• ### Just a math question I thought of?

Best answer: 　 Three points on the curve y = sinx: A (a, sin a), B (b, sin b), C (c, sin c) where 0 ≤ a < b < c ≤ 2π Triangle ABC has base |AC| = √((a − c)² + (sin a − sin c)²) Line AC has slope = (sin a − sin c) / (a − c) Equation of line AC: y − sin a = (sin a − sin c) / (a − c) (x − a) (a − c) (y − sin a) = (sin a −... show more
Three points on the curve y = sinx:
A (a, sin a), B (b, sin b), C (c, sin c)
where 0 ≤ a < b < c ≤ 2π

Triangle ABC has base |AC| = √((a − c)² + (sin a − sin c)²)

Line AC has slope = (sin a − sin c) / (a − c)
Equation of line AC:
y − sin a = (sin a − sin c) / (a − c) (x − a)
(a − c) (y − sin a) = (sin a − sin c) (x − a)
(sin a − sin c) (x − a) + (c − a) (y − sin a) = 0

Triangle ABC has height = distance from B to line AC
= |(sin a − sin c) (b − a) + (c − a) (sin b − sin a)| / √((sin a − sin c)² + (a − c)²)
= |(b−c) sin a + (c−a) sin b + (a−b) sin c| / √((sin a − sin c)² + (a − c)²)

Area (ABC)
= 1/2 * √((a − c)² + (sin a − sin c)²) * |(b−c) sin a + (c−a) sin b + (a−b) sin c| / √((sin a − sin c)² + (a − c)²)
= 1/2 * |(b−c) sin a + (c−a) sin b + (a−b) sin c|

Using WolframAlpha, we find area is maximized when:

a = 0
b = 1.72545
c = 5.32211
Maximum area = 3.33656

http://www.wolframalpha.com/input/?i=max...

Here is a diagram showing the triangle with maximum area:
https://www.desmos.com/calculator/di4j6s...
You can move the points around on the curve.
The area is shown on the second line.
6 answers · 2 days ago
• ### Please demonstrate how to solve this problem step by step using PEMDAS 3-5*(-1)/(1-3*(-1)) Thank you.?

3 + 5/(1+3)
3 + 5/4
12/4 + 5/4
17/4
3 + 5/(1+3)
3 + 5/4
12/4 + 5/4
17/4
6 answers · 2 days ago
• ### Could someone explain what type of Algebra this is: 16x^2 - 80x + 100?

16x² - 80x + 100 = 0 would be a Quadratic Equation.
16x² - 80x + 100 = 0 would be a Quadratic Equation.
6 answers · 2 days ago
• ### How do you simplify?

= (1/cos²(x)) / (sin²(x)/cos²(x))
= 1/sin²(x)
= csc²(x)
= (1/cos²(x)) / (sin²(x)/cos²(x))
= 1/sin²(x)
= csc²(x)
10 answers · 3 days ago
• ### How do you solve this math question?

John rides his bike one-third of the way from his house to school where he meets his friend, Joe. After he rides another mile John is halfway to school. How far is the school from Joe’s house?
John rides his bike one-third of the way from his house to school where he meets his friend, Joe. After he rides another mile John is halfway to school. How far is the school from Joe’s house?
14 answers · 2 days ago
• ### I do not understand where the negative in -sqrt x ^ 6 comes from?

5 answers · 2 days ago
• ### Find the x of (x-6)^2?

Best answer: 　 This is a badly worded question. Are you looking for a root? Or are you looking for the "x" term in the expansion of (x − 6)²? To find root, set expression = 0 and solve for x: (x − 6)² = 0 x − 6 = 0 x = 6 To find the "x" term in the expansion of (x − 6)², use binomial... show more
This is a badly worded question.

Are you looking for a root? Or are you looking for the "x" term in the expansion of (x − 6)²?

To find root, set expression = 0 and solve for x:

(x − 6)² = 0
x − 6 = 0
x = 6

To find the "x" term in the expansion of (x − 6)², use binomial expansion:

"x" term of (x − 6)²
= C(2,1) (x)¹ (−6)¹
= 2 (x) (−6)
= −12x
6 answers · 2 days ago
• ### How much is 1/3 from \$4,000?

12 answers · 2 days ago
• ### Which would you, personally, consider to be easier? Geometry or Algebra?

From what I've heard, Geometry is more of a logic based part of mathematics whereas Algebra is more solid and factual. I'm not entirely sure if that's true, but I would like to hear your opinions. What do you find easier to do?
From what I've heard, Geometry is more of a logic based part of mathematics whereas Algebra is more solid and factual. I'm not entirely sure if that's true, but I would like to hear your opinions. What do you find easier to do?
10 answers · 1 day ago
• ### Math question help?

x² + 1 = 5
x² + 1 = 5
16 answers · 3 days ago
• ### If four times certain number,increased by 6 is equal to 94, what is the number?

12 answers · 2 days ago
• ### Math order of operations?

I have an equation -x^2 + 8 Let s assume x is 4 Is the answer going to be 8 or 24?
I have an equation -x^2 + 8 Let s assume x is 4 Is the answer going to be 8 or 24?
11 answers · 2 days ago
• ### Area of a triangle??? Heron’s formula???? what is the area of a triangular lot with sides of lengths 118, 152, and 201 ft?

S = (A + B + C)/2

A = √[S(S - A)(S - B)(S - C)]

For a triangle with sides 118, 152, and 201 ft:

S = (118 + 152 + 210)/2 = 235.5

A = √[235.5 * (235.5 - 118) * (235.5 - 152) * (235.5 - 210)]
A = √[235.5 * 117.5 * 83.5 * 34.5]
A = √79,713,953.4375

A = 8,928.3 ft²

S = (A + B + C)/2

A = √[S(S - A)(S - B)(S - C)]

For a triangle with sides 118, 152, and 201 ft:

S = (118 + 152 + 210)/2 = 235.5

A = √[235.5 * (235.5 - 118) * (235.5 - 152) * (235.5 - 210)]
A = √[235.5 * 117.5 * 83.5 * 34.5]
A = √79,713,953.4375

A = 8,928.3 ft²
6 answers · 3 days ago
• ### When you perform the following operations, how many significant figures should your answer have? (2.000 cm3 x 15.6 g /cm3) + 1.000 kg?

Best answer: The simple rule of taking the least of all the numbers does not apply when you have addition or subtraction.

2x15.6 = 31.2 g, to 3 places
adding 1000 g, to 4 places gives you 1031.2 g to 4 places, as 1000 g is only good to 4 places. so the answer is 1031 g
Best answer: The simple rule of taking the least of all the numbers does not apply when you have addition or subtraction.

2x15.6 = 31.2 g, to 3 places
adding 1000 g, to 4 places gives you 1031.2 g to 4 places, as 1000 g is only good to 4 places. so the answer is 1031 g
5 answers · 3 days ago
• ### How are the equations y=mx=b and y=kx alike?

Best answer: Assuming that y = mx + b

The equations are both in slope intercept form.
Best answer: Assuming that y = mx + b

The equations are both in slope intercept form.
5 answers · 3 days ago
• ### Is 4 5/8 inches the same as 4.6 inches?

I am very confused with fractions. I'm trying to buy an item online and i want to know its dimensions. So is 4 5/8 inches over 4.5 inches?
I am very confused with fractions. I'm trying to buy an item online and i want to know its dimensions. So is 4 5/8 inches over 4.5 inches?
13 answers · 3 days ago
• ### Is it cheating to use a math calculator in class when allowed to?

I consider people who use a calculator to add and subtract problems to be too easy and I consider it cheating because I think people should know how to do math by hand. If you are in a Math class when you are allowed to use a calculator, isn’t this cheating? Why would some Math classes allow you to use a calculator... show more
I consider people who use a calculator to add and subtract problems to be too easy and I consider it cheating because I think people should know how to do math by hand. If you are in a Math class when you are allowed to use a calculator, isn’t this cheating? Why would some Math classes allow you to use a calculator but others won’t allow you? Isn’t it cheating in general to do math problems by a calculator because you know you are going to be 100 percent right and not have a challenge to do problems by hand with using scratch paper?
15 answers · 3 days ago
• ### True or false: for any real number a and b, a > 0 and b > 0, √(a²+b²) = a + b?

Best answer: 　 For any real number a and b, a > 0 and b > 0, √(a²+b²) = a + b FALSE Any numerical example will show this to be false. For example: a = 3, b = 4 √(a²+b²) = √(9+16) = √25 = 5 a + b = 3 + 4 = 7 The only time √(a²+b²) = a + b is when one of a or b (or both) = 0, and the other is non-negative. However,... show more
For any real number a and b, a > 0 and b > 0, √(a²+b²) = a + b
FALSE

Any numerical example will show this to be false. For example: a = 3, b = 4
√(a²+b²) = √(9+16) = √25 = 5
a + b = 3 + 4 = 7

The only time √(a²+b²) = a + b is when one of a or b (or both) = 0, and the other is non-negative. However, neither a nor b can be = 0 since we are told they are both > 0

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

For real numbers a and b, if a³ = b³, then a = b is always TRUE.
a³ = b³
a³ − b³ = 0
(a − b) (a² + ab + b²) = 0

So either (a − b) = 0 or (a² + ab + b²) = 0

If a² + ab + b² = 0, then
a = (−b ± √(b²−4b²)) / 2 = (−b ± √(−3b²)) / 2
Discriminant −3b² is either negative or 0
Since a is real, discriminant −3b² must be 0 ---> b = 0 ----> a = (−0 ± √0) / 2 = 0
a = b = 0

If a² + ab + b² ≠ 0, then
a − b = 0
a = b

Either way, a = b

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

In the first case, couldn't I just have √(a²+b²) = (a^(2 x 1/2) + b^(2 x 1/2)) = a + b

Exponentiation is distributive over multiplication, but NOT over addition:
For a, b > 0:
(a² + b²)^(1/2) ≠ (a²)^(1/2) + (b²)^(1/2)
but
(a² * b²)^(1/2) = (a²)^(1/2) * (b²)^(1/2) = a^(2*1/2) * b^(2*1/2) = a * b

Check out rules of exponentiation.
You WILL find: (x * y)^n = x^n * y^n
You will NOT find: (x + y)^n = x^n + y^n -----> FALSE

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Here's another way to show that √(a²+b²) ≠ a + b

For a > 0, b > 0

(a + b)² = (a + b) (a + b)
(a + b)² = a (a + b) + b (a + b) -----> using distributive property
(a + b)² = a*a + a*b + b*a + b*b -----> using distributive property again
(a + b)² = a² + 2ab + b²

Now take square root of both sides:

a + b = √(a² + 2ab + b²) ≠ √(a²+b²)

Therefore, a + b ≠ √(a²+b²)
7 answers · 11 hours ago
• ### What is 80,000 a year , monthly ?

12 answers · 3 days ago
• ### What is the next number in this sequence? 1,1,2,4,5,25?

8 answers · 1 day ago