You are at the peak of a mountaim which has an altitude of 8363 feet when
you feel it beginning to erupt.The lava begins to push up from the base
of the mountain at a rate of 5 feet/sec.At the same time, you begin
running down the mountain at a rate of 8 feet/sec with hopes that you
will reach the bottom before the lava reach the top. The slant height
of the mountain forns a 45 degree angle with the peak 1. What is the
slant side of the mountain? 2 what is the lateralarea of the mountain?

This should get you started. The sides of a triangle with two 45
degree angles and one 90 degree angle have the ratio of 1 to the
square root of 2. So, the distance you would travel down the side of
the mountain would be 8363 multiplied by the square root of 2.

8 ft/s is not what I call running.

I don't think 8fps is that fast considering that a decent high school
athlete can run a 100 yard dash (300 ft) in 10 seconds from a standing
start. So, 300/10 = 30 fps. I'll bet that I can run 8fps down a 45
degree slope with molten lava chasing me.

I would open my umbrella and jump, using the umbrella as a parachute.

I bet I would get down before all you others.

Of course, being English, I take my brolly everywhere.

Toot! Toot!

Are we to assume that the base of the mountain is at sea level?