of the parent function: f(x) = 1/x has the following form:
Translating the function two units to the left, give us the Image function:
This function is obtained by adding two units to the denominator.
In conclusion, the solution is -2
At a point on the ground 35 ft from base of a tree, the distance to the top of the tree is 1 ft more than 3 times the height of the tree. Find the height of the tree. The height of the tree is ___. (ft^3, ft^2, or ft)(Simply your answer. Round to the nearest foot as needed)
Answers
At a point on the ground 35 ft from the base of a tree, the distance to the top of the tree is 1 ft more than 3 times the height of the tree. Find the height of the tree
see the attached figure to better understand the problem
Applying the Pythagorean Theorem
(3h+1)^2=h^2+35^2
9h^2+6h+1=h^2+1,225
solve for h
9h^2-h^2+6h+1-1,225=0
8h^2+6h-1,224=0
Solve the quadratic equation
Using a graphing tool
the solution is
h=12 ftA cake is cut into 12 equal slices. After 3 days Jake has eaten 5 slices. What is his weekly rate of eating the cake?
5
36
35
36
cakes/week
cakes/week
1 cakes/week
35
01 cakes/week
Answers
The cake is divided into 12 equal slices. Jake had eaten 5 slices after 3 days. The weekly cake consumption rate is 11.6
What is algebraic expression?An algebraic expression is one that is composed of integer constants, variables, and algebraic operations. 3x2 2xy + c, for example, is an algebraic expression. Algebraic expressions have at least one variable and one operation (addition, subtraction, multiplication, division). 2(x + 8y) is an algebraic expression, for example. An algebraic expression is one that contains constants, variables, and algebraic operations. 3x2 2xy + d, for example, is an algebraic expression. Thus, an algebraic expression is composed of three types of fundamental elements: Coefficient (i.e. numbers) (i.e. numbers)Therefore,
The weekly cake consumption rate is 11.6
we have 7 days.
7days-3days =4
in 3 days he has eaten 5 slices
again 4-3 days=1
so in 6 days, he has eaten 10 slices
we have 1 day left.so if he eats 5 slices in 3 days, how many does he eat slices in 1 day?
5/3=1.6
10+1.6= 11.6
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please help me ASAP!!!
Answers
substitute x = 5 in the above function
[tex]f(5)=\sqrt[]{2(5)^2-3(5)+1}[/tex][tex]=\sqrt[]{2(25)-15+1}[/tex][tex]=\sqrt[]{50-15+1}[/tex][tex]=\sqrt[]{36}=\text{ 6}[/tex]f(5) = 6
All changes saved16. Suppose you invest $7500 at an annual Interest rate of 4.2% compounded continuously. How much will you have in the account after 2 years? Round the solution to the nearest dollar.$8158$17,372$17,373$8157
Answers
We have to use the continuous compound interest formula
[tex]A=P\cdot e^{rt}[/tex]Where P = 7500, r = 0.042, t = 2. Let's replace and solve
[tex]\begin{gathered} A=7500\cdot e^{0.042\cdot2} \\ A\approx8,157 \end{gathered}[/tex]Hence, the answer is $8,157.9. A researcher gathered data on hours of video games played by school-aged children and young adults. She collected the following data:601241215171711409914110131015163915121698131016651717129(a) Complete the frequency distribution for the data.HoursFrequencyRelative Frequency0-23-56-89-1112-1415-17(b) Which of the following is the correct histogram for this data?246810Hours0369121518Frequency[Graphs generated by this script: setBorder(54,40,20,15); initPicture(0,18,0,10);axes(34,2,1,null,2); fill="blue"; textabs([165,0],"Hours","above");line([0,-0.2],[0,0.2]); text([0,0],"0","below");line([3,-0.2],[3,0.2]); text([3,0],"3","below");line([6,-0.2],[6,0.2]); text([6,0],"6","below");line([9,-0.2],[9,0.2]); text([9,0],"9","below");line([12,-0.2],[12,0.2]); text([12,0],"12","below");line([15,-0.2],[15,0.2]); text([15,0],"15","below");line([18,-0.2],[18,0.2]); text([18,0],"18","below");textabs([0,115],"Frequency","right",90);rect([0,0],[3,6]);rect([3,0],[6,4]);rect([6,0],[9,5]);rect([9,0],[12,7]);rect([12,0],[15,6]);rect([15,0],[18,10]);]246810121416Hours061218Frequency[Graphs generated by this script: setBorder(54,40,20,15); initPicture(0,18,0,16);axes(34,2,1,null,2); fill="blue"; textabs([165,0],"Hours","above");line([0,-0.32],[0,0.32]); text([0,0],"0","below");line([6,-0.32],[6,0.32]); text([6,0],"6","below");line([12,-0.32],[12,0.32]); text([12,0],"12","below");line([18,-0.32],[18,0.32]); text([18,0],"18","below");textabs([0,115],"Frequency","right",90);rect([0,0],[6,10]);rect([6,0],[12,12]);rect([12,0],[18,16]);]2468101214Hours061218Frequency[Graphs generated by this script: setBorder(54,40,20,15); initPicture(0,18,0,14);axes(34,2,1,null,2); fill="blue"; textabs([165,0],"Hours","above");line([0,-0.28],[0,0.28]); text([0,0],"0","below");line([6,-0.28],[6,0.28]); text([6,0],"6","below");line([12,-0.28],[12,0.28]); text([12,0],"12","below");line([18,-0.28],[18,0.28]); text([18,0],"18","below");textabs([0,115],"Frequency","right",90);rect([0,0],[6,12]);rect([6,0],[12,14]);rect([12,0],[18,12]);]2468Hours0369121518
Answers
Remember that the frequency refers to the number of times a data shows up. In this case, the frequency is the number of data that falls into each interval.la
To find the relative frequency is calculated by dividing each frequency by 38 (the total number of data).
[tex]\begin{gathered} \frac{6}{38}=0.1579 \\ \frac{4}{38}=0.1053 \\ \frac{4}{38}=0.1053 \\ \frac{8}{38}=0.2105 \\ \frac{7}{38}=0.1842 \\ \frac{9}{38}=0.2368 \end{gathered}[/tex]Let's include the relative frequencies in the table.
On the other hand, the correct histogram has to show the frequencies in the same order. The following histogram shows the correct frequency distribution.
Identify the range of the function shown in the graph. 10 8 4 -10-8-4-2 8 10 O A. -2< y < 2 O B. {-2, 2) O C. y is all real numbers OD. Y > 0
Answers
Answer
Option B is correct.
Range: y is all real numbers.
Explanation
The range of a function refers to the region of values where the function can exist. It refers to the values that the dependent variable [y or f(x)] can take on. It is the region around the y-axis that the graph of the function spans.
For this question, we can see that the graph spans over the entire y-axis.
Hence, the range of this function shown in the graph is all real number.
Hope this Helps!!!
Using this formula and other formulas, find Q1,Q2, Q3 the midquartile, and the interquartile range for the data set.51, 62, 73, 92, 97, 100, 104
Answers
Given:
The given set of data is 51, 62, 73, 92, 97, 100, 104.
The objective is to find Q1,Q2, Q3 the midquartile, and the interquartile range.
Explanation:
The given set of data is already arranged in increasing oder.
To find Q2:
The quartile Q2 represents the middle term of the set of data arranged in increasing order.
The number of terms in the set of data is N = 7.
Then, the middle term of the set of data is 92, which is Q2.
To find Q1:
The quartile 1 represents the middle term of the left side of the Q2.
The left side of Q2 contains 51, 62, 73.
Thus, the middle term of the left side of Q2 is 62, which is Q1.
To find Q3:
The quartile 3 represents the middle temr of the right side of the Q2.
The right side of Q2 contains 97, 100, 104.
Thus, the middle term of the right side of Q2 is 100, which is Q3.
To find midquartile:
The midquartile is termed as the average of highest and lowest value of the set of data.
The highest value in the given set of data is 104 and the lowest value in the given set of data is 51.
Then, the midquartile can be calculated as,
[tex]\begin{gathered} \text{Midquartile}=\frac{104+51}{2} \\ =77.5 \end{gathered}[/tex]To find interquartile:
The
Timothy ran a lemonade stand for 6 days. on the first day he made $5. Each day after that he made $2 more than the previous day. How much money did Marcus make, , after the 6 days?A) $60B) $15C) $12D) $30
Answers
Step `1;
Total number of days = 6
Step 2:
First day = $5
Second day = $5 + $2 = $7
Third day = $7 + $2 = $9
Fourth day + $9 + $2 = $11
Fifth day = $11 + $2 = $13
Sixth day = $13 + $2 = $15
Step 3:
Marcus made = $5 + $7 + $9 + $11 + $13 + $15
= $60
Second method
Use the sum of nth terms of arithmetic progression.
first term a = $5
Common difference = 2
n = 6
[tex]\begin{gathered} S\text{um of the 6 terms = }\frac{n}{2}(\text{ 2a + (n-1)d)} \\ =\text{ }\frac{6}{2}\text{ ( 2}\times5\text{ + (6 -1) }\times\text{ 2)} \\ =\text{ 3( 10 + 5}\times2\text{ )} \\ =\text{ 3( 10 + 10 )} \\ =\text{ 3 }\times\text{ 20} \\ =\text{ \$60} \end{gathered}[/tex]Final answer
Marcus made = $60 Option A
Help me due is tomorrow
Answers
Step-by-step explanation:
5.3g+9=2.3g+15
5.3g-2.3g=15-9
3g=6
3g/3=6/3
g=2
B,5.3(2)+9=2.3(2)+15
10.6+9=4.6+15
19.6=19.6
g = 2
Step-by-step explanation:5.3g + 9 = 2.3g + 15
Subtract 9 from both sides.
5.3g + 9 - 9 = 2.3g + 15 - 9
5.3g = 2.3g + 6
Subtract 2.3g from both sides
5.3g - 2.3g = 2.3g - 2.3g + 6
3g = 6
Divide both sides by 3
g = 2
To check if the value of g is correct, substitute the value of g in the equation above and remember that the both sides should be equal because of the equal sign (=) in the equation.
5.3g + 9 = 2.3g + 15
5.3(2) + 9 = 2.3(2) + 15
10.6 + 9 = 4.6 + 15
19.6 = 19.6
Find the 1st term, last term and the sum for the finite arithmetic series.
Answers
Answer:
Given that,
[tex]\sum ^{30}_{n\mathop=2}(3n-1)[/tex]Simplifying we get,
[tex]\sum ^{30}_{n\mathop{=}2}(3n-1)=\sum ^{30}_{n\mathop{=}2}3n+\sum ^{30}_{n\mathop{=}2}1[/tex][tex]=3\sum ^{30}_{n\mathop{=}2}n+\sum ^{30}_{n\mathop{=}2}1[/tex]we have that,
[tex]\sum ^n_{n\mathop=1}1=n[/tex]If n is from 2 to n we get,
[tex]\sum ^n_{n\mathop{=}2}1=n-1[/tex]Also,
[tex]\sum ^k_{n\mathop=1}n=\frac{k(k+1)}{2}[/tex]If n is from 2 to n we get,
[tex]\sum ^k_{n\mathop=2}n=\frac{k(k+1)}{2}-1[/tex]Using this and substituting in the required expression we get,
[tex]=3\lbrack\frac{30\times31}{2}-1\rbrack+30-1[/tex][tex]=3(464)+29[/tex][tex]=1421[/tex]Answer is: 1421
If I am in San Juan, then
I am in Puerto Rico.
State whether the following
statement is inverse, converse,
contrapositive.
If I am not in San Juan,
then I am not in Puerto
Rico.
Answers
The statement "If I am not in San Juan, then I am not in Puerto Rico." is the inverse and contrapositive statement because it is inverse of "If I am in San Juan, then I am in Puerto Rico."
What is inverse?The inverse function of a function f in mathematics is a function that reverses the operation of f. If and only if f is bijective, then the inverse of f is true. A function that "undoes" another function is called an inverse. In other words, if f(x) produces y, then y entered into the inverse of f produces x. An invertible function is one that has an inverse, and the inverse is represented by the symbol f⁻¹.
What is contrapositive?When you reverse the hypothesis and the conclusion in a statement and reject both of them, you have a contrapositive statement. When the hypothesis and the conclusion are switched in this example and both are negated, the outcome is: If it is not a triangle, then it is not a polygon.
Here,
The statement is "If I am in San Juan, then I am in Puerto Rico."
So the contrapositive and inverse will be:
"If I am not in San Juan, then I am not in Puerto Rico."
Because it is the opposite of "If I am in San Juan, then I am in Puerto Rico," the statement "If I am not in San Juan, then I am not in Puerto Rico" is the inverse and contrapositive statement.
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In gym class, a student can do 40 sit-ups in 60 seconds and 100 sit-ups in 150 seconds.
Graph the proportional relationship.
graph with x axis labeled time seconds and y axis labeled sit ups, with a line from 0 comma 0 going through 90 comma 60
graph with x axis labeled time seconds and y axis labeled sit ups, with a line from 0 comma 0 going through 60 comma 30
graph with x axis labeled time seconds and y axis labeled sit ups, with a line from 0 comma 0 going through 60 comma 50
graph with x axis labeled time seconds and y axis labeled sit ups, with a line from 0 comma 0 going through 90 comma 30
Question 6 (Essay Worth 4 points)
Answers
By using the linear equation y=3x we draw the graph for proportional relationship.
What is Graph?Graph is a mathematical representation of a network and it describes the relationship between lines and points.
A proportional relationship graph between two variables is a relationship where the ratio between the two variables is always the same
the graph of any proportional relationship is characterized by a straight line with data points passing through the origin (0, 0).
By the definition of proportional relationship of graph, we can reduce relationship between the values on the x-coordinate and y-coordinate of the given graph
As they are proportional and represented by euation
y = 3x
Where, x represent the time and y represent the number of sit-ups.
Hence by using the linear equation y=3x we draw the graph for proportional relationship.
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Fine all the missing side lengths and angle measured of each triangle.
Answers
Answer:
[tex]\begin{gathered} AT=8\sqrt[]{3} \\ AC=8 \\ mStep-by-step explanation:To find the missing lengths of the triangle, use trigonometric ratios for right triangles, which are represented by the following equations:
[tex]\begin{gathered} \sin (\text{angle)=}\frac{\text{ opposite}}{\text{ hypotenuse}}_{} \\ \cos (\text{angle)}=\frac{\text{adjacent}}{\text{hypotenuse}} \\ \tan (\text{angle)}=\frac{\text{ opposite}}{\text{ adjacent}} \end{gathered}[/tex]Then, find the opposite and adjacent side given the 60 degrees angle:
[tex]\begin{gathered} \sin (60)=\frac{AT}{16} \\ AT=16\cdot\sin (60) \\ AT=8\sqrt[]{3} \\ \\ \cos (60)=\frac{AC}{16} \\ AC=16\cdot\cos (60) \\ AC=8 \end{gathered}[/tex]Now, since the intern angles of a triangle must add up to 180 degrees, given two of the angles find the missing angle:
[tex]\begin{gathered} mFind the interval in the line below. Use correct symbols to indicate in interval notation. If number is no an integer then round to the nearest hundredth.
Answers
we can see the interval is between -2 and 1. but the -2 isn't included (you can notice by the white circle) and the 1 is included, so in interval notation you get:
(-2,1]
find the intercepts and graph the equation by plotting points. 13^2 + 4y = 52
Answers
ANSWER
[tex]y-intercept:(0,-\frac{117}{4})[/tex]Graph:
EXPLANATION
Given:
[tex]13^2+4y=52[/tex]Desired Results:
Intercepts and graph the equation
Solve for y
[tex][/tex]Which choice is equivalent to the quotient shown here for acceptablevalues of x?25(x - 1) = 5(x - 1)?A.5(x - 1)B. 125(x - 1)C. V25(x - 1) -5(x - 1)?D. V5(x - 1)SUBMIT
Answers
Given the expression:
[tex]\sqrt[]{28(x-1)}\div\sqrt[]{8x^2}[/tex][tex]\frac{\sqrt[]{28(x-1)}}{\sqrt[]{8x^2}}[/tex]Let's determine the inequality that represents all the values of x.
Here, we are to find the domain.
Let's solve for x.
Set the radicand in the numerator and denominator to be greater or equal to zero.
We have:
[tex]\frac{28(x-1)\ge0}{8x^2\ge0}[/tex]For the numerator, we have:
[tex]\begin{gathered} 28(x-1)\ge0 \\ \text{Divide both sides by 28:} \\ \frac{28(x-1)}{28}\ge\frac{0}{28} \\ \\ x-1\ge0 \\ \text{Add 1 to both sides:} \\ x-1+1\ge0+1 \\ x\ge1 \end{gathered}[/tex]For the denominator, we have:
[tex]\begin{gathered} 8x\ge0 \\ x\ge\frac{0}{8} \\ x\ge0 \end{gathered}[/tex]Therefore, the possible x-values for which the quotient is defined is all positive integers greater or equal to 1.
Thus, we have:
[tex]x\ge1[/tex]ANSWER:
[tex]C.x\ge1[/tex]how do you work the problem 3k+16=5k?
Answers
We have the following:
[tex]3k+16=5k[/tex]solving for k
[tex]\begin{gathered} 5k-3k=16 \\ k=\frac{16}{2} \\ k=8 \end{gathered}[/tex]The value of k is 8
I need help with this practice Having trouble solving it The subject is trigonometry
Answers
To solve the problem, we will make use of the identity:
[tex]\cos (\alpha-\beta)=\cos (\alpha)\cos (\beta)+\sin (\alpha)\sin (\beta)_{}[/tex]ANGLE α
The angle lies in the second quadrant. The only positive ratio is the sine.
If we have that:
[tex]\tan \alpha=-\frac{12}{5}[/tex]Displaying this on a triangle for ease of working, we have:
Therefore, the length of the hypotenuse will be:
[tex]\begin{gathered} x=\sqrt[]{12^2+5^2}=\sqrt[]{144+25}=\sqrt[]{169} \\ x=13 \end{gathered}[/tex]Therefore, we have that:
[tex]\begin{gathered} \sin \alpha=\frac{12}{13} \\ \cos \alpha=-\frac{5}{13} \end{gathered}[/tex]ANGLE β
This angle lies in the fourth quadrant. Only the cosine ratio is positive in this quadrant.
We are given in the question:
[tex]\cos \beta=\frac{3}{5}[/tex]Displaying this on a triangle for ease of working, we have:
Therefore, using the Pythagorean Triplets, we have that:
[tex]y=4[/tex]Therefore, we have that:
[tex]\sin \beta=-\frac{4}{5}[/tex]SOLVING THE IDENTITY
Applying the identity quoted earlier, we have:
[tex]\begin{gathered} \cos (\alpha-\beta)=\cos (\alpha)\cos (\beta)+\sin (\alpha)\sin (\beta)_{} \\ \cos (\alpha-\beta)=(-\frac{5}{13})(\frac{3}{5})+(\frac{12}{13})(-\frac{4}{5}) \\ \cos (\alpha-\beta)=-\frac{63}{65} \end{gathered}[/tex]If sin A = 3/5 and cos B = 20/29 and angles A and B are in Quadrant 1, find the valueof tan(A + B).
Answers
Our approach is to use SOHCAHTOA to derive values for sine and cosines of both A and B.
[tex]\begin{gathered} \sin A=\frac{3}{5},\text{ cosA=}\frac{\sqrt[]{5^2-3^2}}{5}=\frac{4}{5} \\ \cos B=\frac{20}{29},\text{ sinB=}\frac{\sqrt[]{29^2-20^2}}{29}=\frac{21}{29} \end{gathered}[/tex][tex]\begin{gathered} \tan (A+B)=\frac{\tan A+\tan B}{1-\text{tanAtanB}}\text{ WHERE} \\ \tan A=\frac{\sin A}{\cos A},\tan B=\frac{\sin B}{\cos B} \end{gathered}[/tex][tex]\begin{gathered} \tan (A+B)=\frac{\frac{\frac{3}{5}}{\frac{4}{5}}+\frac{\frac{21}{29}}{\frac{20}{29}}}{1-\frac{\frac{3}{5}}{\frac{4}{5}}\times\frac{\frac{21}{29}}{\frac{20}{29}}}=\frac{\frac{3}{4}+\frac{21}{20}}{1-\frac{3}{4}\times\frac{21}{20}}=\frac{\frac{9}{5}}{1-\frac{63}{80}}=\frac{\frac{9}{5}}{\frac{17}{80}} \\ \tan (A+B)=8.47 \end{gathered}[/tex]tan (A+B) = 8.47
l show how the distributive property can make the arithmetic simpler in the following problems5(108)
Answers
Firstly Example of Distributive property can be shown below.
GIiven: 6(9 - 4)
6 x 9 - 6 x 4
54 - 24 = 30
a) 3(50.15)
3(50 + 0.15)
3x50 + 3 x0.15
150 + 0.45 = 150.45
(b) 5(108)
5(100 + 8)
5x100 + 5x8
500 + 40 = 540
Lesson 6.07: In a random sample of 74 homeowners in a city, 22 homeowners said they wouldsupport a ban on nonnatural lawn fertilizers to protect fish in the local waterways. The samplingmethod had a margin of error of +3.1%. SHOW ALL WORK!A) Find the point estimate.B) Find the lower and upper limits and state the interval.
Answers
Confidence interval is written in the form,
(point estimate +/- margin of error)
The given scenario involves population proportion
The formula for the point estimate is
p' = x/n
where
p' = estimated proportion of success. p' is a point estimate for p which is the true proportion
x represents the number of success
n represents the number of samples
From the information given,
n = 74
x = 22
p' = 22/74 = 0.297
The formula for finding margin of error is expressed as
[tex]\begin{gathered} \text{margin of error = z}_{\frac{\alpha}{2}}(\sqrt[]{\frac{p^{\prime}q^{\prime}}{n}} \\ q^{\prime}\text{ = 1 - p'} \\ q^{\prime}\text{ = 1 - 0.297 = 0.703} \end{gathered}[/tex]A) The point estimate is 0.297
B) margin of error = +/-3.1% = 3.1/100 = +/- 0.031
Thus,
the lower limit would be 0.297 - 0.031 = 0.266
Expressing in percentage, it is 0.266 x 100 = 26.6%
the upper limit would be 0.297 + 0.031 = 0.328
Expressing in percentage, it is 0.328 x 100 = 32.8%
Thus, the confidence interval is between 26.6% and 32.8%