Over a 4 week period, randy clocks in the following work hours: 68, 71, 66, and 67. How many hours were in Randy’s average work?
What is the measure of
Answer:
∠w = 50°
∠y = 130°
Step-by-step explanation:
Angles ∠w and ∠y are supplementary angles, which means their sum is 180.
4x + 6 + 12x - 2 = 180
Add like terms16x + 4 = 180
Subtract 4 from both sides16x = 176
Divide both sides by 16x = 11
To find the angle measures replace x with 11
∠w = 4x + 6
∠w = 4*11 + 6
∠w = 50°
Now, ∠y
∠y = 12x - 2
∠y = 12*11 - 2
∠y = 130°
The questions below can be answered by collecting data. Data related to which question is most likely to show variability?
Answer:
C
Step-by-step explanation:
Variability - Lack of consistency or fixed pattern; liability to vary or change
Every student most likely won’t have the same amount of letters in their first name.
The length of the longer leg of a 30 60 90 triangle is 6. The shorter leg is 4.
State the solution in Simple Root Form:
State the solution to the nearest tenth:
Step-by-step explanation:
remember the trigonometric triangle inscribed in a circle (the norm circle with radius = 1).
for a larger system the leg lengths are sine and cosine multiplied by the radius (which is the Hypotenuse of the right-angled triangle).
the longer leg is opposite of the larger angle (60°).
the shorter leg is opposite of the smaller angle (30°).
what is the desired solution ? the length of the Hypotenuse ? or what ?
6 = sin(60)×Hypotenuse
Hypotenuse = 6/sin(60) = 6/(sqrt(3)/2) = 12/sqrt(3) =
= 6.92820323... ≈ 6.9
Can anyone show how to solve these two questions. Thank you!
according the given question the exact value of given expression is [tex]$\cos\frac{x}{2} = -\sqrt{\frac{1}{2(1 - \left(-\frac{160}{81}\right)^2)}} = -\sqrt{\frac{81^2}{2(81^2 - 160^2)}} = \boxed{-\frac{81\sqrt{239}}{319}}$[/tex]
First, we need to find [tex]$\sin x$[/tex] using the identity[tex]$\cos^2x + \sin^2x = 1$:$\sin^2x = 1 - \cos^2x = 1 - \left(-\frac{4}{5}\right)^2 = \frac{9}{25}$[/tex]
Since [tex]$\frac{\pi}{2} < x < \pi$[/tex], we know that [tex]$\frac{\pi}{4} < \frac{x}{2} < \frac{\pi}{2}$[/tex]. Therefore, we can use the
identity [tex]$\tan\frac{x}{2} = \frac{\sin x}{1 + \cos x}$[/tex]:
[tex]$\tan\frac{x}{2} = \frac{\sqrt{\frac{9}{25}}}{1 - \frac{4}{5}} = \frac{\frac{3}{5}}{\frac{1}{5}} = \boxed{3}$[/tex]
[tex]If $\tan x = \frac{40}{9}$ and $\pi < x < \frac{3\pi}{2}$, find $\cos\frac{x}{2}$.[/tex]
First, we need to find [tex]$\sin x$[/tex] using the identity [tex]$\tan^2x + 1 = \sec^2x$[/tex]:
[tex]$\sin x = \frac{\tan x}{\sec x} = \frac{\frac{40}{9}}{-\frac{9}{40}} = -\frac{160}{81}$[/tex]
[tex]Since $\pi < x < \frac{3\pi}{2}$, we know that $\frac{\pi}{2} < \frac{x}{2} < \frac{3\pi}{4}$[/tex]. Therefore, we can use the identity [tex]$\cos\frac{x}{2} = \pm\sqrt{\frac{1 + \cos x}{2}}$[/tex]:
[tex]$\cos\frac{x}{2} = -\sqrt{\frac{1 + \cos x}{2}} = -\sqrt{\frac{1 + \frac{\cos^2x}{\sin^2x}}{2}} = -\sqrt{\frac{\sin^2x + \cos^2x}{2\sin^2x}} = -\sqrt{\frac{1}{2(1 - \sin^2x)}}$[/tex]
Plugging in [tex]$\sin x = -\frac{160}{81}$[/tex] , we get:
[tex]$\cos\frac{x}{2} = -\sqrt{\frac{1}{2(1 - \left(-\frac{160}{81}\right)^2)}} = -\sqrt{\frac{81^2}{2(81^2 - 160^2)}} = \boxed{-\frac{81\sqrt{239}}{319}}$[/tex]
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The data shows the number of video streaming hours for10 household in New Hampshire during the month of January and July last year. Which of the following best describes the data?
Please let me know which one is the answer, 100 points! Thank you
Answer: brainliest ?
The correct answer is: The data is dependent, because the hours of video streaming in January and July occurred in the same households.
This is because the data collected is related to the same households and not different people. The same households were monitored during both January and July, which means that any changes in the video streaming hours during those months are likely due to factors within those households, such as changes in habits, routines, or available free time.
If the data were independent, then it would mean that the hours of video streaming in January and July were not related or influenced by any factors from the other month. For example, if the data were collected from two completely different sets of households, then it would be considered independent because there would be no connection or influence between the households in January and July. However, this is not the case in the given scenario, so the correct answer is that the data is dependent.
Step-by-step explanation:
hope its help <:
Gary is making a cake recipe that requires 3/4 cup of flour but Gary wants to make the cake half of the size that the recipe calls for. how much flour should she use?
Answer: 6 tablespoons
These tables represent the relationships between x and y for two different sets of data.
Which statements correctly describe the relationships between x and y for each table?
Responses
Both data sets represent multiplicative relationships. In Table A, y is 3.5 times x, and in Table B, y is 9.2 times x.
Both data sets represent multiplicative relationships. In Table A, , y, is 3.5 times , x, , and in Table B, , y, is 9.2 times , x, .
Both tables represent additive relationships. In Table A, y is 2.5 more than x, and in Table B, y is 8.2 more than x.
Both tables represent additive relationships. In Table A, , y, is 2.5 more than , x, , and in Table B, , y, is 8.2 more than , x, .
Table A represents a multiplicative relationship because y is 3.5 times x, and Table B represents an additive relationship because y is 8.2 more than x.
Table A represents a multiplicative relationship because , y, is 3.5 times , x, , and Table B represents an additive relationship because , y, is 8.2 more than , x, .
Table A represents an additive relationship because y is 2.5 more than x, and Table B represents a multiplicative relationship because y is 9.2 times x.
Table A represents an additive relationship because , y, is 2.5 more than , x, , and Table B represents a multiplicative relationship because , y, is 9.2 times , x, .
Table A
x 1 2 3 4
y 3.5 7 10.5 14
Table B
x 1 2 3 4
y 9.2 10.2 11.2 12.2
Answer:
Both data sets represent multiplicative relationships. In Table A, y is 3.5 times x, and in Table B, y is 9.2 times x.
The correct statement is: Both data sets represent multiplicative relationships. In Table A, y is 3.5 times x, and in Table B, y is 9.2 times x.
100 POINTS NEED HELP ASAP QUESTIONS ARE BELOW
Area of triangle= 24 cm²
Area of sector= 72.243 cm²
Total area= 96.243 cm².
What is triangle?A polygon with three sides and three vertices is called a triangle. It is one of the fundamental geometric forms. Triangle ABC is the designation for a triangle with points A, B, and C. In Euclidean mathematics, any three points that are not collinear produce a distinct triangle and a distinct plane.
What is sector?The portion of a disc enclosed by two radii and an arc is called a circular sector, also known as a circle sector or disc sector. The smaller area is referred to as the minor sector, and the bigger area as the major sector. A sector is referred to as a component of a circle made up of the circular's arc and its two radii.
In this question,
Area of triangle= 1/2 × base × height
= 1/2 × 6 ×8
= 24 cm²
Area of sector = (θ/360°) × πr²
= (82/360) × πr²
Here r=√6²+8²
= 10 cm
Area of sector = (82/360) × π10²
= 0.23 × 314.1
= 72.243 cm²
Total= 24 cm²+72.243 cm²= 96.243 cm².
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9x-7=-7 please answer quick
Answer:
x = 0
Step-by-step explanation:
9x - 7 = -7
Add 7 to both sides.
9x - 7 + 7 = -7 + 7
9x = 0
Divide both sides by 9.
9x/9 = 0/9
x = 0
Answer:
x=0
Step-by-step explanation:
f(x) = x². What is g(x)?
WW
(3, 1)
g(x)
-5
y f(x) = x²
5
Click here for long description
A. g(x) = 3x²
B. g(x) = (x)²
2
C. g(x) = x²
2
D. g(x) = (x)
Finding [tex]g(x)[/tex] given that we know [tex]f(x)[/tex] and the graphs for both functions. The correct option is D:
[tex]g(x) = (x/3)^2[/tex]
How to find g(x)?Looking at the graph on the image, we notice that [tex]f(x)[/tex] and [tex]g(x)[/tex] are two quadratic functions, and [tex]g(x)[/tex] is just a dilation o f[tex]f(x)[/tex].
This means that:
[tex]g(x) = A*f(x)[/tex]
Where A is a real number.
We know that:
[tex]f(x) = x^2[/tex]
By looking at the graph in the image, we know that [tex]g(3) = 1[/tex].
Then we can write:
[tex]g(3) = A*f(3) = A*3^2 = 1[/tex]
We can now solve for A:
[tex]A*3^2 = 1[/tex]
[tex]A*9 = 1[/tex]
[tex]A = 1/9[/tex]
We will have:
[tex]g(x) = (1/9)*f(x) = (1/9)*x^2 = (x/3)^2[/tex]
Therefore, the correct option is D.
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Complete question in image attached.
What is the surface area of an ice cube that has 4 cm sides?
TA
4 cm
4 cm
The surface area of an ice cube with 4 cm sides is 96 square centimeters.
What is surface area?It is a measurement of the sum of all the areas of the faces, sides, and any other surfaces of an object. For example, a cube has six square faces, each with the same area.
According to question:To find the surface area of an ice cube with 4 cm sides, we need to add up the areas of all six faces. Each face is a square with side length 4 cm, so the area of each face is:
Area of one face = (side length)² = 4 cm × 4 cm = 16 cm²
Since there are six faces on a cube, the total surface area of the ice cube is:
Total surface area = 6 × (area of one face) = 6 × 16 cm² = 96 cm²
Therefore, the surface area of an ice cube with 4 cm sides is 96 square centimeters.
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Greg covered the back of the picture with a piece of felt. The picture is 1 1/4 inches shorter than the fram and 1 inch less in width what is the area of the felt
The area of the felt is (5L - 4W - 5) / 4 (inches)².
What is an area?
To find the area of the felt, we need to know the dimensions of the picture and the frame.
Let's say the length of the frame is L and the width of the frame is W.
Then, according to the problem:
The length of the picture is 1 1/4 inches shorter than the frame, so its length is L - 1 1/4 = (4L - 5) / 4 inches.The width of the picture is 1 inch less than the frame, so its width is W - 1 inches.To find the area of the felt, we need to subtract the area of the picture from the area of the frame. The area of the frame is L x W, and the area of the picture is (4L - 5) / 4 x (W - 1). So the area of the felt is:
Felt area = Frame area - Picture area
= L x W - (4L - 5) / 4 x (W - 1)
= (4LW - 4W) / 4 - (4LW - 5L - 4W + 5) / 4
= (5L - 4W - 5) / 4
Therefore, the area of the felt is (5L - 4W - 5) / 4 (inches)². Note that we can't simplify this expression further without knowing the specific values of L and W.
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Complete question is: Greg covered the back of the picture with a piece of felt. The picture is 1 1/4 inches shorter than the fram and 1 inch less in width the area of the felt is (5L - 4W - 5) / 4 (inches)².
The sum of the measures of the angles of a triangle is 180. The sum of the measures of
the second and third angles is five times the measure of the first angle. The third angle
is 26 more than the second. Let x, y, and z represent the measures of the first, second,
and third angles, respectively. Find the measures of the three angles.
The measures οf the three angIes are x = 51.43 degrees, y = 92.85 degrees, and z = 118.85 degrees.
What is Iinear equatiοn?A Iinear equatiοn is a mathematicaI equatiοn that describes a straight Iine in a twο-dimensiοnaI pIane.
We can use the infοrmatiοn given in the prοbIem tο fοrm a system οf three equatiοns with three variabIes. Let x, y, and z represent the measures οf the first, secοnd, and third angIes, respectiveIy.
Frοm the first piece οf infοrmatiοn, we knοw that: x + y + z = 180
Frοm the secοnd piece οf infοrmatiοn, we knοw that: y + z = 5x
Frοm the third piece οf infοrmatiοn, we knοw that: z = y + 26
We can substitute the third equatiοn intο the secοnd equatiοn tο eIiminate z:
y + (y + 26) = 5x
2y + 26 = 5x
2y = 5x - 26
y = (5x - 26)/2
We can substitute this expressiοn fοr y intο the first equatiοn tο eIiminate y and z:
x + (5x - 26)/2 + (5x - 26)/2 + 26 = 180
2x + 5x - 26 + 26 = 360
7x = 360
x = 51.43
We can substitute this vaIue οf x back intο the expressiοn fοr y tο find y:
y = (5x - 26)/2
y = (5(51.43) - 26)/2
y = 92.85
FinaIIy, we can use the equatiοn z = y + 26 tο find z:
z = y + 26
z = 92.85 + 26
z = 118.85
Therefοre, the measures οf the three angIes are x = 51.43 degrees, y = 92.85 degrees, and z = 118.85 degrees.
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Is there anything you want to share? Please answer this asap…No..question but please if you are going through something I’m free and I will answer and take the time. Please write something that happened and that made you happy.. you are worth it and there is a purpose for you… was there anything you need to share???
Answer:
Thank you for your kind words and concern. I appreciate your offer to listen and support others. I think it's important for people to know that they can reach out for help when they're struggling and that there are resources available to them. And as for something that made me happy, I always appreciate it when people show kindness and empathy towards others, and when they're able to make a positive impact in the world.
Answer:hi tysm your literally the person everyone needs in there life and your so kind for that
Step-by-step explanation:
Today me and my friend went to the park with my brother and we got snowcones and played for a while then I got home and took a refreshing nap!
HELP!!!! IM GOING TO FAIL!!!
Answer10 toooooooooooo the third poewedd
Step-by-step explanation:
Someone please help me answer this question
The two statements that are both true are as follows: line
AC is perpendicular to line HB and line AC is parallel to FG. That is option A.
What is a perpendicular line?A perpendicular line is defined as the line that forms angle 90° where it meets with another line in a plane.
A line is said to be parallel to each other when they do not intercept as they are both on the same plane.
From the given diagram, line AC is perpendicular to line HB because they form angle 90° at the point of intersection.
Also, line AC is parallel to FG, because they can never intersect till infinity.
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I please need help matching the angles. I cannot seem to get it right.
The angles and lines that match the angles and lines in the diagram consisting of two lines and their common transversal to the correct description are;
Alternate Interior Angles; 2. ∠4 and ∠8
Consecutive Exterior Angles; 5. ∠1 and ∠6
Alternate Exterior Angles; 3. ∠1 and ∠5
Transversal; line l
Consecutive Interior Angles; 4. ∠3 and ∠4
Corresponding Angles; 1. ∠1 and ∠7
What is a transversal line?A transversal is a line that intersects two or more other lines.
The description of the relationship between the angles in the question are;
Alternate Interior Angles
Alternate interior angles are a pair of angles formed when a transversal intersects two lines. They are located between the two lines on opposite side of the transversal.
Consecutive Exterior Angles
Consecutive Exterior Angles are a pair of angles formed when a transversal intersects two lines. They are located outside the two lines on the same side of the transversal
Alternate Exterior Angles
Alternate exterior angles are a pair of angles formed when a transversal intersects two lines. They are located outside the two lines on the opposite sides of the transversal.
Consecutive Interior Angles
Consecutive Interior Angles, also known as Same-Side Interior Angles are a pair of angles formed when a transversal intersects two parallel lines. They are located between the two parallel lines on the same side of the transversal.
Corresponding Angles
Corresponding angles are a pair of angles formed when a transversal intersects two lines. They are located on the same relative positions with respect to the transversal and the two lines.
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Please help me to answer the question
The range of the function for one day of work is 75 ≤ y ≤ 425. So, correct option is B.
Describe Function?In mathematics, a function is a mathematical object that takes an input (or several inputs) and produces a unique output. It is a relationship between a set of inputs, called the domain, and a set of outputs, called the range.
Formally, a function f is defined by a set of ordered pairs (x, y) where x is an element of the domain, and y is an element of the range, and each element x in the domain is paired with a unique element y in the range. We write this as f(x) = y.
The linear function that models the daily cost of hiring an electrician can be written as:
y = 50x + 75
where x is the number of hours worked by the electrician and y is the cost in dollars.
Since the electrician works a maximum of 7 hours per day, the domain of the function is 0 ≤ x ≤ 7.
To find the range of the function, we can substitute the maximum and minimum values of x into the function and see what values of y we get:
When x = 0 (no hours worked), y = 50(0) + 75 = 75.
When x = 7 (maximum hours worked), y = 50(7) + 75 = 425.
Therefore, the range of the function for one day of work is:
75 ≤ y ≤ 425
So the answer is (B) 75 ≤ y ≤ 425.
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Help! See image below
In the polygon, using sum of exterior angles the value of x = 37°
What is a polygon?A polygon is a shape that has 3 or more sides.
Given the polygon which is a hexagon to find the value of x, we note that the angles are all exterior angles. We know that the sum of the exterior angles of a polygon is 360°.
So, we have the equation as
x + 2x + (x - 1) + 3x + (x + 18) + (x + 10) = 360°
Collecting like terms, we have that
x + 2x + x + 3x + x + x - 1 + 18 + 10 = 360°
9x + 27° = 360°
Subtracting 27° from both sides of the equation, we have that
9x + 27° - 27° = 360° - 27°
9x + 0 = 333°
9x = 333°
Dividing both sides by 9, we have that
x = 333°/9
x = 37°
So, the value of x = 37°
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Marissa ate 4 hot dogs every 16 hours. At that rate, how many would she eat in 12 hours?
Answer: 3
Step-by-step explanation:
Answer: 3
Step-by-step explanation:
16/4=unit rate =4
1 in 4 hour
3 for 12 hours
Lara grows apples in her orchard and sells them at the weekly farmer's market. Each week, she sells the apples for a different price and records the number of apples sold. The scatter plict below
shows the price of one apple and the number of apples that were sold. A line of best fit for these data points, the equation y=-z+32, is also shown on the plot
Apples Number of Apples Sold
Bug S Bug S and Bug F is fast. Both bugs start at 0 on a number line and move in the positive direction. The bugs leave 0 at the same time and move at constant speeds. Four seconds later, F is at 12 and S is at 8. When will F and S be 100 units apart?
Answer:
Let's call the speed of Bug F v_F and the speed of Bug S v_S. Since both bugs started at 0, we can express their positions at any time t as:
Position of Bug F = 12 + v_F * t
Position of Bug S = 8 + v_S * t
To find out when F and S will be 100 units apart, we need to find the time t at which their positions differ by 100 units. In other words, we need to solve the following equation:
|12 + v_F * t - (8 + v_S * t)| = 100
We can simplify this equation by expanding the absolute value and rearranging the terms:
|4 + (v_F - v_S) * t| = 100
Now we can split this equation into two cases:
Case 1: 4 + (v_F - v_S) * t = 100
In this case, we have:
v_F - v_S > 0 (since Bug F is faster)
t = (100 - 4) / (v_F - v_S)
Case 2: 4 + (v_F - v_S) * t = -100
In this case, we have:
v_F - v_S < 0 (since Bug S is faster)
t = (-100 - 4) / (v_F - v_S)
Since we're only interested in positive values of t, we can discard the second case. Therefore, the time at which F and S will be 100 units apart is:
t = (100 - 4) / (v_F - v_S)
t = 96 / (v_F - v_S)
We don't know the values of v_F and v_S, but we can use the fact that Bug F is at 12 and Bug S is at 8, four seconds after they started. This gives us two equations:
12 = 4v_F + 0v_S
8 = 4v_S + 0v_F
Solving these equations for v_F and v_S, we get:
v_F = 3
v_S = 2
Substituting these values into the equation for t, we get:
t = 96 / (3 - 2)
t = 96
Therefore, F and S will be 100 units apart 96 seconds after they start.
Para prepara un pastel Anita tenía un paquete de harina de 1 1/3 si solo uso 3/6 kg del paquete que cantidad de harina le sobró
Para preparar un pastel, Anita tenía un paquete de harina de 1 1/3 kg: 0.83 kg
Primero, debemos convertir 1 1/3 kg a una fracción común. 1 1/3 kg es igual a 4/3 kg o 1.33 kg. Luego, podemos simplificar 3/6 a 1/2.
Ahora podemos restar 1/2 de 1.33 kg para encontrar la cantidad de harina que sobró.
1.33 kg - 1/2 kg = 1.33 kg - 0.5 kg = 0.83 kg
Entonces, Anita le sobró aproximadamente 0.83 kg de harina después de preparar su pastel.
Es importante recordar que la medición y las cantidades de los ingredientes son cruciales al preparar cualquier receta de cocina. Es importante seguir las instrucciones cuidadosamente para obtener resultados exitosos en la cocina.
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solve this please! i've been trying for a while but i cant get it
The three-dimensional object in the figure has a surface area of 52 square yards.
What does surface area dimension mean?Surface area is the two-dimensional measure or area of a three-dimensional space's boundary, just as perimeter is the one-dimensional measure or length of a two-dimensional area's boundary.
The object's rectangular base has a length of 4 yards and a width of 3 yards, as seen in the figure. As a result, the base's area is:
Area of base = length x width = 4 x 3 = 12 square yards
Area of a triangular face = (1/2) x base x height = (1/2) x 4 x 5 = 10 square yards
Since the object has four triangular faces, the total area of the four triangular faces is:
Total area of triangular faces = 4 x 10 = 40 square yards
We add the base's area to the sum of the four triangle sides to determine the object's surface area:
Surface area = Area of base + Total area of triangular faces
Surface area = 12 + 40 = 52 square yards
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A rectangular garden measures 40m by 15m. A 1m flower bed is made round the two shorter sides and one
long side. A circular swimming pool of diameter 8m is constructed in the middle of the garden. Find
correct to the nearest square meter, the area remaining
Answer:
The area remaining, correct to the nearest square meter, is approximately 436 square meters.
Step-by-step explanation:
To find the area remaining, we need to subtract the area of the flower bed and the area of the pool from the total area of the garden.
The total area of the garden is:
40m x 15m = 600 square meters
The flower bed is 1m wide and runs along two shorter sides and one long side of the garden. So the area of the flower bed is:
(40m + 2 x 1m) x (15m + 2 x 1m) - 40m x 15m
= (42m x 17m) - (40m x 15m)
= 714 - 600
= 114 square meters
Now let's calculate the area of the pool. The diameter of the pool is 8m, so the radius is 4m. The area of the pool is:
π x (4m)^2
= 16π
≈ 50.27 square meters (rounded to two decimal places)
So the area remaining is:
600 square meters - 114 square meters - 50.27 square meters
≈ 435.73 square meters
Therefore, the area remaining, correct to the nearest square meter, is approximately 436 square meters.
Solve the following linear programming problem. Maximize: z = 7x + 2y subject to: 7x-y≤ 16 2x+y≥ 10 X≥2 y≤9 The maximum value is
Answer:
To solve the linear programming problem, we need to first graph the feasible region determined by the constraints, and then evaluate the objective function at each corner point of the feasible region to find the maximum value of z.
Plotting the lines corresponding to the inequalities, we get:
Graph of the feasible region:
The feasible region is the shaded polygon in the graph. We can see that the vertices of the feasible region are (2, 9), (2, 12), (4, 7), and (8, 2).
Next, we evaluate the objective function at each of these vertices to find the maximum value of z.
At (2, 9): z = 7x + 2y = 7(2) + 2(9) = 23
At (2, 12): z = 7x + 2y = 7(2) + 2(12) = 31
At (4, 7): z = 7x + 2y = 7(4) + 2(7) = 35
At (8, 2): z = 7x + 2y = 7(8) + 2(2) = 58
Therefore, the maximum value of z is 58, which occurs at the point (8, 2).
Hence, the answer is: the maximum value of z is 58.
An air tank of 500 mL is 80% oxygen and 20% nitrogen. What is the amount of oxygen in milliliters in a 200 ml air tank that contains the same ratio?
Hence, 160 mL of oxygen is contained inside a 200 mL air tank using the same ratio as a 500 mL air tank.
How is ratio calculated?Ratios contrast two figures by ordinarily dividing them. A/B would be your formula if you were trying to compare one piece of data (A) to some other data point (B). We are multiplying information A by data B, as this suggests. In the event that A and B are both 5, for example, your ratio would've been 5/10.
80% of 500 mL of oxygen if indeed the air tank contains 80% oxygen & 20% nitrogen, which is:
0.80 x 500 mL = 400 mL
This means that the air tank contains 400 mL of oxygen and 100 mL of nitrogen.
To find the amount of oxygen in a 200 mL air tank that contains the same ratio, we need to use proportions:
If 500 mL contains 400 mL of oxygen, then 1 mL contains 400/500 = 0.8 mL of oxygen.
Therefore, if 200 mL contains the same ratio of oxygen, it will contain:
0.8 mL x 200 mL = 160 mL
So, the amount of oxygen in a 200 mL air tank with the same ratio as the 500 mL air tank is 160 mL.
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16 Triangle ABC is translated to triangle A'B'C' by
the following motion rule.
(x, y)(x+2y-5)
-8 -6
G
A. (4,-4)
B. (2,-5)
C. (0.6)
D. (-2.5)
N
8
6
B
-2
S
-6
-8
2
What will be the coordinates of A'?
6 8
Answer:
To find the coordinates of A' after the translation, we need to apply the motion rule to the coordinates of A:
(x, y) → (x + 2y - 5, y - 6)
Substituting the coordinates of point A, which is (4, -4), into this motion rule, we get:
A' = (4 + 2(-4) - 5, -4 - 6) = (-3, -10)
Therefore, the coordinates of A' after the translation are (-3, -10).
Solve the given third-order differential equation by variation of parameters.
y''' + y' = cot(x)
Answer: To solve the third-order differential equation y''' + y' = cot(x) by variation of parameters, we first need to find the solution to the associated homogeneous equation, which is:
y''' + y' = 0
The characteristic equation is r^3 + r = 0, which can be factored as r(r^2 + 1) = 0. This gives us the roots r = 0, r = i, and r = -i. Therefore, the general solution to the homogeneous equation is:
y_h = c1 + c2 cos(x) + c3 sin(x)
To find a particular solution to the non-homogeneous equation using variation of parameters, we assume that the solution has the form:
y_p = u1(x) + u2(x) cos(x) + u3(x) sin(x)
where u1, u2, and u3 are functions to be determined.
We can find the derivatives of y_p:
y'_p = u1'(x) + u2'(x) cos(x) - u2(x) sin(x) + u3'(x) sin(x) + u3(x) cos(x)
y''_p = u1''(x) + u2''(x) cos(x) - 2u2'(x) sin(x) - u2(x) cos(x) + u3''(x) sin(x) + 2u3'(x) cos(x) - u3(x) sin(x)
y'''_p = u1'''(x) + u2'''(x) cos(x) - 3u2''(x) sin(x) - 3u2'(x) cos(x) - u2(x) sin(x) + u3'''(x) sin(x) + 3u3''(x) cos(x) - 3u3'(x) sin(x)
Substituting these derivatives into the non-homogeneous equation, we get:
u1'''(x) + u2'''(x) cos(x) - 3u2''(x) sin(x) - 3u2'(x) cos(x) - u2(x) sin(x) + u3'''(x) sin(x) + 3u3''(x) cos(x) - 3u3'(x) sin(x) + u1'(x) + u2'(x) cos(x) - u2(x) sin(x) + u3'(x) sin(x) + u3(x) cos(x) = cot(x)
Grouping the terms with the same functions together, we get:
u1'''(x) + u1'(x) = 0
u2'''(x) cos(x) - 3u2''(x) sin(x) - u2(x) sin(x) + u2'(x) cos(x) + u2'(x) cos(x) = cot(x) cos(x)
u3'''(x) sin(x) + 3u3''(x) cos(x) - 3u3'(x) sin(x) + u3'(x) sin(x) + u3(x) cos(x) = cot(x) sin(x)
The first equation is a first-order differential equation, which can be solved by integrating both sides:
u1'(x) + u1(x) = c1
where c1 is a constant of integration. The solution to this equation is:
u1(x) = c1 + c2 e^(-x)
where c2 is another constant of integration.
Step-by-step explanation: