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
Answer:
Where α is an acute angle (first figure)
The area of the shaded triangle = ((√3)·a²/4)·sin(α)·csc(120 - α))
Where α is an obtuse angle (second figure)
The required area of the shaded region = (√3)·a²/4 + (√3)·a²/4)·sin(α)·sec(α + π/6)
Step-by-step explanation:
Where α is an acute angle (first figure)
The given parameters are;
The given triangle = Equilateral Triangle
Let the sides of the equilateral triangle = 2·a
Therefore;
The measure of each interior angles of the given triangle = 60°
Let c represent the side of the shaded triangle opposite ∠α and b represent the side of the shaded triangle opposite ∠60° and c, represent the third side of the shaded triangle, we have;
The sides of the equilateral triangle = 2·a
By sine rule, we have;
c/sin(α) = b/sin(60°) = a/sin(180 - (60 + α)) = a/sin(120 - α))
b = sin(60°) × a/sin(120 - α)) = (√3)/2 × a/sin(120 - α))
The area of the shaded triangle = 1/2 × a × b × sin(α) = 1/2 × a × (√3)/2 × a/sin(120 - α)) × sin(α) = ((√3)·a²/4)·sin(α)·csc(120 - α))
The area of the shaded triangle = ((√3)·a²/4)·sin(α)·csc(120 - α))
Where α is an obtuse angle (second figure)
The required area of the shaded region = The area of the equilateral triangle - The area of the small unshaded triangle, with base side a and interior angles, (180° - α), 60° and ((180 - (180° - α) - 60°) = ) α - 60°
The area of the unshaded triangle is found as follows;
By sine rule, we have;
c/sin(180° - α) = b/sin(60°) = a/sin(α - 60°)
b = sin(60°) × a/sin(α - 60°) = (√3)/2 × a/sin(α - 60°)
The area of the unshaded triangle = 1/2 × a × b × sin(α) = 1/2 × a × (√3)/2 × a/sin(α - 60°) × sin(α) = -((√3)·a²/4)·sin(α)·sec(α + π/6)
The area of the shaded triangle = -((√3)·a²/4)·sin(α)·sec(α + π/6)
The required area of the shaded region = 1/2×a²·sin(60°) - (-((√3)·a²/4)·sin(α)·sec(α + π/6))
The required area of the shaded region = (√3)·a²/4 + (√3)·a²/4)·sin(α)·sec(α + π/6)
Related Questions
Consider the function f(t) = StartRoot 3 t minus 9 EndRoot. You cannot take the square root of a negative number, so 3t – 9 must be zero.
Answers
Answer:
Greater than or equal to
Step-by-step explanation:
Answer:
1. Consider the function f(t) = .You cannot take the square root of a negative number, so 3t – 9 must be ____ to zero.
answer: C. greater thanless than or equal to✔
2. Set up an inequality showing that the radicand cannot be negative.
answer: A. 3t-9≥ 0
3.What is the domain of the function?
answer: D. Domain: t ≥ 3
hope this helps :)
plzzz help find the missing length indicated????
Answers
Answer:
The missing length is 64
Step-by-step explanation:
In the given figure
∵ The triangle is a right triangle
∵ There is a line drawn from the right angle ⊥ to the hypotenuse
→ There is a relation between the side of length 60, the part of the
length 36, and the length of the hypotenuse
∵ The length of the hypotenuse = x + 36
∴ (60)² = 36 × (x + 36)
→ Multiply the bracket by 36
∴ 3600 = 36(x) + 36(36)
∴ 3600 = 36x + 1296
→ Subtract 1296 from both sides
∵ 3600 - 1296 = 36x + 1296 - 1296
∴ 2304 = 36x
→ Divide both sides by 36 to find x
∴ 64 = x
∴ The missing length is 64
Please help, will mark you as the brainiest! Please do NOT answer if you don’t know it!
Answers
Answer:
if oq or and os are bisectors then MN is 40, also it is a perpendicular angle so it would also be 40
Step-by-step explanation:
Find the value of x.
Answers
Answer:
Step-by-step explanation:
I realize that the time is passed.... but FYI
BD is Geometric Mean
That's mean BD² = 6 × 5 = 30 ⇒ BD = √30
x² = BD² + 6² = 30 + 36 = 66 ⇒ x = √66
y = √55
Answer: 66
Step-by-step explanation:
ΔADC and ΔABD are similar (AAA)
Therefore the cooresponging sides are in proportion: AD/AC = AB/AD
A fireman is on the middle step of a ladder. (There are the same number of steps above him as below him.) If he goes down 4 steps, then up 7 steps, and then down 13 steps, he will be on the first step above the ground. How many steps does the ladder have?
Answers
Answer:
The ladder has 20 steps
Step-by-step explanation:
10 is the direct middle if you go down 4 that is 6, if you go back up 7 you get 13 and if you go back down 13 and you are on the first step 13-13 is 0, so the ladder has 20 steps.
What is the valur of y in the equation 2(3y - 6) = 0?
Answers
Answer:
y=2
Step-by-step explanation:
Distribute the 2 into 3y and -6:
6y-12=0
Move the -12 to the other side by adding both sides by 12:
6y-12+12=0+12
=
6y=12
divide both sides by 6:
y=2
A function is a relationship between two quantities, called the ___ and the ___?
Answers
Answer:
domain and the range
Step-by-step explanation:
please help my sister needs to turn this in soon
Snowy Mountain in Alaska has an elevation of 7,093 feet. Together, Snowy
Mountain and Mammoth Mountain in California have a combined elevation of
18, 152 feet. What is the difference in elevation between Snowy Mountain and
Mammoth Mountain?
The difference in elevation between Snowy Mountain and Mammoth Mountain is
feet.
Answers
Answer:
11,059
Step-by-step explanation:
If you take the combined elevation (18,152) and subtract the elevation of Snowy Mountain (7,093), you should get 11,059.
Answer:
I believe 3966
Step-by-step explanation:
18152-7093=11059
11059-7093=3966
Find the equation of the line.
Use exact numbers.
Answers
Answer:
y=-3/2x+3
Good luck:)
-El papá de Susana es carpintero y en su trabajo utiliza clavos y tornillos de distintas medidas. Ahora, le advirtieron que dentro del muro hay una tubería a 4 centímetros del lugar en el que clavará las maderas. Por lo que Susana quiere saber qué clavos usará su papá. Pero como la medida de los clavos viene en pulgadas, ella quiere conocer la medida en centímetros, para saber qué medida de clavos no dañaría la tubería. Completa la siguiente tabla, convirtiendo la medida de los clavos de pulgadas a centímetros. ¿Cuáles clavos debe usar el Papá de Susana para no dañar la tubería?
Answers
Answer:
la tabla no esta, asi que busque la pregunta por interenet y encontre lo siguiente:
medida de los clavos equivalencia en
en pulgadas centimetros
1 pulgada 2.54 cms
¹/₂ pulgada 2.54 x 0.5 = 1.27 cms
³/₄ pulgada 2.54 x 0.75 = 1.905 cms
¹/₈ pulgada 2.54 x 0.125 = 0.3175 cms
3¹/₂ pulgadas 2.54 x 3.5 = 8.89 cms
The perimeter of a rectangle is 36 cm. The length is 10 cm longer than the
width. What are the length and width of the rectangle?
Answers
Answer:
Below
Step-by-step explanation:
X(x+10)=36
x= -5+sqrt 61 or-5-sqrt 61
since only the first one bigger than0, x is -5+sqrt61 and other side is sqrt61+5
What is the mapping notation that will transform triangle ABC to triangle A'B'C'?
Answers
Answer:
A'B'C' = A'(8,-3) B'(8,-6) C'(2,-6)
Step-by-step explanation:
Hope this helps you.
Answer:
Abc=e A
Step-by-step explanation:
someone explained
How do you find the degree of a polynomial function/ graph in algebra 2??
Answers
Answer:
which way are the arrows facing on the graph? How many times does the fuction change directions?
Step-by-step explanation:
if arrows are going in same direction, degree is even and then count the number of direction changes for exact number
if arrows are going in opposite directions, degree is odd and then count direction changes
Suppose $P$ is the point $(5,3)$ and $Q$ is the point $(-3,6)$. Find point $T$ such that $Q$ is the midpoint of segment $\overline{PT}$.
help qwikly
Answers
Answer:
(1,4.5)
([tex]\frac{5+(-3)}{2} ,\frac{3+6}{2}[/tex])
I’m pretty bad at math :/ cam ya;; help its due today/ What is the slope of the line represented by
y= 12+ 1/2x ?
Answers
Answer:
1/2 (1 , 2)
Step-by-step explanation:
the number before the variable is the slope
A scout troop sold cups of lemonade to raise money for a camping trip. The graph
below represents the situation.
Answers
Answer:
Part A- 3
Part B- 5
Step-by-step explanation:
Answer:
Part A: 3
Part B: 5y
step explanation:
urgent!! hal earns $5 doing chores for his parents. he also earns $10 a week for each dog, d, he walks.
Answers
Answer:
5+10d=x The dogs he walks are the ind variable and the total money he earns is the dep variable
Step-by-step explanation:
If he earns five dollars a week then that is a confirmed amount which means it’s just $5. Then for every dog he walks that’s $10 so if he walks 0 dogs then he gets $0.
The dogs he walks is ind because this variable is the variable that changes and effects the total whereas, the dep is the total because that’s what changes depending on the ind variable
Working alone, Dan can do a certain job in three hours and Stan can do the same job in two hours. At these rates, how long would it take Dan and Stan to do this job working together?
Answers
Answer:
2 hours and thirty minutes
Step-by-step explanation:
3+2=5
5/2=2.5
manuel calculates the business costs and profits to produce and hiking backpacks. manuel's profit is his revenue minus his expenses.
A. Write a function to present the profit Manuel makes selling n backpacks.
B. Graph the profit function. What is a reasonable domain for this function for one year if his revenue is between $4,000 and $30,000? Is the function discrete or continuous? Explain
C. How much is his profit if he sells 43 backpacks? Explain
setup cost is $5,250 assembly cost is $54/ backpack revenue is $159/ backpack
Answers
Answer:
A. P = n × $105 - $5,250
B. Please find attached the required graph
The domain for the range ($30,000 to $4,000) = 25 < n < 189
C. The profit from selling 43 backpacks = -$735
Step-by-step explanation:
A. The given parameters are;
The setup cost = $5,250
The assembly cost = $54/backpack
The revenue = $159/backpack
Let n represent the number of backpack produced
Therefore, we have;
Profit, P = $ 159 × n - $54 × n - $5,250
P = n × ($ 159 - $54) - $5,250 = n × $105 - $5,250
P = n × $105 - $5,250
B. Please fins attached the graph of the profit function, created with Microsoft Excel
The domain for a range of revenue of $30,000 to $4,000 is given as follows;
The domain for the range ($30,000 to $4,000) = 25 < n < 189
C. The amount of profit from selling 43 bags = P(43) = 43 × $105 - $5,250 = -$735
Can someone help with this. Heads up it's ratios
Answers
Answer:
The median side measures 92 yards
Step-by-step explanation:
Let's call x, y, and z the measures of the sides of the triangle.
The ratios between them are:
[tex]\displaystyle \frac{x}{y}=\frac{4}{7}[/tex]
[tex]\displaystyle \frac{x}{z}=\frac{4}{2}=2[/tex]
The first ratio gives the relation:
[tex]\displaystyle y=\frac{7x}{4}[/tex]
The second ratio can be solved for z:
[tex]\displaystyle z=\frac{x}{2}[/tex]
Since the perimeter is 299 yards:
[tex]x+y+z=299[/tex]
Substituting the above relations:
[tex]\displaystyle x+\frac{7x}{4}+\frac{x}{2}=299[/tex]
Multiplying by 4:
[tex]4x + 7x+2x=1196[/tex]
Joining like terms:
[tex]13x=1196[/tex]
Dividing by 13:
[tex]x=1196/13[/tex]
x=92
Now calculate y:
[tex]\displaystyle y=\frac{7\cdot 92}{4}[/tex]
y = 161
[tex]\displaystyle z=\frac{92}{2}[/tex]
z=46
The median side measures 92 yards
A real estate agent earned commission of $8,785 for selling a house. If his rate is 7%, find the selling price of the house.
Answers
Answer:
$125,500
Step-by-step explanation:
Make "x" equal the selling price of the house
7% of x is the commission --->
.07 * x = 8785
x= 8785/ .07
x= 125500
I need it now plzs
Felipe sold t-shirts and hats at a festival. He made a $5 profit for each t-shirt he sold. He also made a profit of $40 from selling hats. If he made a total profit of $125, how many t-shirts did he sell?
Answers
Answer:
17 t-shirts sold
Step-by-step explanation:
If we assume that we don't need t find the number of hats, then we can say that the profit from -tshirts would be $125 minus 40 or $85
Then we can write the following equation of x is the number of t-shirts:
5x = 85
If we solve by diving both sides of the equation by 5:
5x/5 = 85/5
x = 17 t-shirts sold
Help me out plz struggling. Thx
Answers
(02.05 LC) Solve the inequality 6x − 8 > 4x + 26. (5 points) Group of answer choices x 9 x > 17
Answers
The answer is x >1 7
(6x+1)+(-6x+1)=0
What property is this
Answers
Answer:
i believe this is the distriputive property
Step-by-step explanation:
hope this helpss
Suppose x, y, and z are different positive integers such that xy, yz, and zx are all perfect
squares. Must x, y, and z be perfect squares? Why or why not?
Answers
Answer: Yes, and the numbers are 4, 36, and 9
Step-by-step explanation:
Yes, because it x=4, y=36, and z=9 then 4 * 36=144, 144=12*12 4*9=36, 36=6*6, and 9*36=324, 324=18*18. As you can see they are all perfect squares and 4, 36, and 9 are all perfect sqaures too.
Sophia earned $480 in 32 hours. How many dollars did she earn per hour?
Sophia earne
dollars per hour.
Answers
Answer:
15 dollars an hour
Step-by-step explanation:
480/32=15
Answer:
Sophia Earned $15
Step-by-step explanation:
All you have to do is divide the amount she earned by the hours she worked to see how much money she earned per hour, then you get $15 dollars per hour.
help math pls geometry
Answers
Answer:
The point is (8,2)
Step-by-step explanation:
Is this a square or rectangle or rhombus, etc.? Yes, because all the lines are all parallel.
Are AB and FG are parallel to each other right? Yes, so that means whatever you do to F will happen to G and we can start on A, it goes 6 down, then two to the right.
Since all lines are parallel how many steps we do on A to get F, we have to do to B to get G. So, (8,2).
Hope this helps!
What is the value of x if 5^x+2 =5^9
Answers
Answer: 7
Step-by-step explanation: If they equal each other, then that must mean that the exponents equal. Therefore, we can draw out the conclusion that x+2=9, which is x=7
It’s almost your birthday and you are saving up to buy a car. So far, you have $300 in your account. Each month, you will deposit another $200
Answers
Answer:
300+200
Step-by-step explanation:
did not give a equation
Answer:
300 + 200x
Step-by-step explanation:
200 is the one that's changing so it has the x.