• You have given an equal sided triangle with side length a. A straight line connects the center of the bottom side to the border of the triangle with an angle of α. Derive an expression for the enclosed area A(α) with respect to the angle (see drawing).
Answers
9514 1404 393
Answer:
[tex]A(\alpha)=\left\{\begin{array}{ccc}\dfrac{a^2\sqrt{3}\sin{(\alpha)}}{16\sin{(\alpha+\dfrac{\pi}{3})}}&\text{if $\alpha\le\dfrac{\pi}{2}$}\\\\\dfrac{a^2\sqrt{3}}{4}\left(1-\dfrac{\sin{(\alpha)}}{4\sin{(\alpha-\dfrac{\pi}{3})}}\right)&\text{if $\alpha>\dfrac{\pi}{2}$}\end{array}\right.[/tex]
Step-by-step explanation:
If we place the bottom center at the origin and make the sides of the triangle 2 units, then the right side can be described by the equation ...
y = √3(1 -x)
and the terminal side of the angle α can be described by ...
y = tan(α)·x
We want the height of the shaded area, which is the y-coordinate of the point of intersection of these two lines.
y = √3(1 -y/tan(α)) . . . . substituting for x
y = √3·tan(α)/(√3 +tan(α)) . . . . solved for y
Since we have defined the base of the shaded area to be 1, the shaded area is half this value, or ...
A(α) = (√3/2)tan(α)/(√3 +tan(α))
Further manipulation can put this in the form ...
A(α) = (√3/4)sin(α)/sin(α+π/3) . . . . for α ≤ π/2, side length 2
__
For α > π/2, we need to subtract the area A(π-α) from the area of the whole triangle. This gives an area (after some rearranging) of ...
A(α) = √3(1 -sin(α)/(4·sin(α-π/3))) . . . . for α > π/2, side length 2
__
The problem statement defines the triangle as having a side length of 'a', so the final area formula will be a factor of (a/2)² times the above expressions. That is ...
[tex]A(\alpha)=\left\{\begin{array}{ccc}\dfrac{a^2\sqrt{3}\sin{(\alpha)}}{16\sin{(\alpha+\dfrac{\pi}{3})}}&\text{if $\alpha\le\dfrac{\pi}{2}$}\\\\\dfrac{a^2\sqrt{3}}{4}\left(1-\dfrac{\sin{(\alpha)}}{4\sin{(\alpha-\dfrac{\pi}{3})}}\right)&\text{if $\alpha>\dfrac{\pi}{2}$}\end{array}\right.[/tex]
_____
The graph (second attachment) shows the area as a function of α (in degrees) for a unit triangle. There are inflection points in the curve at 30°, 90°, and 150°
Related Questions
El 10% de los huevos de un supermercado están rotos, haya la probabilidad de que un cliente que compra media docena de huevos encuentre como mucho un huevo roto
Answers
Answer:
La probabilidad de encontrar como mucho un huevo roto es 0,8857.
Step-by-step explanation:
Podemos calcular la probabilidad de econtrar un huevo roto usando la ecuación de distribución binomial:
[tex] P (x) = \left(\begin{array}{c}n&x\end{array}\right)p^{x}(1 - p)^{n-x} [/tex]
En donde:
p: es la probablidad de encontrar huevos rotos = 10% = 0,1
x: es el número de éxitos
n: es el número de ensayos = 6 (media docena de huevos)
Ahora, como nos piden la probabilidad de enontrar como mucho un huevo roto, esto quiere decir que debemos encontar la suma de la probablidad de encontar un huevo roto con la probabilidad de encontrar ninguno roto:
[tex] P = P(0) + P(1) [/tex]
[tex] P = \left(\begin{array}{c}6&0\end{array}\right)0,1^{0}(1 - 0,1)^{6-0} + \left(\begin{array}{c}6&1\end{array}\right)0,1^{1}(1 - 0,1)^{6-1} = 0,5314 + 0,3543 = 0,8857 [/tex]
Entonces, la probabilidad de encontrar como mucho un huevo roto es 0,8857.
Espero que te sea de utilidad!
a number b divided by -3 is greater than or equal to 6
Answers
b ≤ −18 or (−∞, −18)
The next section (This is the last five, don't worry xd)
7th grade math
I will give brainliest!
2 = a/6
find the value of a
Answers
Answer:
a = 12
Step-by-step explanation:
Given
2 = [tex]\frac{a}{6}[/tex] ( multiply both sides by 6 to clear the fraction )
12 = a
Answer:
a = 12
Step-by-step explanation:
We can do the inverse of division which is multiplication. 6 x 2 equals 12 and if we substitute 12 for a (12 divided by 6) you get 2 hope that helpss!!!
Someone for the love of god please help me.
Answers
Answer:
Step-by-step explanation:
1l) HAO WENT TO A HOBBY SHOP AND SAW A DISPLAY OF TOY TRAINS. THE SIGN IN FRONT OF
THE DISPLAY SAID THAT THE TOY TRAINS WERE A REPLICA OF THE LONG ISLAND RAILROAD
TRAINS. HAO ESTIMATED THAT ONE OF THE RAILROAD CARS WAS APPROXIMATELY 15
INCHES LONG AND 4 INCHES WIDE. THE SIGN IN FRONT OF THE DISPLAY STATED
"SCALE- 2 /9".
WHAT IS THE LENGTH OF THE RAILROAD CAR ON HE LONG ISLAND RAILROAD?
WHAT IS THE WIDTH OF THE LONG ISLAND RAILROAD CAR?
Answers
Answer:
Length = [tex]57.5\ \text{inches}[/tex]
Width = [tex]18\ \text{inches}[/tex]
Step-by-step explanation:
The scale of the toy train is [tex]\dfrac{2}{9}[/tex]
So, the dimensions of the toy train is [tex]\dfrac{2}{9}[/tex] times the actual dimensions of the train.
The dimensions of actual train is [tex]\dfrac{1}{\frac{2}{9}}=\frac{9}{2}[/tex] times the toy train's dimensions.
Length
[tex]15\times \dfrac{9}{2}=57.5\ \text{inches}[/tex]
Width
[tex]4\times \dfrac{9}{2}=18\ \text{inches}[/tex]
The length of the railroad car on the long island railroad is 57.5 inches.
The width of the railroad car on the long island railroad is 18 inches.
Asher reads 2 books a week. Kyla reads 7 books a month. Sawyer reads 90 book a year. Who
reads more books?
Answers
Answer:
Asher
Step-by-step explanation:
Answer:
Asher reads more books
Step-by-step explanation:
Asher reads 2 books a week wich is about 102 books a year (because there about 52 weeks in a year)
kyla reads 7 books a month wich is 84 books a years
Sawyer reads 90 books a year
Given: ΔDFE is isosceles with base FE; FB ≅ EC. Prove: ΔDFB ≅ ΔDEC Triangle D F E is shown. 2 lines drawn down from point D to points C and B on the base of the triangle to form 3 triangles. Complete the missing parts of the paragraph proof. We know that triangle DFE is isosceles with base FE and that segment FB is congruent to segment EC because . Segment DF is congruent to segment by the definition of isosceles triangle. Since these segments are congruent, the base angles, angles , are congruent by the isosceles triangle theorem. Therefore, triangles are congruent by SAS.
Answers
answer picture answer picture
Answer:
1. it is given
2. DE
3. DFE and DEF
4. DFB and DEC
La mitad de la edad que tenia anahí hace 27 años equivale a la cuarta parte de la edad que tendrá dentro de 5 años. ¿ qué edad tiene actualmente anahí?
Answers
Responder:
59 años
Explicación paso a paso:
Asumir que:
Edad actual = x
Edad 27 años atrás = x - 27
Edad dentro de 5 años = x + 5
La mitad de la edad de hace 27 años = 1/2 (x - 27)
Cuarto de edad dentro de cinco años = 1/4 (x +5)
0,5 (x - 27) = 0,25 (x + 5)
0.5x - 13.5 = 0.25x + 1.25
0.5x - 0.25x = 1.25 + 13.5
0,25 veces = 14,75
Dividir por 0,25
0,25x / 0,25 = 14,75 / 0,25
x = 59
Por lo tanto, la edad de Anahí ahora es de 59 años.
La edad actual de Anahí es de 59 años.
Sabemos por la pregunta que la mitad de la edad que tenía Anahí hace 27 años equivale a la cuarta parte de la edad que tendrá dentro de 5 años.
Para escribir esto como una ecuación, sea y la edad actual;
1/2(s - 27) = 1/4(y + 5)
El MCM de 2 y 4 es 4 por lo tanto;
4/2(y - 27) = 4/4(y + 5)
2(s - 27) =(y + 5)
2y - 54 = y + 5
2y -y = 5 + 54
y = 59
La edad actual de Anahí es de 59 años.
Más información sobre ecuaciones: https://brainly.com/question/2263981
Which pair of complex numbers has a real-number product?
O (1 + 2i) (8i)
0 (1 + 2i) (2 - 5i)
O (1 + 2i) (1 – 2i)
O (1 + 2i)(4i)
Answers
Answer:
(1 + 2i)(1 - 2i)
Step-by-step explanation:
Multiplying a complex number and its conjugate results in a real number.
Thus
1 + 2i and 1 - 2i are conjugates
(1 + 2i)(1 - 2i) ← expand using FOIL
= 1 - 2I + 2I - 4I² [ note i² = - 1 ]
= 1 - 4(- 1)
= 1 + 4
= 5 ← a real number
Answer:
C
Step-by-step explanation:
C
Help me with this one
Answers
Answer:
SSS
Step-by-step explanation:
We can see that 2 sides are equal already, and because they share a side, that side is also equal. So, side side side.
: What is the image after (3,7) is reflected over the x-axis?
Brainly
Answers
Got the question wrong cause the other answer here is the correct answer directly from Delta Math.
(-7, -3)
Enjoy and Happy Holidays!
What is the first step in evaluating the expression shown below?
(12.9-3.1) 6.2-2 + 43
Answers
Answer:
Subtract 3.1 from 12.9
Hope it helped
solve the system by substitution -x+3y=15
y=2x
Answers
Answer:
(3,6)
Step-by-step explanation:
A student finished 45 homework problems in class.If the ratio of problems she finished to problems she still had left was 9:4 how many homework problems did she have total
Answers
Answer:
so in total, I believe that it would be.......65?
Step-by-step explanation:
well 45/9=5 and so you would have to multiply 4x5 and get 20
Please help it’s due in an hour! It’s algebra.
Answers
Answer:
You might wanna double check some of these because i might've forgot to flip some of the < and > signs but I'm pretty sure its correct
1) y = x < -18 and y = x < 18
2) y = -3 < x < 3
3) x = 1 < y < 6 (it just gets simplified)
4) x = -3 > y > -18
5) y = -2 > x and y = x < 3.5
6) y = 8.5 > x > 5.5
7) y = -5.75 < x < -3.5
8) y = 1 > x > 2.25
9) -17 < x < -5
Step-by-step explanation:
Deion divided 1/2 of a liter of plant fertilizer evenly among some smaller bottles. He put 1/8 of a liter into each bottle. How many smaller bottles did Deion fill?
Answers
Answer:
4 bottles
Step-by-step explanation:
Given
[tex]Fertilizer = \frac{1}{2}L[/tex]
[tex]Bottles = \frac{1}{8}L[/tex]
Required
Determine the number of bottles
Represent the number of bottles with N
This implies that N bottles of [tex]\frac{1}{8}L[/tex] fit into [tex]\frac{1}{2}L[/tex] of fertilizer. The relationship between them is:
[tex]Fertilizer = Bottles * N[/tex]
[tex]\frac{1}{2}L = \frac{1}{8}L* N[/tex]
[tex]\frac{1}{2} = \frac{1}{8}* N[/tex]
Multiply through by 8
[tex]8 * \frac{1}{2} = 8 * \frac{1}{8}* N[/tex]
[tex]4 = 1* N[/tex]
[tex]4 = N[/tex]
[tex]N = 4[/tex]
Hence, there are N = 4
What is the c.o.p. that describes the relationship between h and t.
Answers
I need help please
Answers
2. Michael bought a sandwich that cost $4.69, a bag of chips that cost $1.99
and an iced tea for $2.50. How much did he spend in all? *
A. 7.19
B. 4.49
C. 6.68
D. 9.18
Answers
Answer:
9.18
Step-by-step explanation:
4.69
1.99
+2.50
9.18
(x + 10) cm
(x - 3) cm
The area of the rectangle is 48 cm².
Show that x satisfies the equation x2+ 7x - 78=0
Answers
Answer:
area a=lxb
48=(x+10)x(x-3)
48=x2-3x+10x-30
0=x2+7x-30-48
0=x2+7x-78
hence it is proved
The requried, we have shown that x satisfies the equation x² + 7x - 78 = 0.
What is the equation?The equation is the relationship between variables and represented as y = ax + b is an example of a polynomial equation.
Here,
The area of the rectangle is given by the formula:
Area = length x width
Substituting the given expressions for length and width, we have:
Area = (x + 10)(x - 3)
We are also told that the area of the rectangle is 48 cm², so we can set up the equation:
(x + 10)(x - 3) = 48
Expanding the left side using the FOIL method:
x² - 3x + 10x - 30 = 48
Simplifying:
x² + 7x - 78 = 0
Therefore, we have shown that x satisfies the equation x² + 7x - 78 = 0.
Learn more about equations here:
brainly.com/question/10413253
#SPJ7
Evaluate each value expression for the given value of the variable : x+x+5 when x=7
QUICK PLSSSSSS
Answers
Answer:
19
Step-by-step explanation:
x = 7, so we have:
7 + 7 = 14
14 + 5 = 19
Answer:
19
Step-by-step explanation:
you replace the x with a seven. so it will look like
7+7+5= 19
HELP THIS IS IMPORTANT!!!! The table shows the outputs, y, for different inputs, X: Input (x) 6 12 15 25 Output (y) 14 15 15 16 Part A: Do the data in this table represent a function? Justify your answer. (3 points) I Part B: Compare the data in the table with the relation f(x) = 7x – 15. Which relation has a greater value when x = 67 (2 points) Part C: Using the relation in Part B. what is the value of x if f(x) = 67 (5 points) (10 points)
Answers
Answer:
A) sure there is one value of y for every value of x
B) in table y = 8
in function
y = 5(11) - 21 = 55-21 = 34
C) 99 = 5 x - 21
5 x = 120
x = 24
I really need help with this
Answers
Answer:
a. x=11°
b. ABC=106°, BCA=33° and BAC=41°
c. Sorry but I can't see the lines l and m
Step-by-step explanation:
a. the question stated that the sum of the measures of the angles in the triangle is 180°
so therefore equate all the values in the triangle to 180°
(9x+7)+ 3(x) + (3x+8)=180°
15x+ 15=180
15x=180-15
15x=165
Divide both sides by 15
therefore x=11
b. ABC=(9x+7)
where there is x put the x value you solved for there which is 11
therefore ABC=(9×11+7)= 106°
Do the same for BCA and BAC
If the sum of the greatest and smallest of 4 consecutive even integers equals 30, what is the sum of all four integers?
Answers
Answer:
The sum of all four integers is 60.
Step-by-step explanation:
Let the integers be x, x + 2, x + 4, and x + 6.
x + x + 6 = 30
2x = 24
x = 12
So x + x + 2 + x + 4 + x + 6 = 4x + 12 = 60
Answer:
________________________
Duke ate 1.2 kilograms of dog food each day for a total of 8.4 kilograms while his owners were out of town during the month of January. He ate 0.9 kilograms of dog food each day for a total of 7.2 kilograms while his owners were out of town during the month of February. He ate 1.8 kilograms of dog food each day for a total of 10.8 kilograms while his owners were out of town during the month of March. He ate 1.5 kilograms of dog food each day for a total of 13.5 kilograms while his owners were out of town during the month of April. During which month were his owners gone for the shortest period of time?
A.
January
B.
April
C.
February
D.
March
Answers
3. FIHG is a rhombus. Find the value of x.
1
H
ZFIR = 3x + 40
R
ZFGR = x^2
F
G
Answers
Answer:
where is the rhombus
Step-by-step explanation:
If the side opposite the 60 degree angle in a 30-60-90 degree is 12 centimetres, then what length of the side opposite the 30 degree angle?
Answers
Answer:
4*[tex]\sqrt3[/tex]
Step-by-step explanation:
number 10?? please help.
Answers
Answer:
∠GFM= 124°
Step-by-step explanation:
∠GFH= 180° -90° -(5x +9)° (∠ sum of △FGH)
∠GFH= 90° -5x° -9°
∠GFH= (81 -5x)°
∠GFM +∠GFH= 180° (adj. ∠s on a str. line)
25x -1 +81 -5x= 180
20x +80= 180 (simplify)
20x= 180 -80 (-80 on both sides)
20x= 100
x= 100 ÷20 (÷20 on both sides)
x= 5
∴∠GFM
= 25x -1
= 25(5) -1
= 124°
100 points: Sketch a graph of y = 1x| Then sketch a graph of y = -x + 4 on the same coordinate plane. If you can change colors of your graph, make this one red. In the box below, describe the transformation that took place from y=[xl to y = -1x + 4|
Answers
Answer:
i cant like give you the answer off the top of my head but when i did this the video helped me a lot
Step-by-step explanation:
its How to Graph the line y = x by Cowan Academy
Identify the independent variable (domain).
5x+2y= 150
Answers
Answer:
The independent variable is x
Step-by-step explanation:
In the equation y = m x + b
m is the rate of change ⇒ constantb is the initial value ⇒ constantx is the domain ⇒ inputy is the range ⇒ outputx is the independent variable because it does not depend on any value
y is the dependent variable because it depends on the value of x
Let us solve the question
∵ 5x + 2y = 150
→ Put it in the form above by subtracting 5x from both sides
∵ 5x - 5x + 2y = 150 - 5x
∴ 2y = 150 - 5x
→ Divide each term by 2 to make the coefficient of y equal 1
∴ y = 75 - 2.5x
∴ y = -2.5x + 75
→ Compare is by the form above
∴ m = -2.5 and b = 75
∴ x is the independent variable (domain)
∴ y is the dependent variable (range)