A block of wood floats in fresh water with2/3 relative density of its volume V submerged and in oil with 0.92 V submerged.Calculate the density of (1)wood,(2) oil.
Answers
Explanation:
Given: r_{3}=r_{1}+r_{2}+rr
3
=r
1
+r
2
+r
To find: \angle A+\angle B∠A+∠B
Solution:
We know that, for \Delta ABCΔABC ,
r=4R\:sin\frac{A}{2}\:sin\frac{B}{2}\:sin\frac{C}{2}r=4Rsin
2
A
sin
2
B
sin
2
C
r_{1}=4R\:sin\frac{A}{2}\:cos\frac{B}{2}\:cos\frac{C}{2}r
1
=4Rsin
2
A
cos
2
B
cos
2
C
r_{2}=4R\:cos\frac{A}{2}\:sin\frac{B}{2}\:cos\frac{C}{2}r
2
=4Rcos
2
A
sin
2
B
cos
2
C
r_{3}=4R\:cos\frac{A}{2}\:cos\frac{B}{2}\:sin\frac{C}{2}r
3
=4Rcos
2
A
cos
2
B
sin
2
C
where RR is the circum-radius.
Now, r_{3}=r_{1}+r_{2}+rr
3
=r
1
+r
2
+r
\Rightarrow 4R\:cos\frac{A}{2}\:cos\frac{B}{2}\:sin\frac{C}{2}=4R\:sin\frac{A}{2}\:cos\frac{B}{2}\:cos\frac{C}{2}+4R\:cos\frac{A}{2}\:sin\frac{B}{2}\:cos\frac{C}{2}+4R\:sin\frac{A}{2}\:sin\frac{B}{2}\:sin\frac{C}{2}⇒4Rcos
2
A
cos
2
B
sin
2
C
=4Rsin
2
A
cos
2
B
cos
2
C
+4Rcos
2
A
sin
2
B
cos
2
C
+4Rsin
2
A
sin
2
B
sin
2
C
\Rightarrow cos\frac{A}{2}\:cos\frac{B}{2}\:sin\frac{C}{2}=sin\frac{A}{2}\:cos\frac{B}{2}\:cos\frac{C}{2}+cos\frac{A}{2}\:sin\frac{B}{2}\:cos\frac{C}{2}+sin\frac{A}{2}\:sin\frac{B}{2}\:sin\frac{C}{2}⇒cos
2
A
cos
2
B
sin
2
C
=sin
2
A
cos
2
B
cos
2
C
+cos
2
A
sin
2
B
cos
2
C
+sin
2
A
sin
2
B
sin
2
C
\Rightarrow cos\frac{A}{2}\:cos\frac{B}{2}\:sin\frac{C}{2}-cos\frac{A}{2}\:sin\frac{B}{2}\:cos\frac{C}{2}=sin\frac{A}{2}\:cos\frac{B}{2}\:cos\frac{C}{2}+sin\frac{A}{2}\:sin\frac{B}{2}\:sin\frac{C}{2}⇒cos
2
A
cos
2
B
sin
2
C
−cos
2
A
sin
2
B
cos
2
C
=sin
2
A
cos
2
B
cos
2
C
+sin
2
A
sin
2
B
sin
2
C
\Rightarrow cos\frac{A}{2}\:(cos\frac{B}{2}\:sin\frac{C}{2}-sin\frac{B}{2}\:cos\frac{C}{2})=sin\frac{A}{2}\:(cos\frac{B}{2}\:cos\frac{C}{2}+sin\frac{B}{2}\:sin\frac{C}{2})⇒cos
2
A
(cos
2
B
sin
2
C
−sin
2
B
cos
2
C
)=sin
2
A
(cos
2
B
cos
2
C
+sin
2
B
sin
2
C
)
\Rightarrow -cos\frac{A}{2}\:sin(\frac{B}{2}-\frac{C}{2})=sin\frac{A}{2}\:cos(\frac{B}{2}-\frac{C}{2})⇒−cos
2
A
sin(
2
B
−
2
C
)=sin
2
A
cos(
2
B
−
2
C
)
\Rightarrow sin\frac{A}{2}\:cos(\frac{B}{2}-\frac{C}{2})+cos\frac{A}{2}\:sin(\frac{B}{2}-\frac{C}{2})=0⇒sin
2
A
cos(
2
B
−
2
C
)+cos
2
A
sin(
2
B
−
2
C
)=0
\Rightarrow sin(\frac{A}{2}+\frac{B}{2}-\frac{C}{2})=0⇒sin(
2
A
+
2
B
−
2
C
)=0
\Rightarrow sin(\frac{A}{2}+\frac{B}{2}-\frac{C}{2}=sin0⇒sin(
2
A
+
2
B
−
2
C
=sin0
\Rightarrow \frac{A}{2}+\frac{B}{2}-\frac{C}{2}=0⇒
2
A
+
2
B
−
2
C
=0
Hope it helpful