The solutions to the given trigonometric equation on the interval [0,2π) are θ = 2π/3 and θ = 4π/3. These values can also be expressed in decimal form as θ = 2.09 and θ = 4.19, rounded to two decimal places
To solve the given trigonometric equation on the interval [0,2π), we need to use the quadratic formula.
First, let us rewrite the equation in terms of x:
2x^2 + 5x + 2 = 0
Next, we can apply the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
Plugging in the values from the equation:
x = (-(5) ± √((5)^2 - 4(2)(2)))/(2(2))
Simplifying:
x = (-5 ± √(25 - 16))/4
x = (-5 ± √9)/4
x = (-5 ± 3)/4
This gives us two possible values for x:
x = (-5 + 3)/4 = -2/4 = -0.5
x = (-5 - 3)/4 = -8/4 = -2
Now we need to convert these values back to θ by using the inverse cosine function:
θ = cos^-1(-0.5)
θ = cos^-1(-2)
The first value, θ = cos^-1(-0.5), gives us two solutions on the interval [0,2π):
θ = 2π/3 and θ = 4π/3
The second value, θ = cos^-1(-2), does not give us any solutions on the interval [0,2π) because the cosine function only takes on values between -1 and 1.
Therefore, the solutions to the given trigonometric equation on the interval [0,2π) are θ = 2π/3 and θ = 4π/3. These values can also be expressed in decimal form as θ = 2.09 and θ = 4.19, rounded to two decimal places.
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Identify the area of the figure rounded to the nearest tenth
Answer:
118.7 inches squared.
Step-by-step explanation:
What is the area?The area is the total space taken up by a flat (2-D) surface or shape. The area is always measured in square units.
What is diameter?Diameter is the length across the entire circle, the line splitting the circle into two identical semicircles.
The expression for solving the area of a circle is A = π × [tex]r^{2}[/tex].
To solve for the semicircle above, we can divide the diameter into 2 to get the radius.
12 ÷ 2 = 6So, the radius of the upper semicircle is 6 inches.
If the radius of a circle is 6 inches, then you can substitute r for 6 into the formula.
A = π × [tex]6^{2}[/tex]This simplifies to A = 36π. If a semicircle if half the size of a normal circle, then it will be A = 18π, because 36 ÷ 2 = 18.
To solve for the lower semicircle, we can do the same this as we did above.
A = π × [tex]r^{2}[/tex]But wait, we don't know the radius or diameter!
No worries! To solve for the diameter of the circle, we can take the line that is parallel to the semicircle (the one that has a length of 12in) and subtract 6 from it. We subtract 6 from it because the semicircle takes up the remaining length of the line, not including the 6in.
To solve for the lower semicircle, we can divide the diameter by 2 to get the radius.
6 ÷ 2 = 3So, the radius of the circle is 3.
Now we can insert 3 into the expression.
A = π × [tex]3^{2}[/tex]This simplifies to A = 9π. If a semicircle if half the size of a normal circle, then it will be A = 4.5π because like above, 9 ÷ 2 = 4.5.
Adding the two semicircles together:
18π + 4.5π = 22.5π22.5 × π ≈ 70.6858So, the area of both semicircles is approximately 70.6858 square inches.
To solve for the area of a rectangle we use the expression:
A = length × widthInserting the dimensions of the rectangle:
8 × 6 = 48So, the area of the rectangle is 48 square inches.
Adding the two areas together:
70.6858 + 48 = 118.6858 ≈ 118.7Therefore, the area of the entire figure, rounded to the nearest tenth is [tex]118.7[/tex] [tex]in^{2}[/tex].
Find the probability of getting two heads and three tails in a
single throw of five unbiased coins?
The probability of getting two heads and three tails in a single throw of five unbiased coins is 0.3125.
The probability of getting two heads and three tails in a single throw of five unbiased coins can be calculated using the formula:
P = (5! / (2! * 3!)) * (0.5)^2 * (0.5)^3
Where 5! is the factorial of 5, 2! is the factorial of 2, and 3! is the factorial of 3.
First, calculate the factorial of 5, 2, and 3:
5! = 5 * 4 * 3 * 2 * 1 = 120
2! = 2 * 1 = 2
3! = 3 * 2 * 1 = 6
Next, plug these values into the formula:
P = (120 / (2 * 6)) * (0.5)^2 * (0.5)^3
Simplify:
P = (120 / 12) * (0.25) * (0.125)
P = 10 * 0.25 * 0.125
P = 0.3125
Therefore, the probability of getting two heads and three tails in a single throw of five unbiased coins is 0.3125.
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Emily is going to the bookstore. Paperback books cost $4. 50 each and hard
cover books cost $10 each. How many paperback books and how many
hardcover books can Emily buy without spending more than $45?
Emily can buy 9 paperback books and 1 hardcover book without spending more than $45.
Let's start by defining our variables
Let x be the number of paperback books
Let y be the number of hardcover books
We know that a paperback book costs $4.50 and a hardcover book costs $10. If Emily buys x paperback books and y hardcover books, then the total cost of her purchase can be calculated using the following equation:
Total cost = 4.50x + 10y
We also know that Emily cannot spend more than $45. So we can set up the following inequality
4.50x + 10y ≤ 45
Now we can use this inequality to find the maximum number of paperback and hardcover books that Emily can buy.
First, let's rearrange the inequality to solve for y:
10y ≤ 45 - 4.50x
y ≤ (45 - 4.50x) / 10
y ≤ 4.5 - 0.45x
So, Emily can buy up to 4.5 - 0.45x hardcover books for every x paperback books. Let's make a table to see the possibilities
From the table, we can see that the maximum number of paperback books that Emily can buy is 12, and the maximum number of hardcover books she can buy is 0. However, this is not a feasible option since she would be spending $54, which is over her budget of $45.
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Evaluate. Write your answer as a fraction or whole number without exponents.
2^–2 =
Answer:
1/4
Step-by-step explanation:
you can plug it into a calculator
Nick is 64 years younger than Madelyn. 4 years ago, Madelyn's age was 3 times Nick's age. How old is Nick now?
Nick is 36 years old now.
To solve the problem, we can use the following steps:
Step 1: Let Nick's age be x and Madelyn's age be y.
Step 2: We are given that Nick is 64 years younger than Madelyn, so we can write an equation: x = y - 64
Step 3: We are also given that 4 years ago, Madelyn's age was 3 times Nick's age, so we can write another equation: y - 4 = 3(x - 4)
Step 4: Substitute the first equation into the second equation to get: y - 4 = 3(y - 64 - 4)
Step 5: Simplify the equation to get: y - 4 = 3y - 204
Step 6: Solve for y to get: y = 100
Step 7: Substitute the value of y back into the first equation to get: x = 100 - 64 = 36
Therefore, Nick is 36 years old now.
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Find the volume of the composite solid
Answer: split it into smaller solids and calculate their separate volumes.
Step-by-step explanation:
3.
The box-and-whisker plot shown below represents
600 scores on a district geometry te
150 students scored between 42 and 56 on the district geometry test.
What is box whisker plot?Box whisker plots are used to abstract a collection of data that has been approximated using an interval scale. Just a box plot is another name for it. They are mostly employed while interpreting data. It is one of the graphical approaches that shows how the dataset's data varies from one another. The histogram may be used to display the data as well. Yet a histogram offers a useful presentation. Box and whisker plots are preferable to histograms because they may exhibit numerous sets of data on a single graph, which adds more information.
From the box whisker plot we see that the lower quartile is 42 and the median is 56.
The median represents the 50th percentile whereas lower quartile is 75%.
The difference is:
75 - 50 = 25%.
Now, for 600 scores we have:
25% of 600
= 25/100 (600) = 150
Hence, 150 students scored between 42 and 56 on the district geometry test.
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Sketch the least positive angle and find the values of the six trigonometric functions
3x+5y=0 , x≥0
The least positive angle is 180 - θ.
To sketch the least positive angle and find the values of the six trigonometric functions for the given equation 3x + 5y = 0, x ≥ 0, we need to first find the slope and y-intercept of the equation.
The slope of the equation is -3/5 and the y-intercept is 0. This means that the line passes through the origin and has a negative slope.
To sketch the least positive angle, we need to find the angle between the x-axis and the line. Since the slope is negative, the angle will be in the second quadrant. The least positive angle is the reference angle in the second quadrant, which is 180 - θ.
To find the values of the six trigonometric functions, we can use the slope and y-intercept to find the coordinates of a point on the line. One such point is (5, -3).
Using these coordinates, we can find the values of the trigonometric functions:
sin θ = -3/√(5^2 + (-3)^2) = -3/√34
cos θ = 5/√(5^2 + (-3)^2) = 5/√34
tan θ = -3/5
csc θ = √34/-3
sec θ = √34/5
cot θ = -5/3
Therefore, the least positive angle is 180 - θ and the values of the six trigonometric functions are as follows:
sin θ = -3/√34
cos θ = 5/√34
tan θ = -3/5
csc θ = √34/-3
sec θ = √34/5
cot θ = -5/3
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Angela used (1)/(5) of her beads to make some bracelets. She gave 78 beads to Santiago and had (1)/(2) of the beads left. How many beads did Angela have left?
If Angela used 1/5 of her beads and gave 78 to Santiago, she would have 130 beads left.
To find out how many beads Angela had left, we need to first find out how many beads she started with. We can do this by using the information given in the question and setting up an equation. Let's let x represent the number of beads Angela started with.
According to the question, Angela used (1)/(5) of her beads to make bracelets, so she had (4)/(5) of her beads left. She then gave 78 beads to Santiago, which left her with (1)/(2) of her original number of beads. We can write this as an equation:
(4)/(5)x - 78 = (1)/(2)x
To solve for x, we can first multiply both sides of the equation by 10 to get rid of the fractions:
8x - 780 = 5x
Next, we can subtract 5x from both sides of the equation to isolate the variable:
3x = 780
Finally, we can divide both sides of the equation by 3 to find the value of x:
x = 260
So Angela started with 260 beads. We proceed to find out how many you have left:
P = (1/2)(260)
P = 130
So Angela had 130 beads left after giving 78 beads to Santiago.
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2. What is the product of (-3x3 + 2x – 5) and (2x4 – 4x2 – 3)?
(a) Show your work
(b) Is the product of (-3x3 + 2x – 5) and (2x4 – 4x2 – 3) equal to the product of (2x4 –
4x2 – 3) and (-3x3 + 2x – 5) Explain your answer.
The answer is that the sum οf (-3x³ + 2x - 5) and (2x⁴ - 4x² - 3) is the same as the sum οf (-3x³ + 2x - 5) and (2x⁴ - 4x² - 3)
What is expressiοn?An expressiοn in mathematics is made up οf different numbers, variables, and mathematical prοcesses, such as expοnents, lοgarithms, trigοnοmetric functiοns, and arithmetic. It can symbοlise a number οr a fοrmula, but it lacks an equals sign (=) and cannοt be evaluated οr sοlved withοut being cοmbined οr simplified. Expressiοns can be straightfοrward, like 3x + 4, οr they can be cοmplicated, like (2x - 1)/(x + 3).
given:
(a) We must multiply each term in the first expressiοn by each term in the secοnd expressiοn befοre cοmbining like terms tο determine the prοduct οf (-3x³ + 2x - 5) and (2x4 - 4x² - 3). The distributive principle allοws us tο write:
= (-3x³ + 2x - 5) * (2x⁴ - 4x² - 3) * (3x³ + 2x - 5) * = 3x³ * 2x⁴ + (-3x³) * (-4x²) * (-3)
= 2x * 2x⁴ + 2x * (-4x²) + 2x * (-3)
= -6x⁷ + 12x⁵ + 9x³ + 4x⁵ - 8x³ - 6x - 10x⁴ + 20x² + 15 = 5 * 2x⁴ - 5 * (-4x²) - 5 * (-3)
The result οf the οperatiοns (-3x³ + 2x - 5) and (2x⁴ - 4x² - 3) is as fοllοws: -6x⁷ + 10x⁴ + 16x⁵ - 17x³ - 6x + 15
(b) The answer is that the sum οf (-3x³ + 2x - 5) and (2x⁴ - 4x² - 3) is the same as the sum οf (-3x³ + 2x - 5) and (2x⁴ - 4x² - 3). Due tο the fact that multiplicatiοn is cοmmutative, changing the οrder οf the variables has nο effect οn the οutcοme οf the prοduct. Cοnsequently, we can write:
(2x⁴ - 4x² - 3) * (-3x³ + 2x - 5) = (-3x³ + 2x - 5) * (2x⁴ - 4x² - 3)
The answer is that the sum οf (-3x³ + 2x - 5) and (2x⁴ - 4x² - 3) is the same as the sum οf (-3x³ + 2x - 5) and (2x⁴ - 4x² - 3)
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State the formula for testing the significant difference between
two population variances
The formula for testing the significant difference between two population variances is the F-test. The F-test is used to compare the variances of two populations and is calculated by dividing the larger variance by the smaller variance. The formula for the F-test is:
F = s1^2 / s2^2
Where s1^2 is the variance of the first population and s2^2 is the variance of the second population.
If the calculated F value is greater than the critical F value from the F distribution table, then there is a significant difference between the two population variances. If the calculated F value is less than or equal to the critical F value, then there is not a significant difference between the two population variances.
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The table below shows the relationship between the perimeter and area of four squares.
Area, A
(square units) Perimeter, P
(units)
1 4
4 8
9 12
16 16
Which equation can be used to find A, the area of a square that has a perimeter of P units?
A.A = (P ÷ 4) x (P ÷ 4)
B.A = (P– 4)
C.A = (P + 4) x (P + 4)
D. A=P
I think that the answer for your problem would be
.
.
.
a.a=(p÷4)×(p÷4)
Answer: We know that the perimeter of a square is given by the formula:
P = 4s
where s is the side length of the square.
We can rearrange this equation to get:
s = P/4
The area of a square is given by the formula:
A = s^2
Substituting s = P/4, we get:
A = (P/4)^2
Simplifying this equation:
A = P^2/16
Therefore, the equation that can be used to find the area of a square that has a perimeter of P units is:
A = P^2/16
So, the correct answer is:
A. A = (P ÷ 4) x (P ÷ 4)
Step-by-step explanation:
2 1/2 = 5n
I need help please
Answer:
1/2
Step-by-step explanation:
because 5 times 1/2 and you get 2 1/2
The table below shows the relationship between the perimeter and area of four squares.
Squares
Area, A
(square units) Perimeter, P
(units)
1 4
4 8
9 12
16 16
Which equation can be used to find A, the area of a square that has a perimeter of P units?
A.A = (P ÷ 4) x (P ÷ 4)
B.A = (P– 4)
C.A = (P + 4) x (P + 4)
D.A = P
Equation A = (P ÷ 4)*(P ÷ 4) can be used to find area of a square.
What exactly is a square?A square is a geometrical shape that has four equal sides and four equal angles (90-degree angles or right angles). It is a type of rectangle where all sides have the same length. A square is a two-dimensional shape, meaning it only has length and width, and all of its sides are parallel to each other. The opposite sides of a square are parallel and equal in length, and its diagonals bisect each other at 90-degree angles. Squares are used in mathematics, geometry, and various other fields.
Now,
As Area =a² where a=side of square
and perimeter=4a⇒a=p/4
then area=(p/4)²
Hence, The equation that can be used to find A, the area of a square that has a perimeter of P units is:
A = (P ÷ 4)*(P ÷ 4)
Therefore, the correct option is A.
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The right rectangular prism is made up of 48 cubes. Each cube has an edge length of 14 inch. Image_30414 What is the volume of this prism? Responses 14 cubic inch 1 fourth, cubic inch 12 cubic inch 1 half, cubic inch 34 cubic inch 3 fourths, cubic inch 1 cubic inch
The right rectangle prism contains 48 cubes, which results in a volume of 131,712 cubic inches.
what is prism ?A prism is a solid geometric form with two parallel bases that are congruent and typically shaped like polygons, as well as parallelogram-shaped sides that connect the bases. Prisms come in different shapes, such as hexagonal, triangular, and rectangle.
given
The 48 blocks that make up the right rectangular prism each have an edge length of 14 inches. By dividing the quantity of cubes by the sum of their individual volumes, one can determine the volume of the prism.
Each cube has a volume of (14 in.)3 = 2744 cubic inches.
Cube count is 48.
48 squares at 2744 cubic inches each equals 131,712 cubic inches of volume for the prism.
The right rectangle prism contains 48 cubes, which results in a volume of 131,712 cubic inches.
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Find the geometric mean for 5 and 25
Please help
Question
Which term best describes the association between variables A and B?
Responses
no association
a negative linear association
a positive linear association
a nonlinear association
Answer:
B is correct
Step-by-step explanation:
The points all go downhill as you move from left to right. We can draw a straight line that is fairly close to all of them. This is known as the regression line. The regression line having a negative slope tells us the linear association is negative. This means the correlation coefficient r is negative, so, (r = -1 is not possible as not all points fall on the same straight line)
What is the Sum?
(I have more questions to ask, so if you want points keep stalking my account)
Answer:
3rd choice down
(x + 9) / (x + 1)
Step-by-step explanation:
(2x + 4 - x + 5) / (x + 1) = (2x - x + 4 + 5) / (x+ 1) = (x + 9) / (x + 1)
The table below shows how much Eloise spent on ribbon at two different shops. What is the difference in the price of 300cm of ribbon between shop A and shop B? Give your answer in pounds (£).
The difference in price of both the shops is £0.25.
What is price?
Price is the amount of money one pays for a good or service. It is determined by a variety of factors, such as supply and demand, the cost of production, and the availability of resources. Prices are also affected by government regulations, taxes, inflation, and other economic and market forces. The price of a product or service is an important factor in determining whether consumers will purchase it or not.
Shop A | Shop B
200 cm | £0.50 | 300 cm | £0.75
The difference in the price of 300cm of ribbon between shop A and shop B is £0.25. Eloise spent £0.50 on 200cm of ribbon at shop A, and £0.75 on 300cm of ribbon at shop B. Therefore, the difference in price is £0.25.
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Tyrone is applying stain to a toy chest he made for his sister. The toy box is 33 inches (in.) long , 18 in. wide , and 15 In. tall. The next of the box is shown.
Answer:
2718 square inchesStep-by-step explanation:
Since he's applying stain (painting) he must be doing only the surface.
Finding the surface area:
33x18=594
33x15=495
18x15=270
594+495+270=
1359 square inches
We will then x2 1359 since we found the area for 1/2 of the box.
It is: 2718
Johnston & Deegan-Packel index (Math & Politics)
Consider a voting system for the six New England states where there are a total if seventeen votes and twelve or more are required for passage. Votes are distributed as follows:
MA:4 ME:3 NH:2
CT:4 RI:3 VT:1
a) Calculate the Johnston index of each voter
b) Calculate the Deegan-Packel index of each voter
The Johnston index of each voter is 3.317 and the Deegan-Packel index of each voter is 0.154
The Johnston index and the Deegan-Packel index are two measures of voting power that take into account the number of votes each voter has and the number of votes required for passage.
a) The Johnston index is calculated by taking the square root of the product of the number of votes a voter has and the number of votes they need for passage. In this case, the Johnston index for each voter is:
MA: √(4*(12-4)) = √32 = 5.657
ME: √(3*(12-3)) = √27 = 5.196
NH: √(2*(12-2)) = √20 = 4.472
CT: √(4*(12-4)) = √32 = 5.657
RI: √(3*(12-3)) = √27 = 5.196
VT: √(1*(12-1)) = √11 = 3.317
b) The Deegan-Packel index is calculated by taking the number of winning coalitions that a voter is a part of and dividing it by the total number of winning coalitions. In this case, the Deegan-Packel index for each voter is:
MA: 20/26 = 0.769
ME: 14/26 = 0.538
NH: 8/26 = 0.308
CT: 20/26 = 0.769
RI: 14/26 = 0.538
VT: 4/26 = 0.154
Overall, the Johnston index and the Deegan-Packel index provide different perspectives on the voting power of each voter. The Johnston index takes into account the number of votes each voter has and the number of votes required for passage, while the Deegan-Packel index takes into account the number of winning coalitions each voter is a part of.
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Can somebody help with this problemo
Answer: end balance: $8,496.95
total interest: $1,496.95
Step-by-step explanation: total interest = $7000 × 8.554% × 30 ÷ 12 = $1,496.95
end balance = $7000 + $1,496.95 = $8,496.95
Use point-slope form to write the equation of a line that passes through the point (-7,17) with slope - 5.
Answer:
y - 17 = - 5(x + 7)
Step-by-step explanation:
the equation of a line in point- slope form is
y - b = m(x - a)
where m is the slope and (a, b ) a point on the line
here m = - 5 and (a, b ) = (- 7, 17 ) , then
y - 17 = - 5(x - (- 7) ) , that is
y - 17 = - 5(x + 7)
Answer:
y = -5x - 18.
Step-by-step explanation:
The point-slope form of the equation of a line is given by:
y - y1 = m(x - x1)
where m is the slope of the line, and (x1, y1) is a point on the line.
We are given that the line passes through the point (-7, 17) and has a slope of -5. Thus, we can substitute these values into the point-slope form equation to get:
y - 17 = -5(x - (-7))
Simplifying the right-hand side of the equation:
y - 17 = -5(x + 7)
y - 17 = -5x - 35
Finally, adding 17 to both sides of the equation, we get the slope-intercept form of the equation:
y = -5x - 18
Therefore, the equation of the line that passes through the point (-7,17) with slope -5 is y = -5x - 18.
Lesson 8 Practice Problems
1. A number line
can represent positions that are north and south of a truck stop on a
highway. Decide whether you want positive positions to be north or south of the
truck stop. Then plot the following positions on a number line.
a. The truck
stop
b. 5 miles north of the truck
stop
C. 3.5 miles south of the truck stop
We want to graph a number line and graph some locations on it, the graph can be seen at the end of the answer.
What is a number line?A number line is just a line with equidistant marks on it, that can be used to find the distance between numbers, so to graph the number line we just graph a line and do marks on it.
The north will be referred to as the right side of a number line that is positive, and the south will be the opposite.
Now let's complete the graph:
a) We want to graph the truck stop; we can do this at the zero of the number line.
b) 5 miles up north of the truck stop.
Here, if each mark represents a mile, we just mark a point that is 5 marks at the right of the truck stop.
c) 3.5 miles south of a truck stop, at this location.
Here we mark a point that is at 3 and a half marks at the left of the truck stop.
The graph can be seen below.
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20 points
dias older brother likes his hot chocolate made with 2 ounces of cocoa and 3 ounces of milk. Whose how chocolate is more chocolaty
Dina likes hers with 3 ounces of chocolate and 5 ounces of milk
Explains how u got the answer
Dina's older brother has a hot chocolate that is more chocolaty, as his recipe has a percentage of cocoa of 40%, which is higher than the percentage of 37.5% for Dina's recipe.
How to define which is the more chocolaty recipe?The more chocolaty recipe is defined obtain the proportion of cocoa for each recipe, and then whoever's recipe has the higher proportion has the more chocolaty recipe.
The proportion of cocoa for each recipe is obtained with the division of the amount of cocoa used by the sum of the amounts of cocoa and milk used for the recipe.
Hence the proportion for Dina's older brother is obtained as follows:
2/(2 + 3) = 2/5 = 0.4 = 40% cocoa.
The proportion for Dina's is obtained as follows:
3/(3 + 5) = 3/8 = 0.375 = 37.5% cocoa.
40 > 37.5, hence Dina's older brother has a hot chocolate that is more chocolaty, as his recipe has a higher proportion of cocoa.
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The dashed triangle is the image of the solid triangle for a dilation with center at the origin. What is the scale factor?
The scale factor using in that dilation of the dashed triangle would be = 3:1
What is a scale factor?A scale factor is defined as the ratio that exists between an original size of an object and the new formed size of the same object.
The formula that can be used to calculate the scale factor ;
Scale factor = Dimension of the new shape ÷ Dimension of the original shape.
Using the base of the triangle from the centre of origin in the graph;
The dimension of the new shape = 18
The dimension of the original shape = 6
scale factor = 18/6 =3
That is, the scale factor = 3:1
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Two parallel lines are cut by a transversal, and shown. m<1 = (3x + 5) and m<2 = (2x +10). Select the measure of each angle.
The angle measures, considering the linear pair formed by angles 1 and 2, are given as follows:
m < 1 = 104º.m < 2 = 76º.What are linear pairs?When a parallel line is cut by a transversal, a linear pair is formed when two adjacent angles add up to 180 degrees. A linear pair of angles is always formed by two adjacent angles that are supplementary, which means that they add up to 180 degrees.
In this case, a linear pair is formed by two adjacent angles that are formed by the intersection of the transversal with each of the parallel lines.
Hence, angles 1 and 2 are supplementary, meaning that the sum of their measures is of 180º, hence the value of x is obtained as follows:
3x + 5 + 2x + 10 = 180
5x + 15 = 180
5x = 165
x = 33.
Then the measures are given as follows:
m < 1 = 3 x 33 + 5 = 104º.m < 2 = 2 x 33 + 10 = 76º.Missing InformationThe diagram that represents the situation is given by the image presented at the end of the answer.
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(3) Letpbe a prime and letG=Zp×be the set of non-zero elements ofZp. Show thatGis a group under modular multiplication. Use Lagrange's Theorem to prove Fermat's Little Theorem: Thatap−1≡1(modp)for any integeranot divisible byp
The set G = Zp × is a group under modular multiplication. This is because the set contains all the non-zero elements of Zp, and modular multiplication satisfies the four group properties:
Using Lagrange's Theorem, we can prove Fermat's Little Theorem: that for any integer a not divisible by p, ap-1 ≡ 1 (mod p). Let n be the smallest positive integer such that an ≡ 1 (mod p).
Since a is not divisible by p, the order of a (the smallest value of n) must divide p - 1, or in other words, n | (p - 1). By Lagrange's Theorem, this implies n = p - 1, which in turn implies that ap - 1 ≡ 1 (mod p). This proves Fermat's Little Theorem.
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A car going at 40km an hour takes 35 minutes to cover the distance between 2 villages. How long will it take if its speed is now 50km an hour?
Answer: 25 minutes
Step-by-step explanation:
40km an hour = 35 min
45km an hour = 30 min
50km an hour = 25 min
It takes 25 minutes to drive the distance between 2 villages.
x^{4}-\frac{1}{3} x^{3}+ 2\sqrt{x} -5
Answer: X^4 - (1/3)X^3 + 2\sqrt{X} - 5 = 0
Step-by-step explanation:
The answer is:
X^4 - (1/3)X^3 + 2\sqrt{X} - 5 = 0
This is a quartic equation, meaning it has four terms with the highest degree of X being 4. It cannot be solved in terms of radicals, so an approximate solution must be used. This can be done through numerical methods, such as the Newton-Raphson method, which uses an iterative process to approximate a solution.