The Maths and Science Thread - Collection of Problems and Facts

[Varun]

Can someone give me the derivation of this -

Sin2θ= 2SinθCosθ

 

[Gaurav_7]

sin (a+b) = sina.cosb + cosa.sinb

we put a=b=θ

you get your answer then.

 

[Aussie Team]

If you need formula then here it is

sin(A+B) = sinAcosB + cosAsinB

cos(A+B) = cosAcosB − sinAsinB

 

[Varun]

sin (a+b) = sina.cosb + cosa.sinb

Derivation of this ?

 

[Aussie Team]

sin A = cos(pi/2 - A)

sin (A + B) = cos(pi/2 - (A + B))

sin(A + B) = cos ((pi/2 - A) - B)


(Now, you use the following formula :

cos (x - y) = cos x * cos y + sin x * sin y)


sin(A + B) = cos (pi/2 - A) * cosB + sin (pi/2 - A) * sinB

sin(A + B) = sinA * cosB + cosA * sinB

 

[Gaurav_7]

It's a formula which can be directly used. There's a derivation though but it's all inter related.

 

[Varun]

cos (x - y) = cos x * cos y + sin x * sin y)

And derivation of this one ?

varun_rustagi added 0 Minutes and 30 Seconds later...

It's a formula which can be directly used. There's a derivation though but it's all inter related.

I don't want to cram the formulas. Just wanna know their basics.

 

[Gaurav_7]

This will help you. It was a bit too difficult to explain so I found that by Google.

 

[Aussie Team]

cos(x+y) = cos(x)cos(y) - sin(x)sin(y)

replace x and y with x/2

cos(x/2 + x/2) = cos^2(x/2) - sin^2(x/2)

=> cos x = cos^2(x/2) - 1 + cos^2(x/2)
=> cos x = 2cos^2(x/2) - 1
=> 2cos^2(x/2) = 1 + cos (x)
=> cos^2(x/2) = (1/2) [ 1 + cos (x) ]
cos(x/2) = SQRT[1/2 (1 + cos x ) ]

This sums are very difficult

 

[Aussie Team]

Here is my question


You've been asked to buy 100 squeegies, using 100 dollars to do so. You may buy no more or less than 100 squeegies, and the total price must be exactly 100 dollars. There is no sales tax. Red squeegies cost $6.00. Yellow squeegies cost $3.00. Blue squeegies cost $0.10. How many of each must you buy?

 

[P Squared]

Here is my question


You've been asked to buy 100 squeegies, using 100 dollars to do so. You may buy no more or less than 100 squeegies, and the total price must be exactly 100 dollars. There is no sales tax. Red squeegies cost $6.00. Yellow squeegies cost $3.00. Blue squeegies cost $0.10. How many of each must you buy?

Each will differ according to what you like. Say I'll less buy blue's and more reds or vice versa price will be the same when you add them up if you've bought the right amount of them.:drink

 

[Abhas]

Here is my question


You've been asked to buy 100 squeegies, using 100 dollars to do so. You may buy no more or less than 100 squeegies, and the total price must be exactly 100 dollars. There is no sales tax. Red squeegies cost $6.00. Yellow squeegies cost $3.00. Blue squeegies cost $0.10. How many of each must you buy?

Red = 1 = $6
Yellow = 29 = $87
Blue = 70 = $7

Sum:
1+29+70 = 100
6+87+7=100

 

[Varun]

Wow! Abhas solved the question which became my nightmare.
Abhas, did you use Trial and Error or some method ?

 

[Meet]

Great work Abhas,I tried much,but didn't got the answer.
Can you please show the method.

 

[Abhas]

There was definitely a method of using Linear Programming.

We clearly have 2 equations:
6x+3y+0.1z=100
x+y+z=100

But, to solve the 3 unknowns, we need 3 equations. Assumed a value of any of the options, and check for equality.

I used a logic that the Blue should be in multiples of ten.
And, considering the total needed to be 100, a majority had to be from the blue.
I started off by assuming Blue = 60, then 70, then 80.
60 seemed too little, because the remaining 40 needed to be filled by the Yellow and Red, and it was not possible.
70 Seemed right, but so did 80. I had a feeling it should be 70, and tried to find a combination for it.
Putting B=70, I had 30 places left.. and $93. Dividing 93 by 3, I got 31. This meant 29 Yellow, and 1 Red.

Crude, but effective. I managed to solve it quicker than it seems, so not a bad effort