Final answer:
The slope angle or vertical angle corresponding to a zenith angle of 98° 35' 24" is 81° 24' 36". This is found by subtracting the zenith angle from 180 degrees because they are supplementary angles.
Step-by-step explanation:
To solve for the slope angle, or vertical angle, when given a zenith angle, we must understand the relationship between the two measurements. The zenith angle is measured from the point directly overhead (the zenith) down to the celestial object or other point of interest. It can exceed 90 degrees, unlike the altitude angle which is measured from the horizon up to the object and is always less than 90 degrees.
In this case, the zenith angle is given as 98° 35' 24". Since a right angle is 90 degrees, any zenith angle greater than this indicates the object is actually below the horizon. To find the slope angle, we have to subtract the zenith angle from 180 degrees, since the slope angle and zenith angle are supplementary angles (they sum up to 180 degrees).
Let's perform the calculation by converting the minutes and seconds to a decimal equivalent of degrees:
180° - 98° 35' 24" =
180° - (98° + 35'/60 + 24"/3600) =
180° - 98.59° =
81° 24' 36" (approximate)
This is the slope angle or vertical angle that corresponds to the given zenith angle of 98° 35' 24".