Answer:
Step-by-step explanation:
~P is true
q is true
~p v q is true
the hypothesis (or antecedent) is true
the conclusion (or consequent) is false
R is false and ~q is false
the conditional says if true then false
so the conditional is false
Need help multiply fractions and whole numbers
Answer:
Step-by-step explanation:
5. 15 multiplied by 4/5 = 3 multiplied by 4 = 12.
6. 3/11 multiplied by 66 = 3 multiplied by 6 = 18.
Two customers paid the same amount using Uber to travel to work. Andy traveled 10 miles and paid a $12 tip to his drive. Sammy traveled 15 miles and paid a $9 tip to his driver What is the cost per mile to ride the Uber.
If Andy traveled 10 miles and paid a $12 tip to his drive. Sammy traveled 15 miles and paid a $9 tip to his driver. the cost per mile to ride the Uber is $0.60.
How to find the cost per mile?Let the cost per mile to ride the Uber be "x".
Then, Andy paid a total of 10x + 12 dollars for his 10-mile trip.
Similarly, Sammy paid a total of 15x + 9 dollars for his 15-mile trip.
Since both customers paid the same amount, we can set their total costs equal to each other and solve for "x":
10x + 12 = 15x + 9
Subtracting 10x and 9 from both sides:
3 = 5x
Dividing by 5:
x = 0.6
Therefore, the cost per mile to ride the Uber is $0.60.
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expressions that are equivalent to 9(4w)
Using algebraic expressions, the expression that is equivalent to 9(4w) is: 36w.
What are algebraic expressions?Mathematical statements having at least two terms that involve variables or numbers are called algebraic expressions.
Every combination of terms that have undergone operations like addition, subtraction, multiplication, division, etc. is a recognised algebraic expression (or variable expression).
We just join like terms in an algebraic expression to make it simpler. Hence, related variables will be combined. Now, the identical powers will be merged from the similar variables.
Now in the given case,
9(4w)
= 9×4×w
=36w.
Therefore, the expression that is equivalent to 9(4w) is: 36w.
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How many one-thirds are in seven-halves
Answer: 14/3 or 3/4 are in 7/2
Step-by-step explanation:
2. Find eigenvalues and eigenvectors of a given matrix: a = [\begin{array}{ccc}2&2&-2\\1&3&-1\\-1&1&1\end{array}\right]
Calculate A^8 and check that A^8 = T * Λ^8 *T^-1, where Λ= diag{λ1, λ2, λ3} is diagonal matrix composed of eigenvalues λi of matrix A and matrix T is matrix composed of corresponding eigenvectots of matrix A.
A^8 = [\begin{array}{ccc}252&-3&3\\3&252&-3\\-3&3&252\end{array}\right]
To find the eigenvalues and eigenvectors of matrix A, use the characteristic polynomial of A which is given by:
P(λ) = det(A-λI) = λ^3-6λ^2+11λ-6
The roots of the equation are λ1 = 2, λ2 = 3, and λ3 = 1.
The corresponding eigenvectors are:
v1 = [\begin{array}{ccc}-1\\1\\1\end{array}\right]
v2 = [\begin{array}{ccc}1\\-1\\1\end{array}\right]
v3 = [\begin{array}{ccc}1\\1\\1\end{array}\right]
The eigenvalues and eigenvectors of A can be put together in the following matrix:
T = [\begin{array}{ccc}-1&1&1\\1&-1&1\\1&1&1\end{array}\right]
The eigenvalues of A can be put together in the following matrix:
Λ = diag{λ1, λ2, λ3} = [\begin{array}{ccc}2&0&0\\0&3&0\\0&0&1\end{array}\right]
Now, to calculate A^8, you will need to use the eigenvalues and eigenvectors of A. A^8 can be calculated using the following equation: A^8 = T * Λ^8 *T^-1, where Λ= diag{λ1, λ2, λ3} is diagonal matrix composed of eigenvalues λi of matrix A and matrix T is matrix composed of corresponding eigenvectots of matrix A.
Thus, A^8 can be calculated as follows:
A^8 = [\begin{array}{ccc}-1&1&1\\1&-1&1\\1&1&1\end{array}\right] * [\begin{array}{ccc}256&0&0\\0&6561&0\\0&0&1\end{array}\right] * [\begin{array}{ccc}-1&1&1\\1&-1&1\\1&1&1\end{array}\right]^-1
= [\begin{array}{ccc}252&-3&3\\3&252&-3\\-3&3&252\end{array}\right]
Therefore, the result of A^8 is: A^8 = [\begin{array}{ccc}252&-3&3\\3&252&-3\\-3&3&252\end{array}\right]
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Out of 25 26 27 28 29 30 31 and 32 which one is a factor of 27
Answer:
27
Step-by-step explanation:
factors are numbers that divide exactly into a number , leaving no remainder.
then the only factor of 27 from the list is 27
(5 marks) For the following pair of lines in,
R^(3)
determine if they intersect. If so, give the point of intersection. If not, explain.
L1 : x = 2 - t
y = -3 + 5t
z = t
L2 : P = (4,-1,16) + s (1,4,-7)
not intersect.
To determine if two lines intersect in R3, we need to solve for both parameters (t and s) in the two equations and then check if the values are equal.
For Line 1: x = 2 - t, y = -3 + 5t, z = t. We can solve for t by setting all three equations equal to each other, giving t = x - 2 = y + 3 = z.
For Line 2: P = (4,-1,16) + s (1,4,-7). We can solve for s by setting the three equations equal to each other, giving s = (x - 4) / 1 = (y + 1) / 4 = (z - 16) / -7.
If the two values of t and s are equal, then the lines intersect at the point (x, y, z). If they are not equal, then the lines do not intersect.
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Loni wants to open a nail salon. Loni figured out that she can average charging $30 per session. Each session she does will cost her $10 in supplies. Over the first year, she has projected $24,000 in rent and $50,000 in other expenses. The business cost her $15,000to start. How many sessions does she need to do in the first year to break even?
Loni needs to do 4,450 sessions in the first year to break even.
What is total revenue?
Total revenue is the total amount of money earned by a business from the sale of its products or services during a particular period of time.
To break even, the total revenue Loni makes should be equal to the total cost of running the nail salon. Let's calculate the total cost first:
Total Cost = Rent + Other expenses + Start-up cost + Cost of supplies per session x Number of sessions
Total Cost = $24,000 + $50,000 + $15,000 + $10 x Number of sessions
Total Cost = $89,000 + $10 x Number of sessions
Now, to break even, the total revenue should be equal to the total cost:
Total Revenue = Total Cost
$30 x Number of sessions = $89,000 + $10 x Number of sessions
$20 x Number of sessions = $89,000
Number of sessions = $89,000 / $20
Number of sessions = 4,450
Therefore, Loni needs to do 4,450 sessions in the first year to break even.
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a discount of $193.16 to its regular price of $828.15
sales price= $
Answer: $634.99
Step-by-step explanation:
Regular price, R - Discount, D = S
subtract discount from regular price
Answer: $634.99
Step-by-step explanation:
$828.15 - 193.16 would equal 634.99
so the discounted price would be $634.99
Find the missing length indicated.
The length of the missing side is 3 units
How to determine the missing lengthUsing the Pythagorean theorem, we have that;
Square of the hypotenuse is equal to the sum of the squares of the other two sides.
This is written as;
a² = b² + c²
Now, for the larger triangle, substitute the values, we have;
9² = 6² + c²
find the squares
c²= 81 - 36
add the values
c² = 45
c = √45
c = 6, 7 units
For the smaller triangle
6.7² - 6² = x²
x² = 45 - 36
x² = 9
Find the square root
x = 3
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OAB is a sector of a circle as shown below.
Calculate the area of the shaded region.
Give your answer in mm² to 1 d.p.
THIS IS URGENT
Answer:
attached is the answer
Step-by-step explanation:
HI
THIS IS THE ANSWER
I HOPE IT HELPS
GOOD LUCK
If you answered "Categorical" in the previous question, how many categories does the variable of interest consist of? If you answered "Numerical", what is the range of values the variable can take?
The variable consists of five categories: excellent, good, average, poor, and terrible.
How to determine the type of variableThe variable of interest in this scenario is the rating of the service provided by the hotel during the guests' most recent visit.
This variable is a way of measuring guests' perceptions of the quality of the service provided by the hotel.
The variable is categorical because the guests were asked to provide their rating by choosing one of five distinct categories - excellent, good, average, poor, and terrible.
The number of categorical variablesThe categories stated in (a) above represent the different levels of quality the guests perceived in the hotel's service.
Each rating category is non-numerical and non-sequential, meaning they don't have a natural numerical order or a defined distance between each other.
So, the number of categorical variables is 5
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Complete question
A hotel chain sent 2,000 past guests an email asking them to rate the service in the hotel during their most recent visit. Of the 500 who replied, 450 rated the service as excellent.
(a) Give a name to the variable and indicate if the variable is categorical, ordinal, or numerical
(b) If you answered "Categorical" in the previous question, how many categories does the variable of interest consist of? If you answered "Numerical", what is the range of values the variable can take?
Find the final amount for an investment of $5,000 over 5 years at an annual interest rate of 6% if the interest is compounded quarterly.
The final amount for the investment of $5,000 over 5 years at an annual interest rate of 6% compounded quarterly is $ 6,749.29.
Compound interest:To find the final amount for an investment P over t years at an annual interest rate of r compounded quarterly, we can use the formula for compound interest:
[tex]{\displaystyle A=P\left(1+{\frac {r}{n}}\right)^{nt}}[/tex]
Where:
A = Final amount
P = Principal (initial investment)
r = Annual interest rate (as a decimal)
n = Number of times the interest is compounded per year
t = Time period in years
Here we have
P = $5,000,
r = 6% = 0.06,
n = 4 (since the interest is compounded quarterly), and
t = 5 years.
Using the above formula
[tex]A = 5000(1 + 0.06/4)^{(4*5)[/tex]
[tex]A = 5000(1.015)^{20}[/tex]
[tex]A = 5000(1.34985711)[/tex]
[tex]A = $6,749.29[/tex]
Therefore,
The final amount for the investment of $5,000 over 5 years at an annual interest rate of 6% compounded quarterly is $ 6,749.29.
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Karen is planting a garden as shown. She wrote the expression 3(5+4) find the area of her garden. Which expression is equal to Karen's expression? Responses 3(5)+3(4) 3 ( 5 ) + 3 ( 4 ) 3×20 3 × 20 3(5)+4 3 ( 5 ) + 4 3+9
The equation 3(5)+3(4), which also simplifies to 27, is the same as Karen's expression.
what is expression ?A collection of letters and/or numbers that denotes a mathematical quantity or connection is known as an expression in mathematics. It can be evaluated or simplified using mathematical principles and may contain variables, constants, and operators (such as +, -, x,, etc.). Numerical values, relationships between values, and mathematical processes can all be represented using expressions. For instance, the expression "3x + 5" illustrates the connection between the numbers 3 and 5 and the variable x, and it can be assessed for various values of x to yield various numerical outcomes.
given
Karen's formula is 3(5+4), which can be written as 3(9) = 27.
The equation 3(5)+3(4), which also simplifies to 27, is the same as Karen's expression.
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Which equation shows how to use equivalent fractions to evaluate
4/5-2/3
Answer:
[tex] \frac{2}{15} = 0.13[/tex]
Step-by-step explanation:
[tex]1. \: lcd = 15 \\ 2. \: \frac{4 \times 3}{5 \times 3} - \frac{2 \times 5}{3 \times 5} \\ 3. \: \frac{12}{15} - \frac{10}{15} \\ 4. \: \frac{12 - 10}{15} \\ 5. \: \frac{2}{15} = 0.13[/tex]
Javier works each weekend at an animal shelter where he earns $8 an hour. He is saving his earnings to go on a ski trip that costs $190.
A double bar chart for Javiers Work Record. The x axis shows data from two weekends and the y axis shows the number of hours. Each weekend gives data for each day as follows. Weekend 1, Saturday: 5 hours. Sunday: 7 hours. Weekend 2, Saturday: 5 hours. Sunday: 6 hours.
Let h be the number of hours Javier worked.
Select all the true statements.
Javier worked a total of 23 hours over the two weekends (True), Javier earned $184 over the two weekends(True)
Describe Double bar chart?A double bar chart, also known as a grouped bar chart, is a graphical representation of data that compares two or more categories or groups of data. It is similar to a standard bar chart, but with two or more bars grouped together for each category.
In a double bar chart, the horizontal axis represents the categories or groups being compared, while the vertical axis represents the values being measured. The height of each bar represents the value of the data for that category or group, and the bars are usually colored or patterned differently to differentiate between the groups being compared.
Double bar charts are useful for comparing data between two or more groups or categories, such as different time periods, geographic regions, or demographic groups. They can be used to highlight similarities and differences between the groups, and to identify trends and patterns in the data.
Javier worked a total of 23 hours over the two weekends. (True: 5+7+5+6 = 23)Javier earned $184 over the two weekends. (True: 23 hours x $8/hour = $184)Javier needs to work about 24 more hours to save enough money for the ski trip. (False: $190 ÷ $8/hour = 23.75 hours, so Javier actually needs to work about 24 more hours to save enough money for the ski trip.)Javier worked the same number of hours on both weekends. (False: Javier worked 12 hours on Weekend 1 and 11 hours on Weekend 2.)To know more about axis visit:
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The complete question is
Solve the system of equations and choose the correct answer from the list of options. (4 points) x + y = −3 y = 2x + 2 a five over 3 comma 4 over 3 b negative 5 over 3 comma negative 4 over 3 c negative 3 over 5 comma negative 3 over 4 d 3 over 4 comma 3 over 5
The solution to the system of equations is (x, y) = (-5/3, -4/3).
What is system of linear equations?
A system of linear equations is a set of two or more equations with two or more variables that are to be solved simultaneously. Each equation in the system is linear, meaning it can be written in the form of ax + by + cz + ... = d, where a, b, c, and d are constants and x, y, z, and other variables are unknowns.
The goal of solving a system of linear equations is to find the values of the variables that satisfy all of the equations in the system. The solution of a system of linear equations is a set of values for the variables that make all of the equations true.
There are different methods to solve systems of linear equations, such as substitution method, elimination method, and matrix method. These methods involve manipulating the equations in the system to isolate one variable, substitute its value into another equation, and eventually find the values of all the variables.
Systems of linear equations are used in many areas of mathematics, science, engineering, and economics to model real-world situations and solve practical problems.
To solve the system of equations:
x + y = -3 (Equation 1)
y = 2x + 2 (Equation 2)
We can substitute Equation 2 into Equation 1 for y and solve for x:
x + (2x + 2) = -3
3x + 2 = -3
3x = -5
x = -5/3
Now that we know x, we can substitute it into either Equation 1 or Equation 2 to find y. Let's use Equation 2:
y = 2x + 2
y = 2(-5/3) + 2
y = -10/3 + 6/3
y = -4/3
Therefore, the solution to the system of equations is (x, y) = (-5/3, -4/3).
Comparing this solution to the answer choices, we see that option (b) is the correct answer:
(a) 5/3, 4/3
(b) -5/3, -4/3 <--- Correct answer
(c) -3/5, -3/4
(d) 3/4, 3/5
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Let Yı, Y2, ..., Yn be a random sample of size 5 from a normal population with mean 0 and variance 1. Also let Y = 1/5 ∑Yi. Let Y6 be another independent observation from the population. Determine the distribution of the following: (a) L = ∑(Yi - Y)^2 + Y^3 6 (b) J = 2Y6/√U
0 and variance 1.
For (a), L follows a Chi-squared distribution with 6 degrees of freedom. This is because L is a sum of squares of 6 random variables (the 5 Yi - Y plus Y^3).
For (b), J follows a Student's t-distribution with 6 degrees of freedom. This is because it is the ratio of two independent normally distributed random variables with mean 0 and variance 1.
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A shadow 9 feet long is cast by a plum tree that is 12 feet tall. What is the length of the shadow cast by a nearby elm tree that that is 16 feet tall?
Write your answer as a whole number or a decimal. Do not round.
please help mee
The length of the shadow cast by a nearby elm tree that that is 16 feet tall is given as follows:
12 feet.
How to obtain the length of the shadow?The length of the shadow is obtained applying the proportions in the context of the problem.
A shadow 9 feet long is cast by a plum tree that is 12 feet tall, and we want the shadow for a tree that is 16 feet tall, hence the rule of three is given as follows:
9 feet - 12 feet
x feet - 16 feet.
Applying cross multiplication, the shadow is obtained as follows:
12x = 9 x 16
12x = 144
x = 144/12
x = 12 feet.
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Complete the square to solve 0=x^(2)+9x+16.25 and determine the roots. Then sketch the parabola y=x^(2)+9x+16.25 and state the coordinates of the x intercepts and the vertex.
Answer: x intercepts = -5/2 or x = -13/2
the y intercepts y = 16.5
vertex ( -9/2 , -4)
Step-by-step explanation:
Answer:
The roots and x-coordinates of the equation are
[tex]-\frac{5}{2}[/tex] and [tex]-\frac{13}{2}[/tex]
The vertex of the parabola is
[tex]-4.5[/tex]
Step-by-step explanation:
Given
[tex]0=x^2+9x+16.25\\y=x^2+9x+16.25[/tex]
Complete the Square, X intercepts, and Roots
[tex]0=x^2+9x+16.25[/tex]
Write [tex]16.25[/tex] as a fraction.
[tex]x^2+9x+16\frac{1}{4}=0[/tex]
[tex]x^2+9x+\frac{65}{4}=0[/tex]
Factor the left hand side.
Write [tex]x^2[/tex] as a fraction with a common denominator, multiply by [tex]\frac{4}{4}[/tex].
[tex]x^2*\frac{4}{4} +9x+\frac{65}{4}=0[/tex]
Combine [tex]x^2[/tex] and [tex]\frac{4}{4}[/tex].
[tex]\frac{x^2*4}{4} +9x+\frac{65}{4}=0[/tex]
Combine the numerators over the common denominator.
[tex]\frac{4x^2+65}{4} +9x=0[/tex]
To write [tex]9x[/tex] as a fraction with a common denominator, multiply by [tex]\frac{4}{4}[/tex].
[tex]\frac{4x^2+65}{4} +9x*\frac{4}{4} =0[/tex]
Combine [tex]9x[/tex] and [tex]\frac{4}{4}[/tex].
[tex]\frac{4x^2+65}{4} +\frac{36x}{4} =0[/tex]
Combine the numerators over the common denominator.
[tex]\frac{36x+4x^2+65}{4} =0[/tex]
For a polynomial of the form [tex]ax^2+bx+c[/tex], rewrite the middle term as a sum of two terms whose product is [tex]a*c=4*65=260[/tex] and whose sum is [tex]b=36[/tex].
[tex]\frac{36(x)+4x^2+65}{4} =0[/tex]
Rewrite 36 as 10 plus 26.
[tex]\frac{(10+26)x+4x^2+65}{4} =0[/tex]
Apply the distributive property.
[tex]\frac{4x^2+10x+26x+65}{4} =0[/tex]
Group the first two terms and the last two terms.
[tex]\frac{\left(4x^2+10x\right)+26x+65}{4} =0[/tex]
Factor out the GCF from each group.
[tex]\frac{2x\left(2x+5\right)+13\left(2x+5\right)}{4} =0[/tex]
Factor the polynomial by factoring out the GCF.
[tex]\frac{\left(2x+5\right)+\left(2x+13\right)}{4} =0[/tex]
Set the numerator equal to zero.
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
[tex]2x=5=0\\2x=13=0[/tex]
Solving for [tex]x[/tex] in each equation gives us
[tex]-\frac{5}{2}[/tex] and [tex]-\frac{13}{2}[/tex]
The final solution is
[tex]x=-\frac{5}{2} ,-\frac{13}{2}[/tex]
These values of [tex]x[/tex] are the roots of the equation and lie on the x-axis.
Vertex
We can use the formula [tex]\frac{-b}{2a}[/tex] to evaluate the vertex.
This formula is for a polynomial of the form [tex]ax^2+bx+c=0[/tex].
[tex]x^2+9x+16.25=0[/tex]
In this case
[tex]a=1\\b=9[/tex]
Inserting our values into the equation yields
[tex]\frac{-9}{2*1}[/tex]
[tex]\frac{-9}{2}=-4.5[/tex]
You borrow $10,000 at effective rate 2.1%. When will the amount that you owe reach $15,000? Give your answer in years, rounded to the nearest tenth (0.1). [Hint: When using effective rate, what is the number of compounds per year, n?]
The amount that you owe will reach $15,000 after approximately 21.5 years.
Here is a step-by-step explanation of how to calculate this:
1. First, determine the number of compounds per year (n). In this case, since we are using an effective rate, n = 1.
2. Next, use the compound interest formula to find the final amount (A): A = P(1 + r/n)^(nt)
3. Plug in the given values for P ($10,000), r (0.021), n (1), and A ($15,000) and solve for t:
$15,000 = $10,000(1 + 0.021/1)^(1*t)
4. Simplify the equation and isolate t on one side:
1.5 = (1.021)^t
5. Take the natural logarithm of both sides:
ln(1.5) = t*ln(1.021)
6. Solve for t:
t = ln(1.5)/ln(1.021)
7. Use a calculator to find the value of t:
t ≈ 21.5
Therefore, the amount that you owe will reach $15,000 after approximately 21.5 years, rounded to the nearest tenth (0.1).
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Please help me
Write 3 3/4 feet as a single fraction greater than one.
Answer: 9/4
Step-by-step explanation:
Answer:
15/4 feet
Step-by-step explanation:
When given a mixed number like [tex]3\frac{3}{4}[/tex], you multiply the denominator by the whole number and add the numerator.
In this case, multiply 3 x 4 and then add 3. After finding that value, put it over the original denominator, giving 15/4
Use the inverse transform method to generate a random variable having distribution function F(x) = x² + x / 2
The inverse transform method allows us to generate a random variable having a specific distribution function by finding the inverse of the distribution function and plugging in a random value from the uniform distribution on the interval [0, 1].
To generate a random variable having the distribution function F(x) = x² + x / 2 using the inverse transform method, we need to follow the following steps:
Step 1: Find the inverse of the distribution function F(x). This can be done by solving the equation F(x) = u for x, where u is a uniform random variable on the interval [0, 1]. In this case, we have:
x² + x / 2 = u
Step 2: Use the quadratic formula to solve for x:
x = (-1 ± √(1 - 4(1/2)(-u))) / (2(1/2))
x = (-1 ± √(1 + 2u)) / 1
Step 3: Since we want the inverse function, we need to choose the positive solution:
x = (-1 + √(1 + 2u)) / 1
Step 4: Now we have the inverse function F^-1(u) = (-1 + √(1 + 2u)) / 1. To generate a random variable having the distribution function F(x), we simply need to plug in a random value of u from the uniform distribution on the interval [0, 1]:
x = F^-1(u) = (-1 + √(1 + 2u)) / 1
Step 5: This gives us a random variable x having the desired distribution function F(x) = x² + x / 2.
In conclusion, the inverse transform method allows us to generate a random variable having a specific distribution function by finding the inverse of the distribution function and plugging in a random value from the uniform distribution on the interval [0, 1].
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What is the slope that is
perpendicular to the
equation y = -1/2x - 1?
x^2-9,x^3-27,x^2+x-12
Answer:
Step-by-step explanation:
simple 59
Central School has 758 students. There are 2 lunch periods at Central School. Three hundred seventy-one students eat during the first lunch period. Which lunch period has more students? Conjecture: More students eat during the second lunch period. 1. Construct an argument to justify the conjecture. 2. Explain another way you could justify the conjecture
The lunch period that has more students is the second lunch period.
Which period has more students?In order to determine which lunch period that has more students, the number of students that eat in the second lunch period has to be determined. In order to determine this value, the mathematical operation that would be used is subtraction.
Subtraction is the process of determining the difference between two or more numbers. The sign that is used to represent subtraction is -.
Number of students that ear in the second lunch period = total number of students - students that eat in the first lunch period
758 - 371 = 387
387 > 371
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Which number line shows all the values of that make the inequality true? A number line labeled A with the numbers negative 5 through 5 indicated. There is a closed circle at negative 2 and an arrow is drawn from the closed circle extending to the left. A number line labeled B with the numbers negative 5 through 5 indicated. There is an open circle at negative 2 and an arrow is drawn from the open circle extending to the left. A number line labeled C with the numbers negative 5 through 5 indicated. There is a closed circle at negative 2 and an arrow is drawn from the closed circle extending to the right. A number line labeled D with the numbers negative 5 through 5 indicated. There is an open circle at negative 2 and an arrow is drawn from the open circle extending to the right.
The number lines for the given inequalities are drawn as shown in the figures below.
What is a number line?A number line is a diagram of a graduated straight line used to represent real numbers in introductory mathematics. It is assumed that every point on a number line corresponds to a real number and that every real number corresponds to a point. On a number line, inequalities are represented by drawing a straight line and designating the endpoints with either an open or closed circle. An open circle indicates that the value is not included. A closed circle indicates that the value is included.
1) Number line A
The closed circle at -2 means greater than or equal or less than or equal to -2.
Now the arrow is extending to the left.
So the inequality is numbers less than or equal to -2.
2) Number line B
The open circle at -2 means greater than or less than -2.
Now the arrow is extending to the left.
So the inequality is numbers less than -2.
3) Number line C
The closed circle at -2 means greater than or equal or less than or equal to -2.
Now the arrow is extending to the right.
So the inequality is all the numbers greater than or equal to -2.
4) Number line D
The open circle at -2 means greater than or less than -2.
Now the arrow is extending to the right.
So the inequality is all the numbers greater than -2.
Therefore the number lines for the given inequalities are drawn as shown in the figures below.
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Let sine of theta equals the quantity 2 times radical 3 end quantity over 5 and pi over 2 is less than theta is less than pi period
Part A: Determine the exact value of cos 2θ.
Part B: Determine the exact value of sine of the quantity theta over 2 end quantity period
The exact value οf cοs(2θ) is 27/25.
The exact value οf sin(θ/2) is -√[(5 + √13) / 10].
What is Trigοnοmetry?Trigοnοmetry is a branch οf mathematics that deals with the relatiοnships between the sides and angles οf triangles, and the trigοnοmetric functiοns that describe thοse relatiοnships.
Given: sin(θ) = (2√3) / 5, and π/2 < θ < π.
Part A:
cοs(2θ) = 2cοs²(θ) - 1
We can find cοs(θ) using the Pythagοrean identity:
cοs²(θ) + sin²(θ) = 1
cοs²(θ) = 1 - sin²(θ)
cοs(θ) = ±√(1 - sin²(θ))
Since π/2 < θ < π, sin(θ) is pοsitive and cοs(θ) is negative in the secοnd quadrant. Therefοre, we have:
cοs(θ) = -√(1 - (2√3/5)²) = -√(1 - 12/25) = -√13/5
Substituting intο the fοrmula fοr cοs(2θ), we get:
cοs(2θ) = 2cοs²(θ) - 1 = 2(-√13/5)² - 1 = 52/25 - 1 = 27/25
Therefοre, the exact value οf cοs(2θ) is 27/25.
Part B:
We can use the half-angle fοrmula fοr sine tο find sin(θ/2):
sin(θ/2) = ±√[(1 - cοs(θ)) / 2]
Since π/2 < θ < π, sin(θ/2) is negative in the secοnd quadrant. Therefοre, we have:
sin(θ/2) = -√[(1 - cοs(θ)) / 2] = -√[(1 - (-√13/5)) / 2] = -√[(5 + √13) / 10]
Therefοre, the exact value οf sin(θ/2) is -√[(5 + √13) / 10].
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Consider the line x+4y=4
The slope of a line perpendicular to given line is 4.
What is slope of a line?The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.
The given equation of a line is x+4y=4.
Here, 4y=4-x
y=1-x/4
So, slope m is -1/4
The slope of a line perpendicular to given line is m1=-1/m2
Slope of perpendicular to line is -1/(-1/4)
= 4
Therefore, the slope of a line perpendicular to given line is 4.
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The slope of parallel line is,
m = - 1/4
And, Slope of perpendicular line are,
m = 4
What is Equation of line?The equation of line in point-slope form passing through the points
(x₁ , y₁) and (x₂, y₂) with slope m is defined as;
⇒ y - y₁ = m (x - x₁)
Where, m = (y₂ - y₁) / (x₂ - x₁)
Given that;
The equation of line is,
⇒ x + 4y = 4
Now, We can write as;
⇒ x + 4y = 4
⇒ 4y = - x + 4
⇒ y = - 1/4x + 4
Hence, The slope of the line is,
⇒ y = - 1/4x + 4
⇒ dy/dx = - 1/4
⇒ m = - 1/4
Now, We know that;
The slope of parallel line are equal.
Hence, The slope of parallel line is,
m = - 1/4
And, Product of Slopes of perpendicular lines are - 1.
Hence, Slope of perpendicular line are,
m = - 1/ (- 1/4)
m = 4
Thus, The slope of parallel line is,
m = - 1/4
And, Slope of perpendicular line are,
m = 4
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Martha buys a ribbon that is 7/10 of a yard long.
She cuts off 2/5 of a yard.
How many yards of ribbon does Martha have left?
Answer:
7/10 - 2/5 = 7/10 - 4/10 = 3/10
Step-by-step explanation:
Equalize both denominators, as in 2/5 = 4/10.