• 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
Last year, Summerfield Steel sold 6.5×104 pounds of steel to private businesses and 2.5×105 pounds of steel to government agencies. This year, they expect to sell 3 times that amount. How much steel would that be?
Answers
Answer:
9.45 × 10⁵ pounds of steel
Step-by-step explanation:
Last year, Summerfield Steel sold 6.5×10⁴ pounds of steel to private businesses and 2.5×10⁵ pounds of steel to government agencies.
Total amount of steel sold last year = (6.5 × 10⁴ + 2.5 × 10⁵) pounds = 315000 pounds = 3.15 × 10⁵ pounds
This year, they expect to sell 3 times that amount.
This year = 3 × 3.15 × 10⁵ pounds
= 9.45 × 10⁵ pounds of steel
What is the equation of the graphed line written in standard form?
Answers
Answer:
C. y = -2/3x -2
Step-by-step explanation:
We know the slope is negative. You can find slope using rise/run. The y-intercept is -2 because the line meets -2 at the y-intercept.
Answer:
The first answer (top)
Step-by-step explanation:
Standard form is ax+by=c
a and b are the coefficients
Take a point from the given line
I chose (0, -2)
Then, plug in the values of x and y
2*0+3*-2=-6
-6=-6
Since it works the standard form of the line is 2x+3y=-6
2. Now that you have the height of the cone, how can you solve for the
slant height, s?
Answers
Answer:
s^2 = r^2 + h^2.
Step-by-step explanation:
You need to know the radius of the base as well.
Then you use the Pythagoras theorem:
s^2 = r^2 + h^2 where r = radius and h = the height.
how do you solve 18 - (-2)=
Answers
Answer:
the answer would be 20
Step-by-step explanation:
so if 18-(-2)=?
two nwgatives make a positive. So the equation would turn into 18+2=20
Im not 100% if this is right but htis is how I was taught to do it.
What is the length of BC?
O 11
O 23
O 40
O 60
Plz help
Answers
Answer:
BC = 40
Step-by-step explanation:
Answer:
X=23
BC = 40
Step-by-step explanation:
(Find common denominator first.)
3/2 + b=7/4
Answers
Answer:
b = 1/4
Step-by-step explanation:
Take the given fractions : 3/2 and 7/4 and find a common denominator.We will use 4 as a common denominator.
In order for the denominator of 3/2 to get to 4, you need to multiply by 2.3 × 2 = 6 2 × 2 = 4
New fraction: 6/4Now we have the equation 6/4 + b = 7/4
7/4 - 6/4 = 1/4
Which of the following best explains the value of Sine StartFraction pi Over 3 EndFraction on the unit circle below?
A unit circle is shown. A radius with length 1 has an angle measure of StartFraction pi Over 3 EndFraction
sine StartFraction pi Over 3 EndFraction = StartFraction opposite Over hypotenuse EndFractionEquals StartFraction opposite Over 1 EndFraction Equals opposite
The length opposite the angle is the vertical distance from the x-axis on the graph.
sine StartFraction pi Over 3 EndFraction = StartFraction adjacent Over hypotenuse EndFraction = StartFraction adjacent Over 1 EndFraction Equals adjacent
The length adjacent to the angle is the vertical distance from the x-axis on the graph.
Sine StartFraction pi Over 3 EndFraction = StartFraction opposite Over hypotenuse EndFraction = StartFraction opposite Over 1 EndFraction Equals opposite
The length opposite the angle is the horizontal distance from the y-axis on the graph.
Sine StartFraction pi Over 3 EndFraction = StartFraction adjacent Over hypotenuse EndFraction = StartFraction adjacent Over 1 EndFraction Equals adjacent
The length adjacent to the angle is the horizontal distance from the y-axis on the graph.
Answers
Answer:
a. sine StartFraction pi Over 3 EndFraction = StartFraction opposite Over hypotenuse EndFractionEquals StartFraction opposite Over 1 EndFraction Equals opposite
The length opposite the angle is the vertical distance from the x-axis on the graph.
Step-by-step explanation:
sin (π/3) = opposite/hypotenuse = opposite/1; The length opposite the angle is the vertical distance from the x-axis on the graph.
What is the trigonometric ratio of unit circle?
Unit circle is a circle centered at the origin, (0,0) with radius(r) i.,e r= 1.
The angle π/3 forms an arc of length S and the x-axis and y-axis divide the coordinate plane and the unit circle as it is centered at (0.0).
Also, the terminal side intersect the circle at (x, y). Thus;
sin (π/3) = opposite/hypotenuse = opposite/1
Thus;
sin (π/3) = opposite
Also, the length opposite the angle is the vertical distance from the x-axis on the graph.
Read more about Unit circle at; https://brainly.com/question/4637991
1. A line with a slope of 2 passes through (3,-2). Write the equation in Point-Slope Form.
2. A line with a slope of 2 passes through (3,-2). Write the equation in Slope-Intercept Form.
Answers
2. Y=2x-8
Explanation:
Point-Slope Form: y-y1 = m(x-x1)
(x1,y1) => (3,-2)
Y-(-2)=2(x-(3)) —> y+2=2(x-3)
Slope-Intercept Form: y=mx+b
(3,-2). M=2 (slope)
-2=2(3)+b Y=2x-8
-2=6+b
-6
—————
-8=b
What would the equation be if my coordinates were, (-4,3) (2,6) ?
Answers
Answer:
y=1/2x+4
Step-by-step explanation:
change of x=2-(-4)=6
change of y=6-3=3
slope=3/6 or 1/2
y intercept=5
y=1/2x+5
Given the following linear function sketch the graph of the function and find the domain and range. ƒ(x) = -5x + 4
Answers
Answer:
Domain: (-∞,∞)
Range:(-∞,∞)
Step-by-step explanation:
The sketch of the given function has been attached and the domain and range both will be (-∞,∞ ).
What is a linear function?A straight line on the coordinate plane is represented by a linear function.
A function that fluctuates linearly with respect to the changing variable is referred to as being linear.
A linear function always has the same and constant slope.
The formula for a linear function is f(x) = ax + b, where a and b are real values.
Given the function,
ƒ(x) = -5x + 4
Range of the function ;
Since the value of the function is defined for all real x and the value will be unique so
Range = (-∞,∞ ).
The domain of the function ;
Since it is defined for all real x so,
Domain = (-∞,∞ )
For more about the linear function
brainly.com/question/21107621,
#SPJ2
what is positive 1 plus negative 6
Answers
Answer:
-5
Step-by-step explanation:
When answering these problems, I like to think of dirt stacks. (Weird, but it helps.)
Okay, so you have a stack of dirt one foot off the ground. Then you dig six feet down, including the stack. Because the stack will be 5 feet into the ground now, the answer is -5.
Emily is preparing cookies for her friends she is at the market where she sees a 14 ounce bag of chocolate chips for 3.50 the market also sells a 24 ounce bag for 5.28 which size bag is better to buy
Answers
Answer:
Bag 2
Step-by-step explanation:
Given
Bag 1:
[tex]14\ ounce = 3.50[/tex]
Bag 2:
[tex]24\ ounce = 5.28[/tex]
Required
Which is a better buy
To get which buy is better, we first get the unit price of each bag.
[tex]Unit\ Price = Amount/Size[/tex]
For bag 1:
[tex]Unit\ Price = 3.50/14[/tex]
[tex]Unit\ Price = 0.25[/tex]
For bag 2:
[tex]Unit\ Price = 5.28/24[/tex]
[tex]Unit\ Price = 0.22[/tex]
The best buy is the one with smaller unit price
Hence, bag 2 is better to buy
8% of $40 is $
plzzz help
Answers
Answer:
$3.2
Step-by-step explanation:
so there's a strategy to do this its very helpful!
100%-$40
8%-x
~cross multiply~
100x=320
÷100 ÷100
________________
x=$3.2
Hope this helps :D
can someone please help me with these two
Answers
Answer:
1)
x=8 degrees
<B=70 degrees
2)
<MON=57 degrees
Step-by-step explanation:
Detailed ans is in attachment
hope it helps :)
plz mark as brainliest
Graph the equation.
y-1 = -2(x-2)
Answers
Answer:
y-1 = -2(x-2) graph:
Answer:
Graph is below
Step-by-step explanation:
Graph the line using the slope and y-intercept, or two points.
Slope:
−
2
y-intercept:
(
0
,
5
)
x
y
0
5
5
2
0
The amount a basketball coach spends at a sporting goods store depends on the number of basketballs the coach the coach buys. The situation is represented by the function rule a=16b.
Complete the following table of values.
Answers
the sum in the form a(b+c) such that the integers band chave no common factor.
99 +33
Answers
Answer:
33(3+1)
Step-by-step explanation:
Find the GCF of 99 and 33
99=3^2•1133=3•11So the GCF is 3•11=33Factor 33 out
33(3+1)Name a ray with an endpoint of A.
Answers
The height reached by a model rocket can be calculated using the equation h= -16t2 + vt +
where t is the time the rocket is in flight, vis the rocket's velocity, and n is the rocket's beginning
height.
Calculate the height, h at 3 seconds given that the rocket travels 85 feet per second and starts on a
launch pad 15 feet off the ground.
Knowns:
t = 3 seconds
v = 85 feet per second
n = 15 feet
If we plug all those values in, our Formula becomes h= (–16.32) + (85.3) + (15).
Solve for h
Answers
Answer:
Step-by-step explanation:
9.0
The height (h) reached by model rocket will be equal to 126 ft.
It is given that the height reached by a model rocket can be calculated using the equation h = - 16 [tex]t^{2}[/tex] + ( v × t ) + n
We have to calculate the height (h) at 3 seconds.
What is algebra ?
Algebra is the branch of mathematics that deals with various symbols and the arithmetic operations such as multiplication , division , etc.
As per the question ;
h = - 16 [tex]t^{2}[/tex] + ( v × t ) + n ---------- Equation 1
∵ t = time rocket is in flight = 3 seconds
v = velocity at which rocket travels = 85 ft. per second
n = height of rocket at beginning = 15 ft
Putting these values in equation 1 , we get.
h = - 16 × ( 3² ) + (85) × (3) + 15
h = - 16 × 9 + 255 + 15
h = -144 + 270
h = 126 ft.
Thus , the height (h) reached by model rocket will be equal to 126 ft.
To learn more about algebra click here ;
https://brainly.com/question/19253676
#SPJ2
I need help this is my last question (math class)
Answers
Answer:
Step-by-step explanation:
5.00 for 2 loaves is $2.50 a loaf.
She wants 5 loaves for $12.50
5*2.50= $12.50 so she is correct.
since four loaves would be ten dollars. add one more loaf and that would cost $2.50. so $10 + $2.50 is $12.50
Can someone help me please
Answers
whats equivelent to 9a + 12
Answers
Answer:
Multiply both terms of the numbers
hense you would do: 2(9a + 12) = 18a + 24
It doesn't have to be 2 however. If can be any number you want it to be
Step-by-step explanation:
Answer:
12+9a is equivelent to 9a+12 :-)
Step-by-step explanation:
f(0) = 3 and f(1) = 15.
Answers
Answer:
Step-by-step explanat
Given=F(0) = 3 and f(1) = 15.
So f(X)=12x+3
When X=0, f(0)=12x0+3=3
When X=1, f(1) = 12x1+3= 15
So f(X)=12x+3 is the solution o
1) In a survey, Eric, the team manager, asked all 120 soccer players in the league which drink they preferred during
and after the game.
Drink During Game After Game
Sports Beverage 70
10
Juice
10
80
Water
40
30
Directions: For each statement below, tell if it accurately represents the information from the survey AND
explain how you made each decision. (SHOW YOUR THINKING)
a. During the game, players prefer juice to water by a ratio of 4 to 1.
b. 25% of the players prefer water after the game.
c. More than half of the players prefer a sports drink during the game.
Answers
If this helped can I please get brainliest
HELP NOW 15 Points!!!!!!!
Which expression is not equivalent to the other three?
2x + 2+ 2x + 3x – 8
4 + 7x – 2
–2 + 5x + 2x – 4
8x – x – 6
Answers
That is the different one.
Answer:
4+7x-2
Step-by-step explanation:
i dont know how nit is the anwser put i got it right
how do you write 7,870 in scientific notations ?
Answers
Answer:
7.87X10^3
Step-by-step explanation:
^ means to the power of
The equation of the line in the graph is y= (blank) x+ (blank)
Answers
A line passes through (1, –5) and (–3, 7). a. Write an equation for the line in point-slope form. b. Rewrite the equation in slope-intercept form.
Answers
Answer:
Point-slope form: y + 5 = 13/-4 (x-1)
Slope-intercept form:
Step-by-step explanation:
Ok, so first to find the equation in point-slope form, you need to find the slope. So, the slope would be 7- (-5)/-3-1 , which is 13/-4. That is your slope, or m. Now, for point-slope form, you have the equation y - y1 = 13/-4 (x-x1). Now, you take one of your points, say (1, -5), and you put it into the equation. So your point-slope form equation would be y - (-5) = 13/-4 (x - 1). This would simplify to y + 5 = 13/-4(x - 1).
Now for slope-intercept form, you have m, the slope, which is 13/-4. Substituting that into the equation, you have y = 13/-4x + b. To find b, you substitute one of your coordinates into the equation. For example, (1, -5). Then you have -5 = (13/-4 * 1) +b. Simplify this, and you have b = -5+ 13/-4, which is -20/4 + 13/-4, which is -20/4 - 13/4 = -33/4. So your final equation is y = 13/-4x - 33/4.
Hope that helped.
Given m|n, find the value of x.
(8x+6) (9x-30)
Answers
Step-by-step explanation:
(8x+6) (9x-30)
=72x²-240x+54x-180
=72x²-186-180
Answer:
(8x+6) (9x-30)
=72x²-240x+54x-180
=72x²-186-180
Step-by-step explanation:
Guys plz help Answer this QUESTION!!!
Answers
Answer:
I beliEve it is D
Step-by-step explanation: