Answer:
- 30
- COB . . . . or . . . . BOC
Step-by-step explanation:
The reason given on the line of interest is "substitution," so the problem boils down to determining what was substituted for what. The previous statement says ...
AOC + COB = 90
and the first part of the statement we're to complete has BOC + ___.
We see from the given statements that m∠AOC = m∠BOC + 30°, so it appears that is the substitution that has been made: AOC has been replaced by BOC + 30.
This means the first blank is filled with 30.
__
The second part of the previous statement is ...
AOC + COB = 90
so we believe this (COB) should go in the second blank.
__
Then the line of interest would read ...
BOC + 30 + COB = 90
_____
<em>Comment on the problem</em>
There is a curious mix of notations here. Usually, (as in the beginning of this problem) we refer to the measure of an angle using "m∠" in front of the angle designator, and we use a degree symbol to indicate the units of that measure. Part-way through the problem statement written here, those notations were dropped, and we're to assume they are intended. IMO, this is a poor way to demonstrate careful problem solving.
The substitution given for AOC is BOC+30, but the line into which that is substituted has AOC +COB = 90. This means the equation after substitution is ...
BOC +30 +COB = 90
Since BOC and COB are the same angle, we can sort of fudge the "algebra" to get to BOC=30, but if the problem were more carefully written, the angle would be referred to by consistent nomenclature:
m∠AOC + m∠BOC = 90° . . . . . . . . . preferred angle designations
(m∠BOC + 30°) + m∠BOC = 90° . . . . substitution for m∠AOC
2(m∠BOC) = 60° . . . . . . algebra (subtract 30°, collect terms)
m∠BOC = 30° . . . . . . . . algebra (divide by 2)
can be parameterized by
and 
. Then the surface integral can be computed with
