"A flagpole is perpendicular to the horizontal" is another way of saying "the flagpole is vertical".
One way to draw out the diagram:
Points:
A = base of the hill
B = tip of the shadow
C = base of the flagpole
D = top of the flagpole
E = point used to help label the 8.6 degree angle of elevation
Given info:
angle EAB = 8.6 degrees
angle DBC = 24.3 degrees
segment BC = 70 ft = shadow length
The flagpole itself is in red (segment CD).
Next we draw a horizontal line through point C.
Mark points F and G on the new horizontal line and line AC respectively.
Because AE and CF are horizontal parallel lines, we know that
angle EAB = angle FCG = 8.6 degrees
and this is because of the corresponding angles theorem.
Mark point H somewhere to the left of C on the line CF.
I'll place H as the intersection of CF and BD.
It doesn't matter where H is as long as it's to the left of C.
The introduction of point H is to help label the angle HCB which is congruent to angle FCG
angle HCB = angle FCG = 8.6 degrees
since these are vertical angles.
This is the updated diagram with those items added
Then,
angle BCD = (angle HCB)+(angle HCD)
angle BCD = (8.6)+(90)
angle BCD = 98.6 degrees
This is angle C of triangle BCD.
Keeping our focus on triangle BCD only, we can say:
B+C+D = 180
24.3+98.6+D = 180
122.9+D = 180
D = 180-122.9
D = 57.1
This is angle BDC
We'll still keep our focus on triangle BCD only.
Apply the law of sines to find side 'b', aka segment CD.
b/sin(B) = d/sin(D)
b/sin(24.3) = 70/sin(57.1)
b = sin(24.3)*70/sin(57.1)
b = 34.308389211743
b = 34.3
Therefore, segment CD is roughly 34.3 feet long.
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Answer: The flagpole is approximately 34.3 feet tall.
This is roughly comparable to about halfway between a 3 and 4 story building, assuming each floor is 10 ft high.