Answer:
[tex]\mathbf{S =6 \sqrt{1 + \dfrac{\pi^2}{9} }+ \dfrac{18}{\pi} In (\dfrac{\pi}{3}+ \sqrt{1+ \dfrac{\pi^2}{9}})}[/tex]
Step-by-step explanation:
Given that
curve [tex]y = \dfrac{\pi x}{3}, 0 \leq x \leq 3[/tex]
The objective is to find the area of the surface obtained by rotating the above curve about the x-axis.
Suppose f is positive and posses a continuous derivative,
the surface is gotten by the rotating the curve about the x-axis is:
[tex]S = \int ^b_a 2 \pi f (x) \sqrt {1 + (f' (x))^2 } \ dx[/tex]
The derivative of the function [tex]y' = \dfrac{\pi}{3} cos \dfrac{\pi x}{3}[/tex]
As such, the surface area is:
[tex]S = \int ^3_0 2 \pi sin \dfrac{\pi x}{3} \sqrt {1 +(\dfrac{\pi}{3}cos \dfrac{\pi x}{3})^2 } \ dx[/tex]
Suppose ;
[tex]u = \dfrac{\pi}{3}cos \dfrac{\pi x}{3}[/tex]
[tex]du = -( \dfrac{\pi}{3})^2 sin \dfrac{\pi x}{3} \ dx[/tex]
If x = 0 , then [tex]u = \dfrac{\pi}{3}cos \dfrac{\pi (0)}{3} = \dfrac{\pi}{3}[/tex]
If x = 3 , then [tex]u = \dfrac{\pi}{3}cos \dfrac{\pi (3)}{3}[/tex]
[tex]u = \dfrac{\pi}{3}(-1)[/tex]
[tex]u = -\dfrac{\pi}{3}[/tex]
The equation for S can now be rewritten as:
[tex]S = \int^3_0 2 \pi sin \dfrac{\pi x}{3} \sqrt{1+(\dfrac{\pi}{3} cos \dfrac{\pi x}{3})^2 }\ dx[/tex]
[tex]S = 2 \pi \int ^{-\frac{\pi}{3} }_{\frac{\pi}{3}}(-\dfrac{9 \ du }{\pi^2} ) \sqrt{1+u^2}[/tex]
[tex]S = 18 \pi * \dfrac{1}{\pi ^2 } \int ^{-\frac{\pi}{3}}_{\frac{\pi}{3}} \sqrt{1+u^2} \ du[/tex]
[tex]S = \dfrac{18} {\pi} \int ^{-\frac{\pi}{3}}_{\frac{\pi}{3}} \sqrt{1+u^2} \ du[/tex]
[tex]S = \dfrac{18} {\pi} (2 \int ^{-\frac{\pi}{3}}_{0} \sqrt{1+u^2} \ du)[/tex]
since [tex](\int ^a_{-a} fdx = 2\int^a_0 fdx , f= \sqrt{1+u^2} \ is \ even )[/tex]
Applying the formula:
[tex]\int {\sqrt{1+x^2}} \ d x= \dfrac{x}{2} \sqrt{1+x^2}+ \dfrac{1}{2} In ( x + \sqrt{1+x^2})[/tex]
[tex]S = \dfrac{36}{x}[ \dfrac{u}{2} \sqrt{1+u^2}+ \dfrac{1}{2} \ In (u+ \sqrt{1+u^2}) ] ^{\frac{\pi}{3}}_{0}[/tex]
[tex]S = \dfrac{36}{x}[ \dfrac{\dfrac{\pi}{3}}{2} \sqrt{1+\dfrac{\pi^2}{9}}+ \dfrac{1}{2} \ In (\dfrac{\pi}{3}+ \sqrt{1+\dfrac{\pi^2}{9}})-0 ][/tex]
[tex]S =6 \sqrt{1 + \dfrac{\pi^2}{9} }+ \dfrac{18}{\pi} In (\dfrac{\pi}{3}+ \sqrt{1+ \dfrac{\pi^2}{9}})[/tex]
Therefore, the exact area of the surface is [tex]\mathbf{S =6 \sqrt{1 + \dfrac{\pi^2}{9} }+ \dfrac{18}{\pi} In (\dfrac{\pi}{3}+ \sqrt{1+ \dfrac{\pi^2}{9}})}[/tex]
The area of the surface is,[tex]S = 6\sqrt{1+\dfrac{\pi ^2}{3}} + \dfrac{18}{3}\ ln (\dfrac{\pi }{3}+\sqrt{1+\dfrac{\pi ^2} {9}})[/tex].
Given that,
The exact area of the surface obtained by rotating the curve about the x-axis. y = sin πx\3 , 0 ≤ x ≤ 3.
We have to determine,
Area of the surface obtained by rotating the curve.
According to the question,
Suppose f is positive and posses a continuous derivative,
The surface is gotten by the rotating the curve about the x-axis is:
Area of the surface is given by,
[tex]S = \int\limits^b_a {2\pi f(x) .\sqrt{1+ (f'(x))} } \, dx[/tex]
The given curve is x-axis,
[tex]y = \dfrac{sin\pi x}{3}[/tex]
The derivative of the function is,
[tex]\dfrac{dy}{dx} =\dfrac{\pi }{3} \dfrac{ cos\pi x}{3}[/tex]
The surface area is,
[tex]S = \int\limits^b_a {2\pi f(x) .\sqrt{1+(\dfrac{\pi }{3} \dfrac{ cos\pi x}{3})^2} }}} \, dx[/tex]
Substitute the value of f(x),
[tex]S = \int\limits^3_0{2\pi\dfrac{sin\pi x}{3} .\sqrt{1+(\dfrac{\pi }{3} \dfrac{ cos\pi x}{3})^2} }}} \, dx[/tex]
Suppose;
[tex]u = \dfrac{\pi }{3}cos\dfrac{\pi x}{3}dx\\\\\du =( \dfrac{-\pi }{3})^2 sin\dfrac{\pi x}{3}dx\\\\if \ x = 0, \ then \ u = \dfrac{\pi }{3}cos\dfrac{\pi (0)}{3} = \dfrac{\pi }{3}\\\\if \ x = 3, \ then \ u = \dfrac{\pi }{3}cos\dfrac{\pi (3)}{3} = \dfrac{\pi }{3}(-1)= \dfrac{-\pi }{3}[/tex]
Then,
[tex]S = \int\limits^3_0{2\pi\dfrac{sin\pi x}{3} .\sqrt{1+(\dfrac{\pi }{3} \dfrac{ cos\pi x}{3})^2} }}} \, dx\\\\S = 2\pi \int\limits^{\frac{-\pi}{3}}_ \frac{\pi }{3} (\dfrac{-9du}{\pi ^2})\sqrt{1+u^2} \ du\\\\S = 18\pi \times \dfrac{1}{\pi ^2}\int\limits^{\frac{-\pi}{3}}_ \frac{\pi }{3} \sqrt{1+u^2} \ du\\\\S = \dfrac{18}{\pi}\int\limits^{\frac{-\pi}{3}}_ {0} 2\sqrt{1+u^2} \ du\\\\\\[/tex]
Since,
[tex](\int\limits^a_{-a}{f} \, dx = 2\int\limits^a_0 {f} \, dx , \ f = \sqrt{1+u^2}\ is \ even)[/tex]
By applying the formula to solve the given integration,
[tex]\int {\sqrt{1+x^2} } \, dx = \dfrac{x}{2}\sqrt{1+x^2} + \dfrac{1}{2} \ ln(x+\sqrt{1+x^2})\\\\S = \dfrac{36}{2} [ \dfrac{u}{2} \sqrt{1+u^2} + \dfrac{1}{2} \ ln(u+\sqrt{1+u^2})}]^{\frac{\pi }{3}}_0\\\\[/tex]
[tex]S = \dfrac{36}{2} [ \dfrac{\dfrac{\pi }{3}}{2} \sqrt{1+\dfrac{\pi^2 }{9}} + \dfrac{1}{2} \ ln(\dfrac{\pi }{3}+\sqrt{1+\dfrac{\pi ^2} {9}}-0)]\\\\S = 6\sqrt{1+\dfrac{\pi ^2}{3}} + \dfrac{18}{3}\ ln (\dfrac{\pi }{3}+\sqrt{1+\dfrac{\pi ^2} {9}})[/tex]
Hence, The area of the surface is,[tex]S = 6\sqrt{1+\dfrac{\pi ^2}{3}} + \dfrac{18}{3}\ ln (\dfrac{\pi }{3}+\sqrt{1+\dfrac{\pi ^2} {9}})[/tex]
To know more about Integration click the link given below.
https://brainly.com/question/17256859
What is the value of x & y?
Answer:
x=23, y=14
Step-by-step explanation:
The triangles are indeed simular
Enter an inequality that represents the description, and then solve.
Dave has $10 to spend on a $7 book and two birthday cards (c) for his friends. How much can he spend
on each card if he buys the same card for each friend.
Answer:
$1.50 per card
Step-by-step explanation:
Knowns:
$10 to spends, Wants to buy $7 book and two equal priced cards.
10 = 7 + 2x
Subtract 7 from both sides to find out how much money Dave has after he buys his book.
3 = 2x
Divide each side by 2 to find out how much he cana spend on each card
x = 1.50
i hope this helps!
-TheBusinessMan
The answer to 48:(4+4)
Answer:
6:1
Step-by-step explanation:
48:( 4+4 )
= 48:8
= 6:1
please please please
Answer:
don't give up try and you will find the answers
the average of Shondra's test scores in physics is between 88 and 93 what is the inequality
Answer:
The inequality is
88<x<93
Step-by-step explanation:
The average of Shondra's test scores in physics is between 88 and 93.
Let me give out the meaning of some inequality symbols
<= Less than or equal to
>= Greater than or equal to
< Less than
> Greater than
Let the average score be x
In this case , the average score is between 88 and 93
The inequality is
88<x<93
Please help ASAP!!! ILL NAME YOU BRAINLIEST!!!
Answer:
18
Step-by-step explanation:
We can create a ratio of the amount of students who picked green over the total amount of students.
9 students picked the green pattern, and a total of 50 students were surveyed. This means the ratio is [tex]\frac{9}{50}[/tex].
To turn [tex]\frac{9}{50}[/tex] into a percentage, we must divide it's numerator by it's denominator and multiply by 100.
[tex]8\div50=0.18\\\\0.18\cdot100=18[/tex]
So the green bar's height will be 18.
Hope this helped!
I WILL MARK BRAINIEST PLEASE HELP ME ON THIS QUESTION
Lulu Ruby and Emma went shopping went a total of £261. Each of them had different amount of money. Lulu spent 2/3 of her money Ruby spent 1/2 of her money and Emma spent 3/4 of her money. Each of them spent the same amount of money.how much did money did they begin with?
Answer:
Lulu's money = £81
Ruby's money= £108
Emma's money= £72
Step-by-step explanation:
Let
Lulu's money = x
Ruby's money= y
Emma's money= z
x+y+z= 261
Lulu spent 2/3 of her money Ruby spent 1/2 of her money and Emma spent 3/4 of her money.
2/3x = 1/2y= 3/4z
2/3x= 1/2y
4/3x= y
2/3x= 3/4z
8/9x= z
x+y+z= 261
x+4/3x+8/9x= 261
9x+12x+8x= 2349
29x= 2349
X= 81
4/3x= y
4/3(81) =y
108= y
8/9x= z
8/9(81)= z
72= z
Answer:
Step-by-step explanation:
Each of them had different amount of money. Lulu spent 2/3 of her money Ruby spent 1/2 of her money and Emma spent 3/4 of her money.
1 answer
·
Top answer:
Answer:Lulu= £81Ruby=£108Emma=£ 72
4x=x+18 solve for x
Answer:
X = 6
Step-by-step explanation:
Subtract 1 x from both sides so the equation will be 3x = 18. Then divide 3x by 3 and 18 by 3 to get 6
Answer: [tex]x=6[/tex]
Subtract x from both sides
[tex]4x-x=x+18-x\\3x=18[/tex]
Divide both sides by 3
[tex]3x=3=18/3\\x=6[/tex]
Use the number line below, where RS = 7y +3, ST = 5y +8, and RT = 83.
a. What is the value of y?
b. Find RS and ST
a. What is the value of y?
Answer:
Step-by-step explanation:
Given the vectors based on the number line as RS = 7y +3, ST = 5y +8, and RT = 83, the equation RS+ST = RT will be used to get the unknown.
Substituting the given equation into the expression we will have;
7y +3+5y +8 = 83
collect like terms'
7y+5y+3+8 = 83
12y + 11 = 83
12y = 83-11
12y = 72
y = 72/12
y = 6
b) Substitute y = 6 into RS and ST
Given RS = 7y+3
RS = 7(6)+3
RS = 42+3
RS = 45
For ST;
ST = 5y+8
ST = 5(6)+8
ST = 30+8
ST = 38
I’m having trouble doing this
Answer:
0 Is Integer ,
rational and a
whole number.
Hope it helps.
In 5 blank, 278, the difference between the value of the digits in the blank and the blank. What is the blank digit in
Answer: The digit in the thousand place is '9'.
Step-by-step explanation: The given number is 5_278.
The difference between the values of the digits in the thousandth place and the tenth place is 8930.
The digits in the tenth place is already given, which is 7
Let's assume the digits in the thousandth place is x,
How does the value of the 8 in 20,831 compare to the value of the 8 in 20,381?
Answer:
The eight in the first number is ten times larger than the eight in the second number.
Quotient of 9 times x minus 24 and 6
Can somebody please help me write this into an expression?
Answer:
9x - 24 ÷ 6
Step-by-step explanation:
Quotient means that one number will be divided by another, so the division sign will be involved in the expression.
9 times x minus 24 translates to 9x - 24.
Now, put it all together.
9x - 24 ÷ 6 will be the expression.
Hope that helps.
Assume the rate of inflation is 8% per year for the next 2 years. What will be the cost of goods 2 years from now, adjusted for inflation, if the goods cost $280.00 today?
==============================================
Work Shown:
F = future value
P = present value = 280
r = rate of inflation in decimal form = 0.08
t = elapsed time in years = 2
---------
F = P*(1+r)^t
F = 280*(1+0.08)^2
F = 326.592
F = 326.59
Find the slope of the line that passes through each pair of points. Express as a fraction in simplest form. (-6, -2), (-1.5, 5.5)
Answer:
m = slope= 5/3
Step-by-step explanation:
[tex](-6, -2) = (x_1,y_1) \\ (-1.5, 5.5) (x_2 ,y_2) \\ \frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 -x _1} [/tex]
[tex] \frac{y - ( - 2)}{x - ( - 6)} = \frac{5.5 - ( -2 )}{ - 1.5 - ( - 6)} \\ \frac{y + 2}{x + 6} = \frac{5.5 + 2}{ - 1.5 + 6} [/tex]
[tex] \frac{y + 2}{x + 6} = \frac{7.5}{4.5} \\ 4.5(y + 2) = 7.5(x + 6)[/tex]
[tex]4.5y + 9 = 7.5x + 45 \\ 4.5y = 7.5x + 45 - 9 \\ 4.5y = 7.5x + 36[/tex]
Divide through by 4.5
[tex] \frac{4.5y}{4.5} = \frac{7.5x}{4.5} + \frac{36}{4.5} \\ y = \frac{5}{3} x + 8 \\ [/tex]
1. Find the cost of levelling the ground in the form of a triangle having the sides 51 m, 37 m and 20 m at the rate of ₹ 3 per m². 2. Find the area of the isosceles triangle whose perimeter is 11 cm and the base is 5 cm. 3. Find the area of the equilateral triangle whose each side is 8 cm. 4. The perimeter of an isosceles triangle is 32 cm. The ratio of the equal side to its base is 3 : 2. Find the area of the triangle.
1. Find the cost of levelling the ground in the form of a triangle having the sides 51 m, 37 m and 20 m at the rate of ₹ 3 per m².
a = 51 m, b = 37 m, c = 20 m
semiperimeter: p = (51+37+20):2 = 54 m
Area of triangle:
[tex]A=\sqrt{p(p-a)(p-b)(p-c)}\\\\A=\sqrt{54(54-51)(54-37)(54-20)}\\\\A=\sqrt{54\cdot3\cdot17\cdot34}\\\\A=\sqrt{9\cdot2\cdot3\cdot3\cdot17\cdot17\cdot2}\\\\A=3\cdot2\cdot3\cdot17\\\\A=306\,m^2[/tex]
Rate: ₹ 3 per m².
Cost: ₹ 3•306 = ₹ 918
2. Find the area of the isosceles triangle whose perimeter is 11 cm and the base is 5 cm.
a = 5 cm
a+2b = 11 cm ⇒ 2b = 6 cm ⇒ b = 3 cm
p = 11:2 = 5.5
[tex]A=\sqrt{5.5(5.5-3)^2(5.5-5)}\\\\ A=\sqrt{5.5\cdot(2.5)^2\cdot0.5}\\\\ A=\sqrt{11\cdot0.5\cdot(2.5)^2\cdot0.5}\\\\A=0.5\cdot2.5\cdot\sqrt{11}\\\\A=1.25\sqrt{11}\,cm^2\approx4.146\,cm^2[/tex]
3. Find the area of the equilateral triangle whose each side is 8 cm.
a = b = c = 8 cm
p = (8•3):2 = 12 cm
[tex]A=\sqrt{12(12-8)^3}\\\\ A=\sqrt{12\cdot4^3}\\\\ A=\sqrt{3\cdot4\cdot4\cdot4^2}\\\\A=4\cdot4\cdot\sqrt{3}\\\\A=16\sqrt3\ cm^2\approx27.713\ cm^2[/tex]
4. The perimeter of an isosceles triangle is 32 cm. The ratio of the equal side to its base is 3 : 2. Find the area of the triangle.
a = 2x
b = 3x
2x + 2•3x = 32 cm ⇒ 8x = 32 cm ⇒ x = 4 cm ⇒ a = 8 cm, b = 12 cm
p = 32:2 = 16 cm
[tex]A=\sqrt{16(16-8)(16-12)^2}\\\\ A=\sqrt{16\cdot8\cdot4^2}\\\\ A=\sqrt{2\cdot8\cdot8\cdot4^2}\\\\ A=8\cdot4\cdot\sqrt2\\\\ A=32\sqrt2\ cm^2\approx45.2548\ cm^2[/tex]
Express 0.2 degrees without decimals
Use back substitution to solve this problem
Answer:
z = -5
y = 1
x = 4
Step-by-step explanation:
2z = -10 ➡ z = -5
4y -(-5) = 9 ➡ 4y = 4 and y = 1
2x + 3 + (-5) = 6 ➡ 2x = 8 and x = 4
Two lines intersect in a plane in form for angles what are the angles formed fathers intersect is a 53° angle what are the measures of the other three angles explain your answer
Answer:
53°, 127°, 127°
Step-by-step explanation:
Two intersecting lines form two pairs of angles:
Pair of vertical angles. They are opposite to each other and are equalPair of adjacent angles which are supplementary angles and their sum is 180°So if one of the angles is measured 53°, then the other angles are:
53°- vertical angle with the first one180° - 53° = 127°180° - 53° = 127°Find the coordinates of the midpoint of a segment with the endpoints M(6, −41) and N(−18, −27). Then find the distance between the points.
Answer:
Midpoint is (-6,-34)
Step-by-step explanation:
(6+(-18))/2. (-41+(-27))/2
-12/2. (-68)/2
-6. , - 34
Distance =(6--18)^2 +(-14--27)^2
23^2. +13^2
529 +169
= 698
So u will find the square root of 698
The ans u get is the distance
Write an equation that represents the perimeter of the rectangle. The width of a rectangle is 9 less than one-third it's width, when the perimeter is 45.
Answer:
The equation that represents the perimeter of the rectangle is:
[tex]\text{Perimeter} = \frac{2}{3} \times [4l-27][/tex]
Step-by-step explanation:
The perimeter of a rectangle is given by the formula:
Perimeter = 2 × [l + w]
It is provided that the width of a rectangle is 9 less than one-third it's length.
That is:
[tex]w = \frac{1}{3}l-9[/tex]
The perimeter is given as 45.
The equation that represents the perimeter of the rectangle is:
[tex]\text{Perimeter} = 2 \times [l + \frac{1}{3}l-9]\\\\\text{Perimeter} = 2 \times [\frac{3l+l-27}{3}]\\\\\text{Perimeter} = 2 \times [\frac{4l-27}{3}]\\\\\text{Perimeter} = \frac{2}{3} \times [4l-27][/tex]
Thus, the equation that represents the perimeter of the rectangle is:
[tex]\text{Perimeter} = \frac{2}{3} \times [4l-27][/tex]
What is the domain of the function? f(x)=[tex]\frac{x-3}{2x^{2}+x-21 }[/tex] (−∞,3)∪(3,∞) (−∞,−3)∪(−3,72)∪(72,∞) (−∞,−72)∪(−72,∞) (−∞,−72)∪(−72,3)∪(3,∞)
Answer:
(−∞,−7/2)∪(−7/2,3)∪(3,∞)
Step-by-step explanation:
(x-3)
----------------
2x^2 +x -21
First factor the denominator
(x-3)
----------------
( 2x +7) (x-3)
The domain is restricted when the denominator goes to zero
2x+7 =0 x-3 =0
2x = -7 x-3=0
x = -7/2 x =3
This two points are not in the domain
(−∞,−7/2)∪(−7/2,3)∪(3,∞)
a computer printer can print 10 pages per minute.
Answer:
what's the question though ?
A caterer estimates that 2 gallons of fruit punch will serve five people.
How much additional punch is necessary to serve 75 people?
Need all the explanations
Answer:
the answer is 28
Step-by-step explanation:
This is simple
if 5 people need 2 gallons we can use that to figure out how many gallons 75 people need.
75 divided by 5 is 15. so if 5 fits into 75 15 times that means that 75 people need 15 times 2 gallons or 30 gallons of fruit punch.
But the problem ask how much "additional" fruit punch so we subtract 2 from 30 to get 28 additional gallons of fruit punch
What is the length of Line segment A B? Round to the nearest tenth. Triangle A B C is shown. Angle A C B is a right angle and angle C A B is 75 degrees. The length of C A is 10 meters and the length of hypotenuse A B is x. 9.7 m 10.4 m 37.3 m 38.6 m
Answer:
AB = 38.6metres
Step-by-step explanation:
Given the right angled triangle ABC with hypotenuse AB and ∠CAB to be 75°, the opposite side of the triangle will be the side facing the angle ∠CAB directly and this is side BC. The adjacent side of the triangle will be side CA.
To get the length of the hypotenuse AB, we will use the trigonometry identity SOH, CAH, TOA.
According to CAH;
Cos∠CAB = adjacent/hypotenuse = CA/AB
Substituting CA = 10 metres and ∠CAB = 75° into the equation;
Cos75° = 10/AB
AB = 10/cos75°
AB = 10/0.2588
AB = 38.6metres
Hence the length of the hypotenuse AB is 38.6metres.
Answer:
38.6 m
Step-by-step explanation:
In a four year period, about 80,000 acres of coastal wetlands in the United States are lost each year. What integer represents the total change in coastal wetlands? Write an equation and solve. (Hint: They are being lost.)
Answer:
-320000
Step-by-step explanation:
Given
Lost Land = 80,000 acres yearly
Duration 4 years
Required
The total change
Represent the total change with y;
y is calculated as thus;
[tex]y = Lost\ Land * Duration[/tex]
[tex]y = -80,000 acres * 4[/tex]
[tex]y = -320000\ acres[/tex]
Hence, a total of 320000 acres were lost during the period
Find the midpoint of the segment with the following endpoints.
(6,4) and (9,1)
Answer:
[tex]=\left(\frac{15}{2},\:\frac{5}{2}\right)[/tex]
Step-by-step explanation:
[tex]\mathrm{Midpoint\:of\:}\left(x_1,\:y_1\right),\:\left(x_2,\:y_2\right):\quad \left(\frac{x_2+x_1}{2},\:\:\frac{y_2+y_1}{2}\right)\\\\\left(x_1,\:y_1\right)=\left(6,\:4\right),\:\left(x_2,\:y_2\right)=\left(9,\:1\right)\\\\=\left(\frac{9+6}{2},\:\frac{1+4}{2}\right)\\\\=(\frac{15}{2} , \frac{5}{2} )\\\\[/tex]
3
Which function family does f(x)=-1/2x + 7 belong to?
Answer:
Linear
Step-by-step explanation:
Answer:
the answer is Linear
Step-by-step explanation:
9x-2y=-6 5x+4y=12
solve by substitution
Answer:
x = 0
y = 3
Step-by-step explanation:
Step 1: Write systems of equations
9x - 2y = -6
5x + 4y = 12
Step 2: Rewrite 1st equation
9x = 2y - 6
x = 2/9y - 2/3
Step 3: Substitute
5(2/9y - 2/3) + 4y = 12
Step 4: Solve for y
10/9y - 10/3 + 4y = 12
46/9y - 10/3 = 12
46/9y = 46/3
y = 3
Step 5: Plug in y to find x
5x + 4(3) = 12
5x + 12 = 12
5x = 0
x = 0
Suppose a regular n-gon is inscribed in a circle of radius r. Diagrams are shown for n=6, n=8, and n = 12. n = 6 n=8 n = 12 a a VT a Imagine how the relationship between the n-gon and the circle changes as n increases. 1. Describe the relationship between the area of the n-gon and the area of the circle as n increases.
Answer:If an N-gon (polygon with N sides) has perimeter P, then each of the
N sides has length P/N. If we connect two adjacent vertices to the
center, the angle between these two lines is 360/N degrees, or 2*pi/N
radians. (Do you understand radian measure well enough to follow me
this far?)
The two lines I just drew, plus the side of the polygon between them,
form an isosceles triangle. Adding the altitude of the isosceles
triangle makes two right triangles, and we can use one of them to
derive the equation
sin(theta/2) = s/(2R)
where theta is the apex angle (which I said is 2*pi/N radians), R is
the length of the lines to the center (the radius of the circumscribed
circle), and s is the length of the side (which I said is P/N).
Putting those values into the equation, we have
sin(pi/N) = P/(2NR)
so that
P = 2NR sin(pi/N)
gives the perimeter of the N-gon with circumradius R.
Can we see a connection between this formula and the perimeter of a
circle? The perimeter of the circumcircle is 2*pi*R. As we increase
N, the perimeter of the polygon should get closer and closer to this
value. Comparing the two, we see
2NR*sin(pi/N) approaches 2*pi*R
N*sin(pi/N) approaches pi
You can check this out with a calculator, using big numbers for N such
as your teacher's N=1000. If you calculate the sine of an angle in
degrees rather than radians, the formula will look like
N*sin(180/N) --> pi
Step-by-step explanation: