Answer:
x = √17
y = 10.1
Step-by-step explanation:
x² + 8² = 9²
x² = 81 - 64 = 17
x = √17
sin∅ = √17/9
∅ = 27.27°
9/y = cos(27.27)
y = 9/cos(27.27) = 10.13
y = 10.1
The function g is related to one of the parent functions
g(x) = x^2 – 3
The parent function is:
f(x)= x^2
Use function notation to write g in terms of f.
We can write g in terms of f as: g(x) = f(x) - 3 = x² - 3
What is function?
In mathematics, a function is a relationship between two sets of elements, called the domain and the range, such that each element in the domain is associated with a unique element in the range. In simpler terms, a function is a set of rules that takes an input value and produces a corresponding output value.
To write g in terms of f, we can use function composition, which involves plugging the function f(x) into g(x) wherever we see x.
So, we have:
g(x) = f(x) - 3
where f(x) = x².
Substituting f(x) into g(x), we get:
g(x) = (x²) - 3
Therefore, we can write g in terms of f as:
g(x) = f(x) - 3 = x² - 3.
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The test scores of students on a science test are normally distributed with an average score of 78 and a standard deviation of 4. Which statement is true? Responses About 16% of the students scored 74 or below. About 16 percent of the students scored 74 or below. About 32% of the students scored 82 or above. About 32 percent of the students scored 82 or above. About 50% of the students scored 74 or above. About 50 percent of the students scored 74 or above. About 68% of the students scored 82 or below.
The statement "About 16% of the students scored 74 or below" is true for the given normal distribution of test scores with an average of 78 and a standard deviation of 4.
What is standard deviation?Standard deviation is a measure of the amount of variation or dispersion of a set of values from their mean (average) value. It is calculated as the square root of the variance of a set of data. A smaller standard deviation indicates that the values are tightly clustered around the mean, while a larger standard deviation indicates that the values are more spread out.
In the given question,
About 16% of the students scored 74 or below is true. This is because of the empirical rule for normal distributions, which states that approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. So, if the average score is 78 and the standard deviation is 4, a score of 74 falls one standard deviation below the mean, which represents about 16% of the total data.
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Answer:
About 97.5% of the students scored 86 or below.
Step-by-step explanation:
A company is going to make an oil container in the shape of a cylinder. As shown below, the container will have a height of 8 m and a diameter of 10 m. The container will be made from steel (including its top and bottom). Suppose the total cost of the steel will be $13,062.40. How much will the steel cost per square meter? Use 3.14 for it, and do not round your answer.
The per square meter cost of steel will be $32. The solution has been obtained by using the cylinder.
What is a cylinder?
The cylinder, one of the most basic curvilinear geometric shapes, has long been considered to be a three-dimensional solid. It is regarded as a prism with a circle as its basis in elementary geometry.
We are given that the height of cylinder is 8 m and diameter is 10 m.
So, the radius is 5 m.
Now, using the surface area formula, we get
⇒ S = 2πrh + 2π[tex]r^{2}[/tex]
⇒ S = 2π * 5 * 8 + 2π * [tex]5^{2}[/tex]
⇒ S = 2 * 3.14 * 5 * 8 + 2 * 3.14 * 25
⇒ S = 251.2 + 157
⇒ S = 408.2 square meter
Now, it is given that the total cost of the steel will be $13,062.40.
So, per square meter cost will be:
⇒ Cost = [tex]\frac{13,062.40}{408.2\\}[/tex]
⇒ Cost = $32
Hence, the per square meter cost of steel for the cylinder will be $32.
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3. How many ways can you line up 7 books on a shelf?
Answer:
There are 5040 different ways to arrange the 7 books on a shelf.------------------------------
Use the formula for permutations:
n! = n × (n - 1) × (n - 2) × ... × 1, where n is the number of objects and ! denotes a factorial.The number of objects is n = 7 books.
Calculate the factorial:
7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040NASA is conducting an experiment to find out the fraction of people who black out at G forces greater than 6
. In an earlier study, the population proportion was estimated to be 0.33
.
How large a sample would be required in order to estimate the fraction of people who black out at 6
or more Gs at the 95%
confidence level with an error of at most 0.03
? Round your answer up to the next integer.
We need to select at least 754 people for the experiment to estimate the fraction of people who black out at G forces greater than 6 with an error of at most 0.03 and a 95% confidence level.
How to determine how large a sample would be required in order to estimate the fraction of people who black out at 6or more Gs
We can use the formula for the sample size required to estimate a population proportion with a specified margin of error and confidence level:
n = (Z^2 * p * (1-p)) / E^2
where:
Z = the z-score corresponding to the desired confidence level (95% confidence level corresponds to a z-score of 1.96)
p = the estimated population proportion from the earlier study
E = the desired margin of error
Substituting the given values, we get:
n = (1.96^2 * 0.33 * (1-0.33)) / 0.03^2
n = 753.69
Rounding up to the next integer, we need a sample size of 754. Therefore, we need to select at least 754 people for the experiment to estimate the fraction of people who black out at G forces greater than 6 with an error of at most 0.03 and a 95% confidence level.
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1. You go to the ice cream shop with your friends and you can choose an ice cream, a topping
and sprinkles. How many different sundaes can you make when you order one flavor of ice
cream, one topping and one color of sprinkles from the chart below? Show all possible
outcomes in a tree diagram.
Ice Cream
Chocolate
Vanilla
Strawberry
Topping
Fudge
Marshmallow
Sprinkles
Chocolate
Rainbow
How many sample spaces are there? HINT: How many possible combinations?
b. P (Chocolate, Fudge, Rainbow)
Answer:
Step-by-step explanation:
Answer:
You can make 12 possible sundaes with these toppings.
Step-by-step explanation:
Chocolate, Vanilla, and Strawberry all have 4 possible outcomes:
1. Fudge & Chocolate Sprinkles
2. Fudge & Rainbow Sprinkles
3. Marshmallow & Chocolate Sprinkles
4. Marshmallow & Rainbow Sprinkles
-------------------------------------------------------------------------------------------------------------
4 x 3 will equal 12, the total possible sundaes you can make with these toppings and ice cream flavors.
the equation of a parabola is f(x)=x^2-4x-5
the axis of symmetry is x = ?
the vertex of the parabola is (?,?)
A savings account earns an interest rate of 1.625 % compounded quarterly for an account with an initial deposit of $ 2 , 225 . How much money will be in the account after 6 years? Round your answer to the nearest hundredth.
The balance in the account after 6 years is $2,466.69.
What is compound interest?
Compound interest is when you earn interest on both the money you've saved and the interest you earn.
We can use the formula for compound interest to find the balance in the savings account after 6 years:
[tex]A = P(1 + r/n)^{(nt)}[/tex]
where:
A is the balance in the account after t years
P is the initial deposit, which is $2,225
r is the annual interest rate, which is 1.625%
n is the number of times the interest is compounded per year, which is 4 (since it's compounded quarterly)
t is the number of years, which is 6
Plugging in these values, we get:
[tex]A = 2,225(1 + 0.01625/4)^{(4*6)}\\A = 2,225(1.0040625)^{24}[/tex]
A = 2,225(1.108227)
A = 2,466.69
Hence, the balance in the account after 6 years is $2,466.69.
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6. Two out of every five Canadians read at least 10 books a year. What percent of Canadians read at least 10 books a year?
Answer:
40%
Step-by-step explanation:
2/5=x/100
cross multiply and you get 5x=200
by isolation the variable you will get x=40
therefore, 40% of Canadians read at least 10 books a year
Can someone help me with this problem, and explain how to solve it?
Thus, the height of flagpole from the ground is found as 23.34 feet.
Explain about the angle of elevation:The measurement of the angle between a person's eyes' line of sight to anything above and the horizontal line is known as the angle of elevation.
The movement of the observer's eyes determines the elevation angle. The angle of elevation is the angle formed by the line of sight and the horizontal line when a viewer is looking up at an object.
Given data:
height of boy = 5 ftDistance of boy from flagpole = 30 ft angle of elevation = 35 degreesLet the height of pole above the boy be 'h'feet.Using the trigonometric ratios in the right triangle ABE.
tan 35 = h / 30
h = 30* tan(35)
h = 18.34
Height of flagpole = 18.34 + 5 = 23.34 feet
Thus, the height of the flagpole from the ground is found as 23.34 feet.
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Pls help with this question
The plane degrees off course is 6.79 degrees off at a speed of 541. 92 km / hr relative to the ground.
How to find the degrees off course ?To find the airplane's course deviation and its ground speed, we need to first determine the velocity vectors of the airplane and the wind.
W x = W x cos(225) = 90 x cos(225) = -63.64 km/hr
Wy = W x sin(225) = 90 x sin(225) = -63.64 km/hr
Magnitude: |R| = sqrt(Rx^2 + Ry^2) = √(536.36^2 + (-63.64)^2) = 541.92 km/hr
Direction: θ = arctan(Ry / Rx) ≈ arctan (-63.64 / 536.36) = -6.79 degrees
So, the airplane is 6.79 degrees off course, and its ground speed is 541.92 km/hr.
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i have 60 square feet of paper to wrap a box in the shape of a right rectangular prism the height of the box is 1 2/3 feet the width is 4 feet and the length is 5 1/2 feet what percent of the box will remain unwrapped if i use all the paper i has available
The percent of the box that will remain unwrapped by the 60 square feet of paper is about 20.70%
What is a percentage?A percentage is a proportion of a number expressed as fraction of a 100
The dimensions in mixed fractions can be expressed as follows;
Height of the box = 1 2/3 feet = 5/3 feet
Width of the box = 4 feet
Length of the box = 5 1/2 feet = 11/2 feet
Surface area of the box, A = 2 × (5/3) × 4 + 2 × 4 × 11/2 + 2 × (5/3) × 11/2 = 227/3 = 75 2/3 square inches
The percent of the box that will remain unwrapped is therefore;
Percentage = 60/(227/3) × 100 ≈ 79.3%
The percentage of the box that will remain unwrapped is about (100 - 79.3)% = 20.70%
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Please help will mark Brainliest
Step-by-step explanation:
5 cos (307) = 3
5 sin (307) = -4
just plot the point ( 3, -4 ) Done.
Answer:
The point is (3, -4) because 5 cos [307] = 3 and 5 sin [307] = -4
Landon and Maria are meeting at the library to work on their history project. Maria walks 9 blocks east and 3 blocks north to get to the library from her house. Landon walks 5 blocks south and 7 blocks west to get to the library from his house. The map below shows the location of the library and Landon's and Maria's houses. To the nearest block, how far is Landon's house from Maria's house if Maria could walk in a straight line?
To the nearest block, Landon's house is at distance of 12 blocks away from Maria's house if Maria could walk in a straight line.
What is Pythagoras theorem?A basic mathematical theorem relating to the sides of a right-angled triangle is known as Pythagoras' theorem. The square of the length of the hypotenuse, the side that faces the right angle, is said to be equal to the sum of the squares of the lengths of the other two sides, known as the legs, in a right triangle.
This can be written in mathematical notation as:
c² = a² + b²
where c is the length of the hypotenuse, and a and b are the lengths of the legs of the right triangle.
In this case, we can consider the straight line between Maria's house and Landon's house as the hypotenuse of a right triangle, with the distances they walked as the other two sides. We can use the distance formula to find the lengths of those sides:
Distance walked by Maria = √(9² + 3²) = √90 ≈ 9.49 blocks
Distance walked by Landon = √(5² + 7²) = √74 ≈ 8.60 blocks
Now we can use the Pythagorean theorem to find the distance between their houses:
Distance between houses = √(9.49² + 8.60²) ≈ 12.46 blocks
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Hurry help Greg plants a seed and as soon as the plant sprouts, he measures its height each day and records his data for two weeks. If the plant continues to grow the whole two weeks, what would a line graph of Greg's data look like?
It would be flat.
It would move downward.
It would go up and down.
It would move upward.
The line graph of Greg's data wood look like "It would move upward."
What is graph?A graph is a visual representation of data that shows the relationship between two or more variables. There are several types of graphs that can be used to display data
If Greg is measuring the height of the plant each day and the plant is continuing to grow for two weeks, then the line graph of his data would move upward. The graph would show an increasing trend as the plant grows taller each day. Therefore, the correct answer is "It would move upward."
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Cylinder M and cylinder N are similar. The radius of cylinder N is equal to its height, and the ratio of the height of cylinder N to the height of cylinder M is 5: 3. The surface area of cylinder N is 256 square feet greater than the surface area of cylinder M. Find the surface area of each cylinder.
The surface area of cylinder M is approximately 131.95 square feet and the surface area of cylinder N is approximately 319.77 square feet.
What is a cylinder?
A cylinder is a three-dimensional geometric shape that consists of two congruent parallel bases in the shape of circles or ellipses, and a curved surface that connects the bases. The height of a cylinder is the perpendicular distance between the bases. A cylinder is a type of prism, and it can be classified as either a right cylinder or an oblique cylinder depending on whether or not its axis is perpendicular to its bases. Right cylinders have circular bases and their axis is perpendicular to the bases, while oblique cylinders have elliptical bases and their axis is not perpendicular to the bases.
Now,
Let the radius of cylinder M be r and its height be h. Then, the radius and height of cylinder N are both 2r, since the radius is equal to the height.
Since the cylinders are similar, their dimensions are proportional, which means:
(height of N) / (height of M) = 5/3
(radius of N) / (radius of M) = (2r) / r = 2
Using the formula for the surface area of a cylinder, we can write:
Surface area of cylinder M: 2πr² + 2πrh
Surface area of cylinder N: 2π(2r)² + 2π(2r)(5/3)h
We are told that the surface area of cylinder N is 256 square feet greater than the surface area of cylinder M. So we can set up the equation:
2π(2r)² + 2π(2r)(5/3)h = 2πr² + 2πrh + 256
Simplifying and solving for h, we get:
4r² + 20rh/3 = r² + rh + 128
3r² - rh - 128 = 0
(3r + 32)(r - 4) = 0
Since the height of the cylinder cannot be negative, we take the positive solution r = 4. Then, the height of cylinder M is (3/5)(4) = 12/5, and the height of cylinder N is 2(4) = 8.
Using the formulas for surface area, we can find the surface areas of both cylinders:
Surface area of cylinder M: 2π(4)² + 2π(4)(12/5) = 131.95 square feet
Surface area of cylinder N: 2π(2(4))² + 2π(2(4))(8) = 319.77 square feet
Therefore, the surface area of cylinder M is approximately 131.95 square feet and the surface area of cylinder N is approximately 319.77 square feet.
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The diameter of a circle is 5 centimeters. What’s the radius? Give the exact answer in simplest form
Answer:
2.5cm
Step-by-step explanation:
you half the diameter to get the radius
Find the area of a shaded region
Answer:
40 cm²
Step-by-step explanation:
you have to multiply the top of the triangle by the base(40) which is 80 then you divide by two since it's a triangle then you get 40 again since your finding the area you have to put the 2 above the cm
(01.08)
Let f(x) = 3x² + x - 3 and g(x) = x² - 5x +
1. Find f(x) - g(x).
Answer:
f(x) - g(x) = 2x² + 6x - 4
Step-by-step explanation:
f(x) - g(x)
= 3x² + x - 3 - (x² - 5x + 1) ← distribute parenthesis by - 1
= 3x² + x - 3 - x² + 5x - 1 ← collect like terms
= 2x² + 6x - 4
Michelle is building a car for the local soap box derby. The floor boards of her car need to be 2.2 meters long. If she has 13.2 meters of wood, how many floor boards can she make? 20 boardo
Michelle can make 6 floor boards for her car.
How many times does 2.2 meters fit into 13.2 meters?To determine how many floor boards Michelle can make, we need to divide the total length of wood she has (13.2 meters) by the length of each floor board (2.2 meters).
The number of floor which boards Michelle can make is:
= 13.2 meters / 2.2 meters/floor board
= 6 floor boards
Therefore, Michelle can make 6 floor boards for her car using the 13.2 meters of wood she has.
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1. All numbered streets runs parallel to each other. Both 3rd and 4th Streets are intersected by King Ave. as shown:
(a) Suppose a car is traveling east on 4th Street and turns onto King Avenue heading northeast. What is the measure of the angle created by the car's turning? Explain your answer.
(b) Suppose a car is traveling southwest on King Avenue and turns left onto 3rd Street. What is the measure of the angle created by the car's turning? Explain your answer.
(c) Suppose a car is traveling northeast on King Avenue and turns right onto 3rd Street. What is the measure of the angle created by the car's turning? Explain your answer.
When the car travels east on 4th Street and turns onto King Avenue heading northeast, the angle created by the car's turning is a 45-degree angle.
When the car travels southwest on King Avenue and turns left onto 3rd Street, the angle created by the car's turning is a 135-degree angle.
When the car travels northeast on King Avenue and turns right onto 3rd Street, the angle created by the car's turning is a 45-degree angle.
How to get the Angle?(a) When the car travels east on 4th Street and turns onto King Avenue heading northeast, the angle created by the car's turning is a 45-degree angle. This is because the intersection of 4th Street and King Avenue creates a right angle, and the car turns northeast, creating another 45-degree angle with the horizontal 4th Street.
(b) When the car travels southwest on King Avenue and turns left onto 3rd Street, the angle created by the car's turning is a 135-degree angle. This is because the intersection of King Avenue and 3rd Street creates a right angle, and the car turns left, creating an additional 90-degree angle. Therefore, the total angle is 90 degrees + 45 degrees = 135 degrees.
(c) When the car travels northeast on King Avenue and turns right onto 3rd Street, the angle created by the car's turning is a 45-degree angle. This is because the intersection of King Avenue and 3rd Street creates a right angle, and the car turns right, creating another 45-degree angle with the horizontal 3rd Street.
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Hello, I am confused on how to answer this question.
Answer: x=14
Step-by-step explanation:
cb=10.
ac=14
The value of X will be 14.
This is a simple mathematics problem that can be solved by using ratios.
Given, AC : BC : AB = 7 : 5 : 8
AC = X ; BC = 10 ; AB = X + 2
AC/ BC = X/ 10 = 7/5 (Ratios can also be represented as fractions)
X/10 = 7/5
On Transposing,
5X = 70
X = 70/5
X = 14
Alternatively,
BC/AB = 5/8 = 10/ (X + 2)
5/8 = 10/ (X + 2)
On Transposing,
80 = 5X + 10
5X = 70
X = 14
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3√b in exponential form
Determine which relation is a function.
A: {(–3, 2), (–1, 3), (–1, 2), (0, 4), (1, 1)}
B: {(–3, 2), (–2, 3), (–1, 1), (0, 4), (0, 1)}
C: {(–3, 3), (–2, 3), (–1, 1), (0, 4), (0, 1)}
D: {(–3, 2), (–2, 3), (–1, 2), (0, 4), (1, 1)}
The relation that is a function is D: {(–3, 2), (–2, 3), (–1, 2), (0, 4), (1, 1)}.
What is a relation?A function is a relation between a set of inputs and a set of possible outputs with the property that each input is related to exactly one output. In other words, for a relation to be a function, each input can only be related to one output.
In relation D, each input (x-value) has a unique output (y-value), meaning that each x-value is only paired with one y-value. In contrast, relations A, B, and C have repeated x-values with different y-values, which means that they are not functions.
In relation A, the input -1 is paired with two different outputs (2 and 3). In relation B, both inputs 0 and -1 are each paired with two different outputs. In relation C, both inputs 0 and -3 are each paired with two different outputs. Therefore, only relation D satisfies the definition of a function.
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A rectangular solid has edges that are 10, 8, and 3 units long.
1. Draw the solid, showing the 10 x 8 face as the base. Find:
a. The lateral area
b. The total area
c. The volume
2. Repeat Exercise 1, but show the 10 x 3 face as the base.
a.
b.
c.
We are adjusting the original rectangular solid and here is what we have:
1. Rectangular solid showing 10x8 face as the base(a) The lateral area is the sum of the areas of the four sides, which are all rectangles. The two sides with dimensions 8 x 3 have an area of 8 x 3 = 24 square units each, and the two sides with dimensions 10 x 3 have an area of 10 x 3 = 30 square units each. Therefore, the lateral area is:
2(24) + 2(30) = 48 + 60 = 108 square units
(b) The total area is the sum of the areas of all six faces. The two faces with dimensions 10 x 8 have an area of 10 x 8 = 80 square units each, the two faces with dimensions 8 x 3 have an area of 8 x 3 = 24 square units each, and the two faces with dimensions 10 x 3 have an area of 10 x 3 = 30 square units each. Therefore, the total area is:
2(80) + 2(24) + 2(30) = 160 + 48 + 60 = 268 square units
(c) The volume is the product of the length, width, and height of the rectangular solid. Therefore, the volume is:
10 x 8 x 3 = 240 cubic units
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^4√p7
in exponential form.
Answer:1111
Step-by-step explanation:
In the coordinate plane, a square has vertices (4, 3), (-3, 3), (-3, - 4). What is the location of the fourth vertex?
The two possible locations for the fourth vertex are (4, -4) and (-3, -1).
What is quadratic equation?
it's a second-degree quadratic equation which is an algebraic equation in x. Ax2 + bx + c = 0, where a and b are the coefficients, x is the variable, and c is the constant term, is the quadratic equation in its standard form.
To find the location of the fourth vertex of the square, we need to use the fact that a square has four equal sides and four right angles.
The first two vertices given are (4, 3) and (-3, 3), which lie on a horizontal line segment of length 7.
The third vertex is (-3, -4), which is 7 units away from the first two vertices and lies on a vertical line segment.
Since the square has four equal sides, the distance between the third vertex and the fourth vertex must also be 7 units.
And since the square has four right angles, the fourth vertex must be located on a vertical line passing through the first two vertices or on a horizontal line passing through the third vertex.
So, there are two possible locations for the fourth vertex:
(4, -4): This point is located 7 units below the first vertex (4, 3) on the vertical line passing through it.
(-3, -1): This point is located 7 units to the right of the third vertex (-3, -4) on the horizontal line passing through it.
Therefore, the two possible locations for the fourth vertex are (4, -4) and (-3, -1).
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If P(A)=0.3, P(B)=0.2 and P(A∩B)=0.2 determine the following probabilities:
(a) P(A')
(b) P(A∪B)
(c) P(A'∩B)
(d) P(A∩B')
(e) P[(A∪B)']
The probabilities are as follows:
(a) P(A')= 0.7
(b) P(A∪B) = 0.3
(c) P(A'∩B) = 0
(d) P(A∩B') = 0.1
(e) P[(A∪B)'] = 0.7
What is probability?
Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to happen.
(a) P(A') = 1 - P(A) = 1 - 0.3 = 0.7
(b) P(A∪B) = P(A) + P(B) - P(A∩B) = 0.3 + 0.2 - 0.2 = 0.3
(c) P(A'∩B) = P(B) - P(A∩B) = 0.2 - 0.2 = 0
(d) P(A∩B') = P(A) - P(A∩B) = 0.3 - 0.2 = 0.1
(e) P[(A∪B)'] = 1 - P(A∪B) = 1 - 0.3 = 0.7
Note: P(A) represents the probability of event A occurring, P(B) represents the probability of event B occurring, and P(A∩B) represents the probability of both events A and B occurring simultaneously. The symbol '∪' represents the union of two events, and the symbol '∩' represents the intersection of two events. The complement of an event A is represented by A'.
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coordinate plane with points at A 0 comma 2 and B 2 comma 0 intersected by line f Dilate line f by a scale factor of one half with the center of dilation at the origin to create line f′. Where are points A′ and B′ located after dilation, and how are lines f and f′ related? The locations of A′ and B′ are A′ (0, 2) and B′ (0, 0); lines f and f′ intersect at point A. The locations of A′ and B′ are A′ (0, 1) and B′ (1, 0); lines f and f′ are parallel. The locations of A′ and B′ are A′ (0, 0) and B′ (2, 0); lines f and f′ intersect at point B. The locations of A′ and B′ are A′ (0, 2) and B′ (2, 0); lines f and f′ are the same line.
The answer of the given question based on the graph is , The locations of A′ and B′ are A′ (0, 1) and B′ (1, 0); lines f and f′ are parallel.
What is Scale factor?A scale factor is a number that scales, or multiplies, a quantity by some factor. It is used in mathematics to describe the relationship between corresponding measurements of two similar figures, such as triangles or rectangles.
To dilate line f by scale factor of one half with center of dilation at origin, we multiply coordinates of each point on line f by 1/2.
The equation of line f can be found by using the points A and B:
slope of line f = (0 - 2)/(2 - 0) = -1
y-intercept of line f = 2
Therefore, the equation of line f is y = -x + 2.
To find the coordinates of A' and B' after dilation, we can apply the dilation factor to each point:
A' = (0, 2)*1/2 =(0, 1)
B' = (2, 0)*1/2 =(1, 0)
So A' is located at (0, 1) and B' is located at (1, 0) after dilation.
Now let's analyze the relationship between lines f and f'. The dilation was centered at the origin, so the origin is a fixed point of the dilation. This means that the point where lines f and f' intersect must be the origin.
If we plug in x = 0 into the equation of line f, we get y = 2. This means that point A is located at (0, 2) and intersects with line f at y = 2. After dilation, point A' is located at (0, 1), which means that lines f and f' intersect at point A.
To determine the relationship between lines f and f', we can compare their equations. The equation of f' can be found by using the points A' and B':
slope of f' = (0 - 1)/(1 - 0) = -1
y-intercept of f' = 0
Therefore, the equation of f' is y = -x.
Comparing the equations of f and f', we can see that they have the same slope of -1, which means they are parallel. Therefore, the correct answer is: The locations of A′ and B′ are A′ (0, 1) and B′ (1, 0); lines f and f′ are parallel.
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. (Adapted from Terence Blows, Northern Arizona University.) Classify as in Exercise 6 but for the following circumstances. a. The amount of caffeine in the bloodstream decreases by 50% every 5 hours or so after stopping drinking coffee. b. The amount of trash in a landfill increases by 350 tons per week. c. The amount of alcohol in the bloodstream decreases by 10 grams (the amount in a standard drink) per hour after stopping drinking. d. Your age increases every day.
The statements are classified as Exponential decay and Linear growth.
What are the classification of the statements?
a. Exponential decay: The amount of caffeine in the bloodstream decreases by 50% every 5 hours, which indicates an exponential decay process where the quantity decreases at a constant percentage rate over time.
b. Linear growth: The amount of trash in a landfill increases by 350 tons per week, which indicates a linear growth process where the quantity increases by a constant amount over time.
c. Exponential decay: The amount of alcohol in the bloodstream decreases by 10 grams (the amount in a standard drink) per hour after stopping drinking, which indicates an exponential decay process where the quantity decreases at a constant percentage rate over time.
d. Linear growth: Your age increases every day, which indicates a linear growth process where the quantity (age) increases by a constant amount (1 day) over time.
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