Answers
Using cosine addition formula, the value of cos(a + b) is -16/25
What is the exact values of cos(a + b)Using cosine addition formula, the exact value of cos (a + b) is calculated as;
cos(a + b) = cos a × cos b - sin a × sin b
Our given data;
cos a = 4/5, 0 < a < π/2
cos b = 5/13, -π/2 < b < 0
Using Pythagorean identity;
sin a and sin b;
sin² x + cos² x = 1
To solve for a;
sin a = √(1 - cos² a) = √(1 - (4/5)²) = √(1 - 16/25) = √(9/25) = 3/5
Solving for b;
sin b = √(1 - cos² b) = √(1 - (5/13)²) = √(1 - 25/169) = √(144/169) = 12/13
Substituting the values into cosine addition formula
cos(a + b) = (4/5) * (5/13) - (3/5) * (12/13)
cos(a + b) = (20/65) - (36/65)
cos(a + b) = -16/65
Therefore, the exact value of cos(a + b) is -16/65.
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Related Questions
Suppose you own 125 shares of a stock that pays a dividend of $0.29 per
share per year. How much will you receive in dividends over 6 years,
assuming the dividends stay the same and you buy no more stock?
OA. $161.25
OB. $217.50
C. $174.00
D. $36.25
Answers
Answer:
To calculate the total dividends received over 6 years, we can use the formula:
Total dividends = Annual dividend per share x Number of shares x Number of years
In this case, the annual dividend per share is $0.29, the number of shares is 125, and the number of years is 6.
Total dividends = $0.29 x 125 x 6 = $217.50
Therefore, the answer is (B) $217.50.
Step-by-step explanation:
Which of the following ordered pairs represents the point plotted and labeled B?
(3, −1)
(−1, 2)
(3, 1)
(1, 2)
Answers
(3,1)
Ordered pairs are always ordered as (x,y). Point B has an x-value of 3 and a y-value of 1.
Samuel was given the diagram below and asked to prove that AC ED. What
would be the reasoning for the third step of the proof?
Given: Quadrilateral ABCD is a rectangle.
Prove: AC & BD
Statements
1. Quadrilateral ABCD is a rectangle
2. AB & CD
3. ZBAD ZCDA
4. AD AD
5. ABAD ACDA
6. AC BD
B
A
Reasons
Given
Opposite sides of a rectangle are equal
Reflexive Property of Equality
SAS Triangle Congruence Theorem
Corresponding parts of congruent
triangles are congruent
C
O
Answers
The reasoning for the third step of the proof is the SAS Triangle Congruence Theorem, which establishes the congruence of the two triangles involved.
In the given proof, we are trying to prove that AC is equal to ED in a rectangle ABCD. The reasoning for the third step of the proof can be explained as follows:
Statement 3: ZBAD = ZCDA
Reason: SAS Triangle Congruence Theorem
Explanation: The SAS (Side-Angle-Side) Triangle Congruence Theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
In this case, we are considering triangles ZBA and ZDA. We know that AB is congruent to AD (statement 2, "AB & CD") and that the angles ZBA and ZDA are right angles since ABCD is a rectangle (statement 1, "Quadrilateral ABCD is a rectangle").
Therefore, by the SAS Triangle Congruence Theorem, we can conclude that triangle ZBA is congruent to triangle ZDA. And since corresponding parts of congruent triangles are congruent (CPCTC), we can say that angle ZBAD is congruent to angle ZCDA.
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select the fraction that has a numerator of 7
Answers
B
7/10 has a 7 in the numerator
Answer:
B, 7/10
Step-by-step explanation:
The numerator is the number being divided in the fraction (the number on the top)
The table shows the poplulation of a town from 1996 to 2004. Assume that t is the number of
years since 1996 and P is measured in thousands of people.
Year Population
0
22.8
1
25
2
26.5
3
27.1
4
27.8
28.1
27.9
26.9
26.1
5) What model would best fit this data?
5
6
7
8
7) Use your equation from #6 to determine
what year the population will be less than
24,000 (round to the nearest whole
number if necessary).
6) Use a calculator to find the regression
equation (round to the nearest hundredths
if needed).
8) Use your equation from #6 to find the
population in 2008 (round to the nearest
whole number if necessary).
Answers
To determine the best model for fitting the given data, we can analyze the trend and pattern in the population values over time. One common approach is to perform linear regression to find the equation of the line that best represents the data.
To find the regression equation, we'll use the given data points (t, P) as follows:
t: 0, 1, 2, 3, 4
P: 22.8, 25, 26.5, 27.1, 27.8
Using a calculator or statistical software, we can perform linear regression to find the equation of the line that best fits the data. The equation will have the form:
P = mt + b
where m represents the slope of the line and b represents the y-intercept.
Performing the regression calculation, we find that the equation of the line that best fits the data is approximately:
P = 1.155t + 22.625
To determine the year when the population will be less than 24,000, we can substitute P = 24 into the equation and solve for t:
24 = 1.155t + 22.625
1.155t = 24 - 22.625
1.155t = 1.375
t ≈ 1.19
Rounding to the nearest whole number, we find that the year when the population will be less than 24,000 is 1 (1997).
To find the population in 2008, we substitute t = 12 into the equation:
P = 1.155(12) + 22.625
P ≈ 36.18
Rounding to the nearest whole number, the estimated population in 2008 is 36,000.
In summary:
6) The linear regression equation that best fits the data is P = 1.155t + 22.625.
The population will be less than 24,000 in the year 1997.
The estimated population in 2008 is 36,000.
<33
Find the range and mean of 1,3,5,7,9
Answers
Answer:
Step-by-step explanation:
Range: what the numbers range from the smallest to largest. Subtract the largest number and smallest.
Range = 9 - 1
Range = 8
Median: Middle number. If you listed numbers in order from smallest to largest, it is the middle number
Median = 5
Use f(x) = 1
2
x and f -1(x) = 2x to solve the problems.
f(2) =
1
f−1(1) =
⇒ 2
f−1(f(2)) =
⇒ 2
f−1(−2) =
1
⇒ -4
f(−4) =
2
⇒ -2
f(f−1(−2)) =
2
⇒ -2
In general, f−1(f(x)) = f(f−1(x)) =
Answers
Using the function f(x) = 1/2 x and the inverse of the function f⁻¹(x) = 2x, the solutions of the given are :
f(2) = 1, f⁻¹(1) = 2, f⁻¹(f(2)) = 2
f⁻¹(-2) = -4, f(-4) = -2, f(f⁻¹(-2)) = -2
Given a function,
f(x) = 1/2 x
And an inverse function,
f⁻¹(x) = 2x
We have to find the values of the following.
f(2) = 1/2 × 2 = 1
f⁻¹(1) = 2 × 1 = 2
f⁻¹(f(2)) = f⁻¹(1) = 2 × 1 = 2
f⁻¹(-2) = 2 × -2 = -4
f(-4) = 1/2 × 4 = -2
f(f⁻¹(-2)) = f (-4) = 1/2 × 4 = -2
So, in general, f⁻¹(f(x)) = f(f⁻¹(x))
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A bag contains 3 gold marbles, 8 silver marbles, and 30 black marbles. Someone offers to play this game: You randomly select one marble from the bag. If it is gold, you win $3. If it is silver, you win $2. If it is black, you lose $1.
What is your expected value if you play this game?
Answers
Answer: To calculate the expected value, we need to multiply each possible outcome by its probability and then add up the results.
The probability of selecting a gold marble is 3/41, the probability of selecting a silver marble is 8/41, and the probability of selecting a black marble is 30/41.
The winnings/losses associated with each outcome are $3 for gold, $2 for silver, and -$1 for black.
Step-by-step explanation:
Therefore, the expected value can be calculated as follows:
(3/41) x $3 + (8/41) x $2 + (30/41) x (-$1)
= $0.073
The expected value is $0.073, which means that on average, you can expect to win $0.073 per game if you play this game many times.
Therefore, if you play this game, you can expect to win a small amount of money on average, but it is not a guaranteed win.
References:
Grinstead, C.M. and Snell, J.L. (2006). Introduction to Probability. American Mathematical Society.
"Expected Value," Investopedia, accessed May 14, 2023, https://www.investopedia.com/terms/e/expectedvalue.asp.
2.3y+1.2x=0
1.1x+0.1y=1.2
Linear or nonlinear
Answers
Write the coordinates of the vertices after a dilation with a scale factor of 1/3 considered at the origin
Answers
The coordinates of the vertices after a dilation with a scale factor of 1/3 considered at the origin include:
Q' (-3, -3).
R' (1, -3).
S' (3, 3).
T' (-1, 3).
What is a dilation?In Geometry, a dilation is a type of transformation which typically changes the side lengths of a geometric object, but not its shape.
In this scenario and exercise, we would dilate the coordinates of the vertices by applying a scale factor of 1/3 that is centered at the origin as follows:
Coordinate Q (-9, -9) → Q' (-9 × 1/3, -9 × 1/3) = Coordinate Q' (-3, -3).
Coordinate R (3, -9) → R' (3 × 1/3, -9 × 1/3) = Coordinate R' (1, -3).
Coordinate S (9, 9) → S' (9 × 1/3, 9 × 1/3) = Coordinate S' (3, 3).
Coordinate T (-3, 9) → T' (-3 × 1/3, 9 × 1/3) = Coordinate T' (-1, 3).
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expand and simplify (x+2)(x-2)-(x+3)²
Answers
The Simplified form of the expression (x+2)(x-2) - (x+3)² is -13 - 6x.
The expression (x+2)(x-2) - (x+3)², we'll follow the distributive property and simplify each term.
First, let's expand (x+2)(x-2):
(x+2)(x-2) = x(x) + x(-2) + 2(x) + 2(-2)
= x² - 2x + 2x - 4
= x² - 4
Next, let's expand (x+3)²:
(x+3)² = (x+3)(x+3)
= x(x) + x(3) + 3(x) + 3(3)
= x² + 3x + 3x + 9
= x² + 6x + 9
Now, we can substitute these expansions into the original expression:
(x+2)(x-2) - (x+3)² = (x² - 4) - (x² + 6x + 9)
To simplify further, let's distribute the negative sign to every term inside the parentheses:
(x² - 4) - (x² + 6x + 9) = x² - 4 - x² - 6x - 9
Now, we can combine like terms:
(x² - x²) + (-4 - 9) - 6x = -13 - 6x
Therefore, the simplified form of the expression (x+2)(x-2) - (x+3)² is -13 - 6x.
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The table and graph show the number of rabbits in a national park over an 18-year
Answers
I apologize, but it seems that the table and graph you mentioned did not come through in your message. Could you please provide the table and graph or describe the information in more detail? That way, I can assist you with your question.
if two distinct numbers are chosen randomly from the set {-2, -6/5, -1/3, 0, 1/2, 5/6, 3}, find the probability that they will be the slopes of two perpendicular lines
Answers
To find the probability that two distinct numbers chosen randomly from the set {-2, -6/5, -1/3, 0, 1/2, 5/6, 3} will be the slopes of two perpendicular lines, we need to determine the total number of possible pairs of distinct numbers and the number of pairs that result in perpendicular slopes.
Total number of possible pairs:
Since we are choosing two distinct numbers from a set of seven, the total number of possible pairs is given by the combination formula:
C(7, 2) = 7! / (2!(7 - 2)!) = 7! / (2! * 5!) = 21
Pairs resulting in perpendicular slopes:
For two lines to be perpendicular, the product of their slopes must be -1. So, we need to count the number of pairs whose product is -1.
The set contains two negative numbers {-2, -6/5} and one positive number {3}. None of these numbers have a reciprocal within the set that gives a product of -1. Therefore, no pairs can be formed from these numbers that result in perpendicular slopes.
Hence, the number of pairs resulting in perpendicular slopes is 0.
Therefore, the probability of randomly choosing two distinct numbers from the set {-2, -6/5, -1/3, 0, 1/2, 5/6, 3} that will be the slopes of two perpendicular lines is 0/21, which simplifies to 0.
Cindy bought a 48 ounce can of papaya juice for $12. What is the unit price per ounce?
Answers
Answer:
The unit price is $.25 per ounce.
Step-by-step explanation:
To find the unit price per ounce, take the total cost and divide by the number of ounces
12 dollars/ 48 ounces
.25
The unit price is $.25 per ounce.
Which side lengths form a right triangle?
Choose all answers that apply:
A 5,√8,33
B 8, 15, 17
√2, √2,2
Answers
Answer:
B and C
Step-by-step explanation:
Answer: B and C
Step-by-step explanation:
Pythagorean Theorem:
c²=a²+b² >where a and be are the shorter legs and c is the
longer/hypotenuse
Let's plug each into the pythagorean theorem
A) Not a right triangle
33² = 5² + (√8)²
1089 = 25 +8
1089 = 33 >Not equal, The statement is not true, so this is not a
right triangle.
B) Yes, it is a right triangle
17²=8²+15²
289=64+225
289=289 They are equal, true statement, so yes it is a right triangle
C) No, it is not a right triangle
2² = (√2)²+(√2)²
4 = 2+2
4=4 They are equal, true statement, so yes it is a right triangle
a bag has equal numbers of magenta turquoise and black marbles. corey picked a marble from the bag, replaced it, then picked another marble. both times he did not select a magenta marble. if he repeats this process 250 times predict how many repetitions will result in no magenta marbles?
Answers
The repetitions that result in no magenta marbles will be 62.5.
The expected value is the predicted average outcome of a random variable, calculated by multiplying each outcome by its probability.
The expected value is given below.
E(x) = np
Where n is the number of samples and p is the probability.
The number of magenta turquoise and black marbles is equal. Thus, the probability will be equal. Then the probability of not selecting a magenta marble is calculated as,
P = (1/2) x (1/2)
P = 1/4
The expected value is calculated as,
E = 1/4 x 250
E = 62.5
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19
For an ideal gas, it is assumed that no attractive force exists between particles.
This is not true for reall gas. What effect does this attraction have on the
properties of a real gas when compared to those of an ideal gas under
equivalent conditions?
A
The pressure of the real gas is higher.
B
The temperature of the real gas is higher.
C
The pressure of the real gas is lower.
D
The temperature of the real gas is lower.
Answers
Réduire l'expression suivante :
4
�
+
8
+
(
6
�
−
7
)
−
(
8
�
+
9
)
4x+8+(6x−7)−(8x+9)
Answers
To simplify the given expression: 4x + 8 + (6x - 7) - (8x + 9), we can follow the order of operations (parentheses, exponents, multiplication and division, and addition and subtraction) to simplify it step by step.
First, let's simplify the expression inside the parentheses:
6x - 7 becomes 6x - 7
8x + 9 becomes 8x + 9
Now, let's simplify the entire expression:
4x + 8 + 6x - 7 - 8x - 9
Combining like terms:
(4x + 6x - 8x) + (8 - 7 - 9)
2x - 8
Therefore, the simplified form of the expression 4x + 8 + (6x - 7) - (8x + 9) is 2x - 8.
conner has 3 times as many trading cards as max. If connor has 90 trading cards, how many trading cards does max have?
Answers
Answer:
Max has 30 cards
Step-by-step explanation:
C:M
3x:x
Conner has 90 trading cards
3x=90
divide by 3
x=30
this means that Max has 30 cards
what is the slope of the line represented by the equation below?
y= 1/3x +4
Answers
Step-by-step explanation:
y = 1/3 x +4
is the equation of a line in the form
y = mx+b where m =slope and b = the y-axis intercept
so m = slope = 1/3 for this line
Answer:
[tex]\huge\boxed{\sf Slope = 1/3}[/tex]
Step-by-step explanation:
Given equation:[tex]\displaystyle y=\frac{1}{3} x + 4[/tex]
General form of slope-intercept equation:y = mx + c
Where m is slope and c is y-intercept.
So,
By comparing both the equations, we get:
m = 1/3
Slope = 1/3[tex]\rule[225]{225}{2}[/tex]
Whitney is planning to bake two types of brownies. One type needs cup of
sugar. The other needs cup of sugar. How much sugar in total does Whitney
need to bake all the brownies?
Answer:
Explain:
Answers
Whitney needs 5/4 cup of sugar in total to bake all the brownies.
To calculate the total amount of sugar needed to bake all the brownies, we need to add the quantities of sugar required for each type of brownie.
According to the information given, one type of brownie requires 1/2 cup of sugar and the other type requires 3/4 cup of sugar.
To find the total amount of sugar needed, we add these two quantities:
1/2 cup + 3/4 cup
To add fractions, we need to have a common denominator.
The common denominator is 4:
(1/2) x (2/2) cup + (3/4) x (4/4) cup
This simplifies to:
2/4 cup + 12/16 cup
Now, we can add the fractions:
(2/4) cup + (12/16) cup = (4/8) cup + (12/16) cup
Next, we find a common denominator, which is 16:
(4/8) x (2/2) cup + (12/16) x (1/1) cup
This simplifies to:
8/16 cup + 12/16 cup
Finally, we add the fractions:
8/16 cup + 12/16 cup = 20/16 cup
Since 20/16 is an improper fraction, we can simplify it by dividing both the numerator and denominator by their greatest common divisor, which is 4:
(20/4)/(16/4) cup = 5/4 cup
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A high school is voting on a new mascot. Lion Eagle Bee Ninth Grade 45% 32% 23% Tenth Grade 36% 40% 24% Eleventh Grade 50% 28% 22% Twelfth Grade 40% 25% 35% Match the two-way table to the segmented bar graph.
Answers
The steps to create a bar graph are explained below.
To match the two-way table to the segmented bar graph, we need to create a segmented bar graph that represents the data in the table. The table shows the percentage of students in each grade who voted for each mascot option.
Here is how we can match the two-way table to the segmented bar graph:
For the Lion mascot option, the segmented bar graph would have the largest segment for 11th grade, followed by 9th grade, 10th grade, and 12th grade.For the Eagle mascot option, the segmented bar graph would have the largest segment for 10th grade, followed by 11th grade, 9th grade, and 12th grade.For the Bee mascot option, the segmented bar graph would have the largest segment for 12th grade, followed by 9th grade, 10th grade, and 11th grade.By creating a segmented bar graph that represents the data in the two-way table, we can visually compare the percentage of students who voted for each mascot option across different grade levels.
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12. A rectangle has a perimeter of 12y² - 2y +18 and has a width of 4y²-y + 6. What is the length of the rectangle?
Answers
Answer:
2y² + 3
Step-by-step explanation:
width of one side is 4y² - y + 6.
two equal sides of rectangle = 2(4y² - y +6) = 8y² - 2y + 12.
perimeter = 2 lengths + 2 widths
12y² - 2y + 18 = 2 lengths + 8y² - 2y + 12
2 lengths = (12y² - 2y + 18) - (8y² - 2y + 12)
= 4y² +6
length = 1/2 (4y² +6)
= 2y² + 3
If the three digits of a three-digit number are multiplied ou get 135. Which result to you get, by adding the three digits?
Answers
The result obtained by adding the three digits is 17.
Let's denote the three-digit number as XYZ, where X, Y, and Z represent the individual digits.
Given that the product of the three digits is 135, we have the equation
X × Y × Z = 135.
To find the result obtained by adding the three digits (X + Y + Z), we need to determine the individual values of X, Y, and Z that satisfy the given equation.
We can start by listing all possible combinations of three digits that multiply to 135:
1 × 3 × 45
1 × 5 × 27
1 × 9 × 15
3 × 5 × 9
Out of these combinations, we can observe that the sum of the digits in each combination is:
1 + 3 + 45 = 49
1 + 5 + 27 = 33
1 + 9 + 15 = 25
3 + 5 + 9 = 17
Therefore, the result obtained by adding the three digits is 17.
In conclusion, by adding the three digits of the three-digit number, we get the result of 17.
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Long division by single digit (no remainder)
Grade 4 Division Worksheet
Find the quotient.
1.
3 561
2.
2 308
3.
8 288
4.
7 616
5.
5 365
6.
9 990
7.
6 294
8.
6 180
9.
5 205
Answers
Sure! Here are the solutions for the long division problems:
3 561 ÷ 7 = 509
2 308 ÷ 4 = 577
8 288 ÷ 9 = 920
7 616 ÷ 8 = 952
5 365 ÷ 5 = 1 073
9 990 ÷ 3 = 3 330
6 294 ÷ 7 = 899
6 180 ÷ 6 = 1 030
5 205 ÷ 9 = 578
Please note that the results are rounded down to the nearest whole number since you specified "no remainder" in the problem statement.
what is the reason behind Russia and ukarine war?
Answers
Answer:
Step-by-step explanation:
Ukraine wants to join the european union.Due to this Russia realize that if ukraine join the european union then if war started between Ukraine and any other country like Russia it means Ukraine and european union both attack on this country.
Solve (y + 2)² +5=50, where y is a real number.
Round your answer to the nearest hundredth.
Answers
Answer:
y = 4.71 and y = -8.71
Step-by-step explanation:
Step 1: Subtract 5 from both sides:
((y + 2)^2 + 5 = 50) - 5
(y + 2)^2 = 45
Step 2: Take the square root of both sides, subtract 2 from both sides, and round to the nearest hundredth to solve for y.
Remember that taking the square root will give us both a negative and positive answer on the right-hand side of the equation. This is because squaring both a positive and negative number gives us. For example 2^2 = 2 * 2 = 4 but (-2)^2 = -2 * -2 = 4 alsoPositive answer:
√(y + 2)^2 = √45
(y + 2 = √45) - 2
y = √45 - 2
y = 4.708203932
y = 4.71
Thus, one answer for y (rounded to the nearest hundredth) is approximately 4.71.
Negative answer:
√(y + 2)^2 = -√45
(y + 2 = -√45) - 2
y = -√45 - 2
y = -8.708203932
y ≈ - 8.71
Thus, the other answer for y (rounded to the nearest hundredth) approximately. -8.71
Optional Step 3: We can check our work by plugging in 4.71 and -8.71 for y and seeing whether we get 50. Because they're rounded, you won't get exactly 50, but the answer should be close enough for us to trust our approximations:
Plugging in 4.71 for y:
(4.71 + 2)^2 + 5 = 50
(6.71)^2 + 5 = 50
45.0241 + 5 = 50
50.0241 > 50
Our answer is close enough to 50 that we can trust our approximation.
Plugging in -8.71 for y:
(-8.71 + 2)^2 + 5 = 50
(-6.71)^2 + 5 = 50
45.0241 + 5 = 50
50.0241 > 50
Again, our answer is close enough to 50 that we can trust our approximation.
The area of parallelogram is 36 square meters. If the height is 4 meters, what is the length of the base
Answers
Answer:
The area of a parallelogram is given by the formula:
Area = base * height
We are given that the area is 36 square meters and the height is 4 meters. Substituting these values into the formula, we get:
36 = base * 4
36 / 4 = base
9 = base
Therefore, the length of the base is 9 meters.
find the scale factor
Answers
Answer:
a. 2
Step-by-step explanation:
We Know
Side AB = 7
Side XY = 14
7 → 14, we time 2. So, the scale factor is 2.
a seller has a house that is 2100 ft. The neighborhood comps show the line of best fit to be y = 0.074x + 50.48
Answers
Based on the line of best fit, the predicted price for the seller's house with a size of 2100 square feet is approximately $205.88.
In the given scenario, the seller has a house that is 2100 square feet. The line of best fit for the neighborhood comps is expressed as y = 0.074x + 50.48. Here, "x" represents the square footage of a house, and "y" represents the predicted price based on the square footage.
To determine the predicted price of the seller's house, we substitute the square footage (x) with 2100 in the equation:
y = 0.074 * 2100 + 50.48
Calculating the expression, we have:
y = 155.4 + 50.48
y = 205.88
Hence, the predicted price for the seller's house, based on the line of best fit, is approximately $205.88.
It's important to note that the line of best fit is derived from neighborhood comps and may not perfectly predict the actual price of the seller's house. It serves as an estimation based on the trend observed in the neighborhood.
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