Mathematics High School

let f be the function given by f(x)=1(2 x). what is the coefficient of x3 in the taylor series for f about x = 0 ?

Answers

Answer 1

The coefficient of x^3 in the Taylor series for f(x) is 0, since there is no term involving x^3.

To find the Taylor series of the function f(x) = 1/(2x) about x = 0, we can use the formula:

[tex]f(x) = f(0) + f'(0)x + (1/2!)f''(0)x^2 + (1/3!)f'''(0)x^3 + ...[/tex]

where f'(x), f''(x), f'''(x), etc. denote the derivatives of f(x).

First, we need to find the derivatives of f(x):

f'(x) = -1/(2x^2)

f''(x) = 2/(x^3)

f'''(x) = -6/(x^4)

f''''(x) = 24/(x^5)

Next, we evaluate these derivatives at x = 0 to get:

f(0) = 1/(2(0)) = undefined

f'(0) = -1/(2(0)^2) = undefined

f''(0) = 2/(0)^3 = undefined

f'''(0) = -6/(0)^4 = undefined

f''''(0) = 24/(0)^5 = undefined

Since the derivatives are undefined at x = 0, we need to use a different method to find the Taylor series. We can use the identity:

1/(1 - t) = 1 + t + t^2 + t^3 + ...

where |t| < 1.

Substituting t = -x^2/a^2, we get:

1/(1 + x^2/a^2) = 1 - x^2/a^2 + x^4/a^4 - x^6/a^6 + ...

This is the Taylor series for 1/(1 + x^2/a^2) about x = 0. To get the Taylor series for f(x) = 1/(2x), we need to replace x with ax^2:

f(x) = 1/(2(ax^2)) = 1/(2a) * 1/(1 + x^2/a^2)

Substituting the Taylor series for 1/(1 + x^2/a^2), we get:

f(x) = 1/(2a) - x^2/(2a^3) + x^4/(2a^5) - x^6/(2a^7) + ...

Therefore, the coefficient of x^3 in the Taylor series for f(x) is 0, since there is no term involving x^3.

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Related Questions

Question 1
Which equation represents the relationship shown in the table
at the right?

Answers

Answer:

Step-by-step explanation:

b = a + b table chart what equation represents the relationship between a and b shown in the table .

slope and why intercept?

Answers

Answer:

Slope: -1/3

Y-intercept: 0

Step-by-step explanation:

The lines passes the y-axis at 0, and since the line is going downwards, the slope will be negative.

I hope it helps! Have a great day!

bren~

Given that DRWH is a parallelogram, determine the values of x and y.

Answers

Answer:

the opposite sides of the parallelogram are equal.

so DR = HW mean : 2 y + 2 = 3 y - 9

3y - 2y = 9+2

y = 11

RW = y+4 = 11+4 = 15

DH = RW

so, 3 x + 6 = 15

3x = 15-6 = 9

x = 3

A rectangular field is to be enclosed by 400m of fencing. Use quadratic to
determine what dimensions will result in the maximum area inside the fence. Show
all your work.

Answers

Work:

First it's a rectangle so its area is equal to the product (multiplication) of both sides of the rectangle (dimensions). A = a x b = ab

NOW: knowing the the circumference or perimeter is equal to 400m, we can say that P = 400 = 2(a+b) = 2a + 2b since the given polynom is rectangle. 2a + 2b = 400 <==> 2a = 400 - 2b <==> a = 200 - b.

We gave an expression of a in function of b. Now we can replace the variable a by 200 - b in the first expression of the area.

A = ab = (200-b)b = 200b-b^2 = -b^2 + 200b

A is now a quadratic equation. We note A(b) the epression -b^2 +200b so:

A(b) = -b^2 +200b

We can already see that A is a quadratic equation of the form:

ax^2 + b + c. The a coefficient is negative which will lead to closed parabola when looking from the top. Now we need to find the maximum of the function by using the derivatives:

A(b) = -b^2 +200b <==> A'(b) = -2b + 200b

and -2b + 200b = 0 <==> b = 100;

So the derivative function crosses the x-axis at (100; 0).

So is increasing over ]-∞; 100] and decreasing over [100; ∞[.

We obtain a maxima on (100; x) with A. To find it we need to replace 100 by b in the function A.

A(b) = -b^2 + 200b

<==> A(100) = -100^2 + 200 x 100 = - 10000 + 20000 = 10000

Now let's find a: if b = 100 what equals a ?

We know that ab = 10000 <==> 100a = 10000 <==> a = 100;

And we verify that ab = 100 x 100 = 10000.

The rectangle needs to be a square to reach the maximum area of 10000m^2 or also 0.01 km^2.

Q.E.D.

Mr. Krueger wanted a square platform built in the drama room at
school. He provided the carpenter with the desired area of the platform
to be built. Which function can he use to determine the side length of
the square platform?

F(x)=x²

F(x)=√x

F(x) = -=-=

F(x) = |x|

Answers

Answer: F(x) = sqrt(x)

Explanation: inverse of square is square root. Since the area is the side length squared, take the square root of the area to find the side length.

PLEASE HELP IM SERIOUSLY STUCK ON THIS

Answers

The purchase price of the new car if the sales tax is $3500 is $50000

Direct variation

Let the amount of sales tax be A and the purchase price be p

If the amount of sales tax is directly proportional to the purchase price, this is expressed as:

S = kp

If S = $1750 and p = $25000, the;

k = S/p

k = 1750/25000

k = 175/2500

If the sales tax is $3500, the purchase price will be given as:

S = kp
3500 = 175/2500 p

175p = 3500*2500
175p = 8750000

p = 8750000/175

p = 50000

Hence the purchase price of the new car if the sales tax is $3500 is $50000

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Please help me on this geometry question

Answers

Answer:

? = 3

Step-by-step explanation:

given a line parallel to a side of the triangle and intersecting the other 2 sides, then it divides those sides proportionally , that is

[tex]\frac{?}{6}[/tex] = [tex]\frac{2}{4}[/tex] ( cross- multiply )

4? = 12 ( divide both sides by 4 )

? = 3

Answer:

Step-by-step explanation:

Because the triangles are similar

6/4 = (?+6)/(2+4)

6/4 = (?+6)/6

Cross multiply

36 = 4?+24

4?=12

?=3

PLEASE ANSWER QUICKLY

Answers

Answer:

Step-by-step explanation:

See the attached image.

find the area of the diagram

Answers

Answer:

13500 ft²

Step-by-step explanation:

Area of rectangle:

We can find the width of the rectangle using Pythagorean theorem.

AD² + DC² = AC²

180² + DC² = 195²

32400 + DC² = 38025

DC² = 38025 - 32400

= 5625

DC = √5625

= 75 ft

length = 75 ft

Width = 180 ft

[tex]\sf \boxed{\text{\bf Area of rectangle = length * width}}[/tex]

= 75 * 180

= 13500 ft²

The picnic breakfast cost $12. Jasmine left a tip that was 15 percent of the
cost of the meal. How much money was the tip that Jasmine left?

Answers

12 x 0.15 = $1.80

0.15 = 15%

Determine an equation that describes the number of bacteria in both the foods when they are mixed.

Answers

The equation that describes the number of bacteria in both the foods when they are mixed is; Option C: 35T₂ + 55T + 450

How to simply quadratic Equations?

We are given the equations that describes the number of bacteria in both the foods when they are mixed.

Equation for first bacteria is;

N₁(T) = 15T₂ + 60T + 300

Equation for second Bacteria is;

N₂(T) = 20T² - 5T + 150

The equation that describes the number of bacteria in both the foods when they are mixed is;

N₁(T) + N₂(T) = 15T₂ + 60T + 300 + 20T² - 5T + 150

⇒ 35T₂ + 55T + 450

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Round 5 358 708 to the nearest million

Answers

Answer:5 million

Step-by-step explanation: Because 300 thousand is below 500 thousand so if its below 500 thousand it cant round up it rounds down.

Find the area of a circle with radius, r =
7.4m.
Give your answer rounded to 2 DP.
r
C
The diagram is not drawn to scale.

Answers

Answer:

172.03 m²

Step-by-step explanation:

Hello!

Area of a circle: [tex]A = \pi r^2[/tex]

A = areaπ = pir = radius

Plug in the radius into the formula to find the area.

Find the Area[tex]A = \pi r^2[/tex][tex]A = \pi (7.4)^2[/tex][tex]A = 54.76\pi[/tex][tex]A = 172.0336137106 \approx 172.03[/tex]

The area is approximately 172.03 m².

To calculate the radius of this circle, we must take into account that:

Data:

R = 7.4 π = 3.14

To calculate the area of the circle, we apply the following formula:

A = π * r², where

a = area

π = pi

r = radius

Substituting the values of the equation:

[tex]\large\displaystyle\text{$\begin{gathered}\sf A=3.14*(7.4 \ m)^{2} \end{gathered}$}[/tex]

Taking the square root

[tex]\large\displaystyle\text{$\begin{gathered}\sf A=3.14*54.76 \ m^{2} \end{gathered}$}[/tex]

Multiplying

[tex]\boxed{\large\displaystyle\text{$\begin{gathered}\sf A=171.95 \ m^{2} \end{gathered}$}}[/tex]

Answer: The approximate area of the circle is 171.95 m².

Consider the volume of the region shown below, which shows a right circular cone with a top radius of 3 cm and height 9cm

Answers

Answer:

84.82 cm

Step-by-step explanation:

Volume of a cone: [tex]\frac{1}{3} \pi r^2h[/tex]

[tex]\frac{1}{3} \pi r^2h\\\\\\\\= \frac{1}{3}\pi (3)^2(9)\\\\= \frac{1}{3}\pi(9)(9)\\\\\\\= \frac{1}{3} \pi (81)\\\\= 27\pi \\[/tex]

≈ 84.82 cm

deeksha made cuboid of size 2 cm x 3 cm x 4 cm. how many such
cuboids will be required to make a cube?
b) in a right triangle pqr,

Answers

The number of cuboids of dimension 2 cm x 3 cm x 4 cm, required to make a cube is 9.

Dimensions of the cuboid Deeksha made are given as 2 cm x 3 cm x 4 cm.

Thus, the volume of this cuboid = 2*3*4 cm³ = 24 cm³.

Using the formula for the volume of cuboid as the product of the three sides.

Deeksha wants to make a cube combining some number of these cuboids.

Assuming the side length of the cube to a, the volume of the cube = a³.

Assuming the number of cuboids required to make 1 cube to be n, we can write that a³ = n*(24 cm³).

To make this relation true, we need the right-hand side to be a perfect cube.

Prime factorizing the volume of the cuboid, we get 24 cm³ = 2³ * 3 cm³.

To make it a perfect cube, we need to multiply 3², by it.

Thus, n = 3² = 9.

Thus, the number of cuboids of dimension 2 cm x 3 cm x 4 cm, required to make a cube is 9.

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The provided question is incorrect. The correct question is:

"Deeksha made cuboid of size 2 cm x 3 cm x 4 cm. how many such

cuboids will be required to make a cube?"

Instructions: Given the coordinates, Complete the translation (x,y)→(x+2,y−2) C(−5,−1) D(−4,3) E(−3,2) F(−2,0)

Answers

the translations is,

C(-5,-1) → (-3,-3)

D(-4,3) → (-2,1)

E(-3,2) → (-1,0)

F(-2,0) → (0,-2)

What is Translation ?

The changes in the value of the x and y coordinates is known as translation.

Calculation:

The given points are C(-5,-1), E(-4,3), F(-3,2), F(-2,0).

according to the question (x,y)→(x+2,y-2)

this means x coordinate changes by addition of 2

And the y coordinate changes by subtraction of 2

Addition and Subtraction in the Coordinates,

C(-5,-1) → (-5+2,-1-2) → (-3,-3)

D(-4,3) → (-4+2,3-2) → (-2,1)

E(-3,2) → (-3+2,2-2) → (-1,0)

F(-2,0) → (-2+2,0-2) → (0,-2)

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PLEASE HELP WILL GIVE YOU ALL MY POINTS In the picture, t || s, mz1 = 10x + 2, and mz2 = 12x - 22. Find the measure of angle 2.

Answers

Answer:

[tex]\huge\boxed{\sf < 2 = 122\°}[/tex]

Step-by-step explanation:

From the figure,

∠1 = ∠2 (Alternate angles are equal)

We know that:

∠1 = 10x + 2∠2 = 12x - 22

So,

10x + 2 = 12x - 22

Add 22 to both sides

10x + 2 + 22 = 12x

10x + 24 = 12x

Subtract 10x to both sides

24 = 12x - 10x

24 = 2x

Divide 2 to both sides

12 = x

x = 12

Given that,

∠2 = 12x - 22

Put x = 12

∠2 = 12(12) - 22

∠2 = 144 - 22

∠2 = 122°

[tex]\rule[225]{225}{2}[/tex]

Answer: 122°

Step-by-step explanation:

∠1 an ∠2 are alternate interior angles, which are angles that are inside both parentheses and are on alternate sides of the transversal. The Alternate Interior Angles Theorem states that if two lines are parallel and cut by a transversal, then the alternate interior angles formed are congruent.

Since the lines are parallel as given in the question, ∠1 and ∠2 are of equal measure. We can solve for ∠2 by first solving for x, then plugging it in ∠2's measure.

[tex]10x+2=12x-22[/tex]

[tex]-2x=-24[/tex]

[tex]x=12[/tex]

Putting x in the expression 12x-22, we get

[tex]12(12)-22\\144-22\\122[/tex]

Hence, the value of ∠2 is 122°.

Based on the graph of the general solution to the differential equation dy over dx equals 2 times x minus 2 times y comma which of the following statements is true

Answers

Statements is TRUE

Based on the graph of the general solution to the differential equation dy over dx equals 2 times x - 2 times y = dy/dx=2x-2y.

What is general solution to the differential equation?

A differential equation's solution is an expression for the dependent variable in terms of one or more independent variables that satisfy the relationship.

The statement which is true is the slopes are all positive in quadrant I.

Given the differential equation is dy/dx=2x-2y

A differential equation is an equation that contains at least one derivative of an unknown function, either a normal differential equation or a partial differential equation.

Given dy/dx=2x-2y

now slope=2x-2y

Along x-axis, y=0. So, slope=2x+0.

Since it depends upon x hence the slope along the y-axis are not horizontal.

Along y-axis, x=0. So, slope -2y+0.

The slope along the x-axis are also not horizontal.

In quadrant I:

x,y≥20

So, dy/dx ≥20

Therefore, the slopes are all positive in quadrant I. In quadrant IV,

x≥0,y≤0

so, dy/dx is not always positive.

The slope are not all positive in quadrant IV:

Therefore, the slope are all positive in quadrant I for the differential equation dy/dx=2x-2y.

General solution to the differential equation =

dy/dx=2x-2y.

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Complete Question:

Based on the graph of the general solution to the differential equation dy over dx equals 2 times x plus y comma which of the following statements is true?

The slopes along the y-axis are horizontal.

The slopes along the x-axis are horizontal.

The slopes are all positive in Quadrant 1.

The slopes are all positive in Quadrant 4.

Air pressure may be represented as a function of height (in meters) above the surface of the Earth, as shown below. In this function, Po is the air pressure at the surface of the Earth, and h is the height above the surface of the Earth, measured in meters. Which expression best represents the air pressure at an altitude of 3,000 meters above the surface of the Earth

Answers

The expression that best represents the air pressure at an altitude of 3,000 meters above the surface of the Earth is 3,000ρg.

Air pressure at an altitude above the surface of the Earth

The air pressure at an altitude of 3,000 meters above the surface of the Earth is calculated as follows;

Po = ρgh

where;

ρ is density of airg is acceleration due to gravityh is the altitude

Substitute the value of altitude into the equation;

Po = 3,000ρg

Thus, the expression that best represents the air pressure at an altitude of 3,000 meters above the surface of the Earth is 3,000ρg.

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The figure shows an example of a multiplication problem where the numbers are coded by letters. Same letter represents same digits, and different letters represent different digits. Which digit does the letter x represent?

Answers

Based on the information provided, it can inferred that letter A is equivalent to 4.

What is the value of letter B?

The image shows B + B+ A = A. This is only possible if letter adding 2 letters B is equal to 10. Now 10/2 = 5. This means B = 5.

What is the value of the letter A?

It is known 5 + A + A = A. This is only possible is A is 4. 5 + 4+ 4 + 1 (from the previous addition) = 4 (+2 that is added in the next column).

Note: This question is incomplete below I add the missing image.

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Given that f(x) = -6x and g(x) = x + 4, multiply the functions (f •g)(x).

Answers

Answer:

(f • g)(x) = -6x² - 24x

Step-by-step explanation:

The expression (f • g)(x) can be written in an expanded form, f(x) • g(x).

(f • g)(x) <----- Original expression

f(x) • g(x) <----- Rewritten expression

(- 6 x) • (x + 4) <----- Insert functions

-6x² - 24x <----- Multiply -6x and x, and multiply -6x and 4

the final answer is 6x and 24

A rectangle has a width of 2.45 feet and a length of 6.5 feet. how will the area of the rectangle change if each side is increased by a factor of 5? the area will be one-fifth the original. the area will be startfraction 1 over 25 endfraction the original. the area will be 25 times the original. the area will be 5 times the original.

Answers

The area of the new rectangle will be 25 times that of the original

Area of a rectangle

A rectangle is a 2D shape that has 4 sides and equal interior angles

If rectangle has a width of 2.45 feet and a length of 6.5 feet, then;

Width = 2.45 feet

Length = 6.5feet

A = 2.45 * 6.5

A = 15.925 square feet

If each side of the rectangle is increased by a factor of 5, hence;

W1 = 5(2.45) = 12.25feet

L1 = 5(6.5) = 32.5 feet

Determine the area of the new rectangle

A = 12.25 * 32.5

A = 398.125 square feet

Ratio of the area

A1/A = 398.125/15.925

A1 = 25A

Hence the area of the new rectangle will be 25 times that of the original

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Which graph represents the solution to x<6?

Answers

The answer is last one

. In your rectangular backyard, you know the width of the yard is three less than four times the length. If the perimeter of your yard is 24 yards, what is the width?

Answers

The width of the rectangular backyard is 9 yards.

How to find the width of the rectangle?

The width of the yard is three less than four times the length.

Therefore,

w = 4l - 3

Perimeter of a rectangle = 2l + 2w

where

l = length

w = width

Hence,

24 = 2l + 2(4l - 3)

24 = 2l + 8l - 6

24 = 10l - 6

24 + 6 = 10l

30 = 10l

l = 30 / 10

l = 3 yards

Hence,

w = 4(3) - 3

w = 12 - 3

w = 9 yards

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Find the volume of this sphere.

Answers

Answer:

4ft^3

Step-by-step explanation:

1.333333*3*1^3=4

Calculate the volume of the sphere

To start finding the volume of this sphere, we get the following data:

π = 3r = 2 ft²

To find the volume we apply the following formula:

[tex]\boldsymbol{\sf{V= \dfrac{3}{4}\pi r^{2} }}[/tex], wherev = volumeπ = pir = radius

We substitute our data in the formula and solve:

Substituting values into the equation

[tex]\boldsymbol{\sf{V=\dfrac{3}{4}*3*(2 \ ft)^{3} }}[/tex]

Calculate exponent cubed

[tex]\boldsymbol{\sf{V=\dfrac{3}{4}*3*8 \ ft^{3} }}[/tex]

Multiplying

[tex]\boxed{\boldsymbol{\sf{V=32 \ ft^{3} }}}[/tex]

Therefore, the volume of the sphere is 32 ft³.

PLEASE HELP IM STUCK

Answers

Answer:

The 18th term would be "-70"

Answer:

-70

Step-by-step explanation:

pls help math problem pls

Answers

Answer: The fourth choice

Step-by-step explanation:

(f - g)(x) = f(x) - g(x) = 4[tex]x^{2}[/tex] - 5x - (3[tex]x^{2}[/tex] + 6x - 4) = [tex]x^{2}[/tex] - 11x + 4

Lines b and c are parallel. What is the measure of angle 6

Answers

The measure of ∠6 is 54 degrees.

How to find the angles in a parallel lines?

When parallel lines are cut by a transversal, angle relationship are formed. Therefore,

∠6 = 5x + 9 (vertically opposite angles)

∠6 = ∠2

Hence,

∠2 + 13x + 9 = 180

5x + 9 + 13x + 9 = 180

18x = 162

x = 162 / 18

x = 9

Hence,

∠6 = 5(9) + 9 54°

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find the domain and range of f^-1 where f(x)=x+1/6x+7

Answers

Answer:

Step-by-step explanation:

hello here is an solution

The domain and range are (-∞,1/6)∪ (1/6, -∞), {x/x≠1/6} and (-∞,-7/6)∪ (-7/6, ∞), {y/y≠-7/6} respectively.

What is a function?

A relation is a function if it has only One y-value for each x-value.

The given function is f(x)=(x+1)/6x+7

Let f(x)=y

(x+1)/6x+7=y

x+1=y(6x+7)

x+1=6xy+7y

6xy-x=7y-1

x(6y-1)=7y-1

Divide both sides by 6y-1

x=7y-1/6y-1

So f⁻¹(x)=1-7x/6x-1

The domain is (-∞,1/6)∪ (1/6, -∞), {x/x≠1/6}

The range is (-∞,-7/6)∪ (-7/6, ∞), {y/y≠-7/6}

Hence, the domain and range are (-∞,1/6)∪ (1/6, -∞), {x/x≠1/6} and (-∞,-7/6)∪ (-7/6, ∞), {y/y≠-7/6} respectively.

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Please look at the picture, I need help ASAP.

Answers

See below for the proof that the areas of the lune and the isosceles triangle are equal

How to prove the areas?

The area of the isosceles triangle is:

[tex]A_1 = \frac 12r^2\sin(\theta)[/tex]

Where r represents the radius.

From the figure, we have:

[tex]\theta = 90[/tex]

So, the equation becomes

[tex]A_1 = \frac 12r^2\sin(90)[/tex]

Evaluate

[tex]A_1 = \frac 12r^2[/tex]

Next, we calculate the length (L) of the chord as follows:

[tex]\sin(45) = \frac{\frac 12L}{r}[/tex]

Multiply both sides by r

[tex]r\sin(45) = \frac 12L[/tex]

Multiply by 2

[tex]L = 2r\sin(45)[/tex]

This gives

[tex]L = 2r\times \frac{\sqrt 2}{2}[/tex]

[tex]L = r\sqrt 2[/tex]

The area of the semicircle is then calculated as:

[tex]A_2 = \frac 12 \pi (\frac{L}{2})^2[/tex]

This gives

[tex]A_2 = \frac 12 \pi (\frac{r\sqrt 2}{2})^2[/tex]

Evaluate the square

[tex]A_2 = \frac 12 \pi (\frac{2r^2}{4})[/tex]

Divide

[tex]A_2 = \frac{\pi r^2}{4}[/tex]

Next, calculate the area of the chord using

[tex]A_3 = \frac 12r^2(\theta - \sin(\theta))[/tex]

Recall that:

[tex]\theta = 90[/tex]

Convert to radians

[tex]\theta = \frac{\pi}{2}[/tex]

So, we have:

[tex]A_3 = \frac 12r^2(\frac{\pi}{2} - \sin(\frac{\pi}{2}))[/tex]

This gives

[tex]A_3 = \frac 12r^2(\frac{\pi}{2} - 1)[/tex]

The area of the lune is then calculated as:

[tex]A = A_2 - A_3[/tex]

This gives

[tex]A = \frac{\pi r^2}{4} - \frac 12r^2(\frac{\pi}{2} - 1)[/tex]

Expand

[tex]A = \frac{\pi r^2}{4} - \frac{\pi r^2}{4} + \frac 12r^2[/tex]

Evaluate the difference

[tex]A = \frac 12r^2[/tex]

Recall that the area of the isosceles triangle is

[tex]A_1 = \frac 12r^2[/tex]

By comparison, we have:

[tex]A = A_1 = \frac 12r^2[/tex]

This means that the areas of the lune and the isosceles triangle are equal