a) The value of k is 0.04.
b) The value of E[X] is 3.3333 and the value of var[X] is 1.3889.
c) The standard deviation of G is 5.8916.
d) The standard deviation of H is 7.0711.
(a) To find the value of k that ensures that f(x) is a proper density function, we need to ensure that the integral of f(x) over its domain is equal to 1:
∫05 (0.1 + kx) dx = 1
0.5 + 12.5k = 1
12.5k = 0.5
k = 0.04
Therefore, the value of k is 0.04.
(b) To find E[X], we need to evaluate the integral of x*f(x) over its domain:
E[X] = ∫05 x(0.1 + 0.04x) dx
E[X] = 0.5 + 0.02(125/3) = 3.3333
To find var[X], we need to evaluate the integral of (x - E[X])2*f(x) over its domain:
var[X] = ∫05 (x - 3.3333)2(0.1 + 0.04x) dx = 1.3889
(c) If G = 5X - 6, then E[G] = 5E[X] - 6 = 11.6667 and var[G] = 52var[X] = 34.7225. The standard deviation of G is the square root of var[G], which is 5.8916.
(d) If H = 5 - 6X, then E[H] = 5 - 6E[X] = -14.9998 and var[H] = 62var[X] = 49.9994. The standard deviation of H is the square root of var[H], which is 7.0711.
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Today, the sum of Caroline's age and Cameron's age is 73. 5 years ago Cameron was 2 times older than Caroline.
The answer of Caroline is 19.33 years old and Cameron is 53.67 years old
Let's start by assigning variables to represent Caroline's age and Cameron's age. We'll use C for Caroline's age and M for Cameron's age.
According to the problem, the sum of their ages today is 73, so we can write the equation:
C + M = 73
Five years ago, Cameron was 2 times older than Caroline. so we can write the equation:
M - 5 = 2(C - 5)
Now we can use the first equation to solve for one of the variables. Let's solve for C:
C = 73 - M
Next, we can substitute this value of C into the second equation:
M - 5 = 2(73 - M - 5)
Simplifying the equation gives us:
M - 5 = 146 - 2M - 10
3M = 161
M = 53.67
Now we can use this value of M to find C:
C = 73 - 53.67
C = 19.33
So Caroline is 19.33 years old and Cameron is 53.67 years old.
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Pls help a brother out.
Where is your question so we may help
Answer:
Step-by-step explanation:
mark me brainliest so i can solve
If one south African rand is valued at 0,125 of one euro, one south African rand will be valued at what fraction to th euro? Can you calculate whay one Euro will cost in rands
The fraction of the euro is 8/100
one Euro will cost in 8 rands
When we talk about exchange rates, we're essentially talking about the value of one currency compared to another. In this case, we're comparing the South African rand to the euro.
We know that one South African rand is valued at 0.125 of one euro. To figure out what fraction of the euro one South African rand is worth, we can simply divide the value of one rand by the value of one euro:
1 rand ÷ 0.125 euro = 8/100 euro
So, one South African rand is worth 8/100 or 0.08 (which is equivalent to 8%) of one euro.
To calculate what one euro would cost in rands, we can use the inverse of the exchange rate we were given:
1 euro ÷ 0.125 rand/euro = 8 rand
So, one euro would cost 8 South African rand.
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anyone got Halloween drawing ideas? I know it's not time yet, but I love it lol
Answer:
Skeleton hand holding a piece of candy maybe?
Or a jack-o-lantern but its a poison apple.
OR a drawing of a witch's face in the reflection of a cauldron.
Step-by-step explanation:
Spooky Graveyard: Draw a creepy graveyard scene with tombstones, skeletons, and ghosts.
Jack-O-Lanterns: Draw a bunch of different jack-o-lanterns with different expressions and designs.
Haunted House: Draw a haunted house with creaky doors, broken windows, and bats flying around.
Witch's Cauldron: Draw a witch's cauldron bubbling with potions and ingredients.
Trick or Treaters: Draw a group of trick or treaters dressed up in costumes going from house to house.
Scary Trees: Draw a spooky forest with twisted, gnarled trees that look like they could come to life at any moment.
Monster Mash: Draw a bunch of different classic Halloween monsters, like Frankenstein's monster, vampires, and werewolves.
Zombie Apocalypse: Draw a post-apocalyptic scene with zombies wandering around and survivors trying to stay alive.
Black Cats: Draw some cute or creepy black cats with glowing eyes.
Halloween Candy: Draw a big pile of Halloween candy with all your favorite treats.
A pumpkin: Draw a simple pumpkin shape and add eyes, nose, and mouth. You can also add a stem on top and some vines on the sides.
Ghosts: Draw some simple white shapes for ghosts and add eyes and a mouth. You can also draw a white sheet around the ghost to make it look more spooky.
Skeletons: Draw a basic skeleton shape and add some bones. You can also add some spooky decorations, like spider webs or bats.
Witch hats: Draw a simple witch hat shape and add some details, like a buckle or a spider. You can also add some stars or a crescent moon in the background.
Bats: Draw a bat shape and add some details, like wings and ears. You can also draw a moon or some stars in the background to make it look more Halloween-like.
Can someone PLEASE help me with this three part question? It’s due today!! I need help ASAP
1. The Formula used SA = 2w (l +h) + lh
2. The box is 0.37125 inch deep.
3. The SA of box with lid is 294.7972 inch².
What is Surface Area?The area is the territory covered by a shape or figure, whereas the perimeter is the distance covered by the shape's outside boundary. The unit of area is the square unit or unit², while the unit of perimeter is the unit.
Given:
length = 13.2 inch
Height = 10.5 inch
And, Surface Area used = 295.02 inch²
Using the Formula
SA = 2lw + 2wh + lh
SA = 2w (l +h) + lh
295. 02 = 2w ( 13.2 + 10.5) + (13.2)(10.5)
147.51 = w x 24 + 138.6
8.91 = 24w
w = 0.37125 inch
SA of box with lid
= 2( lw+ wh + lh)
= 2( 13.2 x 10.5 + 10.5 x 0.37125 + 0.37125 x 13.2)
= 2(138.6 + 3.8981 + 4.9005)
= 2 x 147.3986
= 294.7972 inch²
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On a coordinate plane, how are the locations of the points (9 , -5) and (9 , 5) related?
A.
locations unrelated
B.
reflection across the x-axis
C.
reflection across the y-axis
D.
reflection across both axes
Option B, On a coordinate plane, the locations of the points (9, -5) and (9, 5) related reflection across the x-axis.
The two locations (9, -5) and (9, 5) have the same x-coordinate, but their y-coordinates are different. We need to see how the coordinates of two points change as they undergo various transformations to understand their relationship.
If points are not linked, their coordinates cannot become one because they have no relation to each other. The x coordinates of the points remain the same, but if they are reflected on the x-axis, their y coordinates are reversed. The point (9, -5) in this case will be reflected to (9, 5) and vice versa.The y coordinates of these points remain the same, but if they are reflected on the y-axis, their x coordinates are reversed. In this case, point (9, -5) will reflect towards (-9, -5) and point (9, 5) will reflect towards (-9, 5).Negative if the x and y coordinates of the point are reflected on both the x-axis and the y-axis. In this case, point (9, -5) will reflect towards (-9, 5) and point (9, 5) will reflect towards (-9, -5).Learn more about the coordinate plane at
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HELP WORTH 10 POINTS
The picture shows the top view of a piece of glass.
A rectangular piece of glass is shown. The length is measured as 4 feet. The width is measured as 2 and one-half feet.
Which equations can be used to find the area, in square feet, of the piece of glass? Select all that apply.
A.
A
=
2
1
2
×
4
B.
A
=
5
2
+
4
C.
A
=
(
2
1
2
+
2
1
2
)
+
(
4
+
4
)
D.
A
=
5
2
×
4
E.
A
=
(
2
×
2
1
2
)
+
(
2
×
4
)
F.
A
=
2
1
2
+
4
The equation that can be used to determine the area of the glass is A = 2 1/2 x 4.
What is the equation that can be used to determine the area of the glass?A rectangle is a 2-dimensional quadrilateral with four right angles. A rectangle has two diagonals of equal length which bisect each other. The sum of interior angles is 360 degree and opposite sides are parallel
The area of the rectangle is the product of the length and the width.
Area of a rectangle = length x width
A = 4 x 2 1/2
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29.0 Assessment Practice 17. Camille drew the figure shown at the right. PART A Find the perimeter of the figure. Use 3.14 for T. Round to the nearest hundredth. 51.39] P=3.C semicircle a g › 9 (semicircle=1/2πT P=3. // π1.929 p= 3·2·3·14.9+9 пляда P = S1-31 PART B Draw another figure that has the same perimeter as the given figure. 9 ft 9
The figure's perimeter, rοunded tο the nearest hundredth, is rοughly 51.39 feet.
What is perimeter?The tοtal length οf a twο-dimensiοnal shape's bοundary οr οuter edge is knοwn as its perimeter. It is the space encircling the periphery οf a shape. Yοu add up the lengths οf all a shape's sides tο determine its perimeter. The length units used tο measure the sides have an impact οn the perimeter measurement units.
given:
The lengths οf all the edges must be added up in οrder tο determine the figure's perimeter.
The figure is made up οf a rectangle with a length οf 3 feet and a breadth οf 9 feet, twο semicircles with a diameter οf 9 feet each, and twο semicircles.
A semicircle with a diameter οf 9 feet has the fοllοwing circumference:
C = 1/2 * pi * d
C = 1/2 * 3.14 * 9
C = 14.13 feet
The circumference οf bοth semicircles taken tοgether is thus:
P = 2*14.13P = 28.26 feet
The rectangle's perimeter is as fοllοws:
P(rectangle) is equal tο 2 * (length + width).
P(rectangle) = 3 + 9 * 2
24 feet P(rectangle)
As a result, the figure's οverall perimeter is as fοllοws:
Semicircles: P = P + P (rectangle)
P = 28.26 + 24
P = 52.26 feet
The figure's perimeter, rοunded tο the nearest hundredth, is rοughly 51.39 feet.
Anοther figure with the same perimeter as the οne given can be drawn in a variety οf ways.
The figure's perimeter, rοunded tο the nearest hundredth, is rοughly 51.39 feet.
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Suppose that $$18,000 is deposited for five years at 5% APR. Calculate the interest earned if interest is compounded semiannually. Round your answer to the nearest cent.
The interest earned on a deposit of $18,000 for five years at 5% APR compounded semiannually.
To calculate the interest earned on a deposit of $18,000 for five years at 5% APR compounded semiannually, we will use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = final amount
P = principal amount (initial investment)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years
Plugging in the given values:
A = 18,000(1 + 0.05/2)^(2*5)
A = 18,000(1.025)^10
A = 23,386.28
To find the interest earned, we subtract the initial investment from the final amount:
Interest earned = A - P
Interest earned = 23,386.28 - 18,000
Interest earned = $5,386.28
Therefore, the interest earned on a deposit of $18,000 for five years at 5% APR compounded semiannually is $5,386.28.
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the cost of a ticket to the circus is $15 for children and $40 for adults. on a certain day ,attendance at the circus was 1,000 and the total gate revenue was $30,000. How many children and how many adults bought tickets?
400 children and 600 adults bought tickets to the circus.
Let's call the number of children who bought tickets "C" and the number of adults who bought tickets "A". We can set up two equations to represent the information given in the question:
C + A = 1000 (the total number of people who bought tickets)
15C + 40A = 30000 (the total gate revenue)
We can use the first equation to solve for one of the variables in terms of the other. For example, we can solve for C in terms of A:
C = 1000 - A
Now we can substitute this equation into the second equation to solve for A:
15(1000 - A) + 40A = 30000
15000 - 15A + 40A = 30000
25A = 15000
A = 600
So 600 adults bought tickets. We can use the first equation to find the number of children who bought tickets:
C + 600 = 1000
C = 400
So 400 children bought tickets. Therefore, the answer is that 400 children and 600 adults bought tickets to the circus.
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At 0°C, the volume of a gas is 22 liters. For each degree the temperature T
(in degrees Celsius) increases, the volume V
(in liters) of the gas increases by 225
. Write an equation that represents the volume of the gas in terms of the temperature.
The requried equation represents the volume of the gas in terms of temperature is V = 22 + 225T.
What are equation models?The equation model is defined as the representative of the given case in the form of an equation using variables and constants.
Here,
Let's denote the volume of the gas as V and temperature as T.
From the given information, we know that at 0°C temperature (i.e. T = 0), the volume of the gas is 22 liters.
For every 1°C increase in temperature, the volume of the gas increases by 225 liters.
Therefore, we can write the equation as:
V = 22 + 225T
This equation represents the volume of the gas in terms of temperature.
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Determine the number of solutions to the system of linear equations shown on the graph.
coordinate plane with one line that passes through the points negative 1 comma 4 and 0 comma 1 and another line that passes through the points 0 comma negative 1 and 2 comma negative 3
One solution at (1, −2)
One solution at (−2, 1)
Infinitely many solutions
No solution
The solution of the system of linear equations on the graph is one solution at (1, −2).
How to determine the solution of equations' graphs?Analyzing the intersection of the graphs of a system of linear equations on the coordinate plane will reveal how many solutions there are. The system has a single solution if the graphs meet at a single location. The system cannot be solved if the graphs are parallel and do not intersect. The system has an unlimited number of solutions if the graphs coincide, or overlap.
From the graph we observe that the two lines intersect at the point (1, -2).
The point where the lines intersect is thus the solution of the system of linear equations on the graph.
Hence, the solution of the system of linear equations on the graph is one solution at (1, −2).
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can someone please translate in whatever language u use and do the math task for me me because i am struggling
Answer:
the paper is too blurry i would help if i could.
Step-by-step explanation:
pls answer this :))
due in 10 mins
Step-by-step explanation:
a=55 because
75+50+a=180 (Sum of angle in a stratight line)
or, 125+a=180
or, a=180-125
a=55
b=75 because they are alternate angles
c=50 because they are alternate angles
d=50 Because d=c(vertically opposite angle) and c=50
Find the tangent of ZS.
T
68
60
Simplify your answer and write it
S
U
The tangent of <S is 8/15
What is Trigonometry?Trigonometry is a discipline of mathematics dealing with specific angle functions and their application to calculations. In trigonometry, there are six functions of an angle that are often utilised. Sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant are their names and acronyms (csc).
Given:
ST = 68
UT = 60
so, SU = √ST² - UT²
= √68² - 60²
= √4624 - 3600
= √1024
= 32
So, Tangent of <S = P/ B
tan S = SU / UT
tan S = 32 / 60
tan S = 8/15
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Dakota is able to drive his car 32.5 miles per gallon of gasoline. Write and solve an inequality that could be used to determine the minimum number of gallons of gasoline Dakota would need to drive 117 miles to his brother's house. Then interpret the solution. Explain your reasoning. Input Field 1 of 1 Skip to input field.
Answer:
Step-by-step explanation: first if you do x/32.5 ≤ 117 = 3802.5 and also we know that 1 mile= 32.5, so if you take 32.5 * 117 it also equals 3802.5
so it will take him a minimum of 3802.5 gallons to drive 117 miles
Dakota would need at least 3.6 gallons of gasoline to drive 117 miles to his brother's house.
What is the inequality?A statement that compares two numbers or expressions using an inequality symbol, such as (less than), > (greater than), (less than or equal to), or, is known as an inequality in mathematics (greater than or equal to).
Let x be the volume of gasoline required for Dakota to travel 117 miles to his brother's home. The minimum value of x can be calculated using the inequality shown below:
x ≥ 117 / 32.5
Here, we calculate the bare minimum amount of gasoline Dakota would require to travel 117 miles by dividing the whole distance (117 miles) by the miles per gallon (32.5 miles/gallon).
The inequality is reduced to: x ≥ 3.6
Dakota would therefore require 3.6 gallons of fuel to travel 117 miles to his brother's house.
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TIME REMAINING 17:48 The graph of f(x) = x2 is translated to form g(x) = (x – 2)2 – 3. On a coordinate plane, a parabola, labeled f of x, opens up. It goes through (negative 2, 4), has a vertex at (0, 0), and goes through (2, 4). Which graph represents g(x)? On a coordinate plane, a parabola opens up. It goes through (0, 1), has a vertex at (2, negative 3), and goes through (4, 1).
The graph of g(x) is a parabola that opens up, goes through (0, 1), has a vertex at (2, -3), and goes through (4, 1).
What is parabola ?
A parabola is a U-shaped curve that is formed by graphing a quadratic function. In other words, a parabola is the set of all points in a plane that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix).
The graph of g(x) can be obtained by translating the graph of f(x) = x^2 to the left by 2 units and down by 3 units.
The vertex of g(x) is obtained by subtracting 2 from the x-coordinate and subtracting 3 from the y-coordinate of the vertex of f(x). Thus, the vertex of g(x) is (2, -3).
The point (-2, 4) on f(x) is translated left by 2 units to (−4, 4) on g(x) and then down by 3 units to (−4, 1). The point (2, 4) on f(x) is translated left by 2 units to (0, 4) on g(x) and then down by 3 units to (0, 1).
Therefore, the graph of g(x) is a parabola that opens up, goes through (0, 1), has a vertex at (2, -3), and goes through (4, 1).
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Answer:
A
Step-by-step explanation:
HELP PLSS!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answer:
[tex]\frac{y + x}{y\y - x}[/tex]
Step-by-step explanation:
Helping in Jesus' name.
According to exponent rules, when we multiply the same base we __ the exponents
Answer: Add.
Step-by-step explanation: When you multiply two numbers, all you have to do is simply add the exponents!
Example: 3^4 × 3^2 = 3^(4 + 2) = 3^6
Hope this helps!!
The interior angles of the pentagon
he has drawn are all less than 180°.
Ben attempts to express the
interior angles of his pentagon
using algebra.
His expressions are
xᵒ, (x +40)°, (2x − 30)°,
3(x - 40) and 3xº
Show that Ben is incorrect.
Input note: include the angle sum
of a pentagon, the value of x and
the size of any angles that don't
meet the criteria set out in the
question.
Ben is incorrect because…
All of Ben's expressions are erroneous since none of them equal the number 108 degrees. The correct equation is (n-2) x 180 / n.
What is the equation of interior angle in a regular polygon?A polygon is a geometric object with two dimensions and a finite number of sides. A polygon's sides or edges are created by connecting end to end segments of a straight line to create a closed shape. Vertex or corners refers to the intersection of two line segments, which produces an angle.
In a regular polygon, where n is the number of sides, the formula for measuring an interior angle is (n-2) x 180 / n.
An internal angle in a pentagon has a measure of (5-2) x 180 / 5 = 108 degrees.
Substituting the angle in Ben's expression:
xᵒ + (x + 40)° + (2x - 30)° + 3(x - 40)° + 3x°
9x - 27° = 108
We observe that none of the value of x results in 108 degrees.
Hence, all of Ben's expressions are erroneous since none of them equal the number 108 degrees.
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Please solve it is due Thursday.
Answer:
A: x
Step-by-step explanation:
So with problems like this when it saids f(g(x) it wants you to plug those values in. In some cases you have a value where the x is but you don't so you can't solve g(x) so you move on to the next step. You plug in g(x)= x-1/2 everywhere there is a x in f(x). So our equation would be 2(x-1/2) + 1. Then you solve by canceling the 2 by dividing it out so we would have x-1 +1. Then you combine like terms which -1 +1 is 0 so our final answer would be x.
Ina makes cakes in a pan shaped like a rectangular prism. The base of the pan is an 8-inch by 12-inch rectangle, and the volume of the pan is 288 cubic inches. Find the surface area of a cheesecake baked in this pan
Answer:
312 square inches
Step-by-step explanation:
To find the surface area of a cake baked in a rectangular pan with an 8-inch by 12-inch base and a volume of 288 cubic inches, we need to find the dimensions of the pan. We can do this by dividing the volume by the area of the base, which gives us the height of the pan. Once we know the height, we can use the dimensions of the base and the height to find the surface area of the cake.
Suzie's Desserts offers its customers 5 dessert options. The prices are: $5.00 $9.00 $9.00 $6.00 $6.00 $9.00 $9.00 What is the mean absolute deviation of the prices? If the answer is a decimal, round it to the nearest ten cents
We can write the mean absolute deviation as -
1.63.
What is absolute deviation?Absolute deviation or mean absolute deviation is the measure of how far a given data element is from a given mean value of the data.
Given is that Suzie's Desserts offers its customers 5 dessert options. The prices are: $5.00 $9.00 $9.00 $6.00 $6.00 $9.00 $9.00.
The formula for absolute deviation is -
[tex]$\frac {1}{n} \sum \limits_{i=1}^n |x_i-m(X)|[/tex]
We can calculate the mean as -
(5 + 9 + 9 + 6 + 6 + 9 + 9)/7 = 7.57
We can write the absolute deviation as -
Absolute deviation =
1/7(5 - 7.57 + 9 - 7.57 + 9 - 7.57 + 6 - 7.57 + 6 - 7.57 + 9 - 7.57 + 9 - 7.57) = 1.63
Therefore, we can write the mean absolute deviation as -
1.63.
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Determine the formula for an exponential function f(x) = a -bpasses through the points (1,4.5) and (-1,0.5); i.e., determine the values of a and b, and write the equation for the associated exponential function
The equation for the exponential function is f(x) = (4.5)((0.5/4.5)-1/2)x. The equation for an exponential function is f(x) = abx, where a and b are constants.
To determine the values of a and b, we can use the two points given in the question, (1,4.5) and (-1,0.5).
Let's substitute the point (1,4.5) into the equation.
f(1) = a*b1
4.5 = a*b
Now let's substitute the point (-1,0.5) into the equation.
f(-1) = a*b-1
0.5 = a*b-1
We can now solve for a and b.
a = 4.5 / b
b-1 = 0.5 / a
b-1 = 0.5 / (4.5/b)
b-1 = 0.5b/4.5
b-2 = 0.5/4.5
b = (0.5/4.5)-1/2
Thus, the equation for the exponential function is f(x) = (4.5)((0.5/4.5)-1/2)x
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Cameron is at a location with a coordinate of (-3, 2) on a coordinate plane, Neveah is at a location with a coordinate of (2, -3), Jude is at a location with coordinates (2,3) and Jose is at (-2, -3). Who is in quadrant IV?
a) Cameron
b) Neveah
c) Jude
d) Jose
Jose is in the quadrant 4 as his coordinates are (-2, -3).
A coordinate plane is what?The x-axis and y-axis in a coordinate plane stand in for the horizontal and vertical axes, respectively. Using top right as quadrant I, topmost left as quadrant II, bottom left as quadrant III, and bottom right as quadrant IV, the four quadrants are numbered anticlockwise.
The given coordinates are:
Cameron = (-3, 2)
Neveah = (2, -3),
Jude = (2,3) and
Jose is at (-2, -3).
We must find the point whose x-coordinate is positive and y-coordinate is negative in order to know which point is in quadrant IV.
Hence, Jose is in the quadrant 4 as his coordinates are (-2, -3).
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Find out what X and Y equal.
x =
y =
The measures of the two interior angles are:
x = 62°
y = 118°
How to find the values of x and y?We can see a cuadrilateral, if the sides are parallel like in this case, opposite interior angles have the same measure, then:
x = 62°
And we know that adjacent angles should add up to 180°, then:
y+ 62° = 180°
y = 180° - 62° = 118°
These are the measures of the two interior angles.
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Jessica’s financial advisor believes that she should spend no more than 28% of her gross monthly income for housing . She has determined that amount is $1,400 per month. Based on this amount and her advisor’s recommendation, what is Jessica’s annual salary?
please explain in a sentence
y=-3x-7 y=x+9 solve by substitution how do I do this, I'm so lost!
Solve for the first variable in one of the equations, then substitute the result into the other equation.
Answer:
Point Form:
(−8,−17)
Equation Form: x=−8,y=−17
Answer:
x=-4, y=5
Step-by-step explanation:
Because y=-3x-7 and y=x+9, so y=y, and -3x-7=x+9
-3x-7=x+9,
so 4x=-16
x=-4
so we put x=-4 to y=-3x-7
y=(-3)* (-4)-7
y=12-7
y=5
Molly is verifying if the two functions are inverses of each other? Her answer is as follows:
f(g(x))=x+5 g(f(x))=x+5
She stated that they are not inverses of each other. Is she correct and why? Explain.
Answer:
Molly is correct in stating that the two functions are not inverses of each other.
To be inverses of each other, two functions must satisfy the property that when they are composed in either order, they result in the identity function, which is represented by f(x) = x.
In this case, we have:
f(g(x)) = (x + 5) + 5 = x + 10
g(f(x)) = (x + 5) + 5 = x + 10
Since both compositions of the functions result in x + 10, which is not equal to x, the two functions are not inverses of each other.
Step-by-step explanation:
Answer:
Molly is correct. The fact that $f(g(x)) = g(f(x)) = x+5$ indicates that the two functions, $f$ and $g$, are symmetric about the line $y=x$, which means that they are not inverses of each other.
To determine whether two functions are inverses of each other, we need to show that their composition results in the identity function. That is, if $f(x)$ and $g(x)$ are two functions, then $f(g(x)) = g(f(x)) = x$ for all $x$ in the domain of $f$ and $g$.
In this case, we see that $f(g(x)) = g(f(x)) = x+5$, which is not the identity function. Therefore, the two functions are not inverses of each other.
1. Find the center of mass of the solid bounded by x = y 2 and the planes x = z, z = 0, and x = 1 if the density is rho(x, y, z) = k ∈ R is constant
2. The electric charge distributes over the disk x 2 + y 2 ≤ 1 such that the charge density at any point (x, y) is rho(x, y) = x + y + x 2 + y 2 (in coulombs per square meter). Find the total charge Q on the disk.
3. Find the center of mass of the triangular region with vertices (0, 0), (2, 0) and (0, 2) if the density is given by rho(x, y) = 1 + x + 2y
1) The center of mass of the solid bounded by x = y^2 and the planes x = z, z = 0, and x = 1 the center of mass of the solid is (1/3, 2/15, 1/3). 2) The total charge on the disk is 4/3 coulombs. 3) The center of mass of the triangular region is (2/3, 2/3).
Center of Mass = (∫xyzρdV)/(∫ρdV).
Here, V is the volume of the solid. Since the density is constant, we can pull it out of the integral:
Center of Mass = k*(∫xyzdV)/(∫dV).
We can now use the volume formula for the solid which is V = ∫xyzdxdyz. Plugging this in the above formula, we get:
Center of Mass = k*[(∫x∫ydxdyz)/(∫dxdyz)]
Evaluating the integrals, we get the x coordinate of the center of mass to be (1/3), the y coordinate to be (2/15) and the z coordinate to be (1/3). Thus, the center of mass of the solid is (1/3, 2/15, 1/3).
2. To find the total charge Q on the disk x^2 + y^2 ≤ 1 such that the charge density at any point (x, y) is rho(x, y) = x + y + x^2 + y^2 (in coulombs per square meter), we need to use the following formula:
Q = ∫∫rho(x, y)dxdy
Evaluating the integral, we get Q = (1/3) + (1/3) + (1/3) + (1/3) = 4/3. Thus, the total charge on the disk is 4/3 coulombs.
3. To find the center of mass of the triangular region with vertices (0, 0), (2, 0) and (0, 2) if the density is given by rho(x, y) = 1 + x + 2y, we need to use the following formula:
Center of Mass = (∫xyρdA)/(∫ρdA).
Here, A is the area of the triangle. Evaluating the integral, we get the x coordinate of the center of mass to be (2/3) and the y coordinate to be (2/3). Thus, the center of mass of the triangular region is (2/3, 2/3).
To learn more about Center of Mass here:
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