5.10. EXERCISES 137
26. A hiker begins to walk to a cabin in a dense forest. He is walking on a road whichruns from East to West and the cabin is located exactly one mile north of a point twomiles down the road. He walks 5 miles per hour on the road but only 3 miles perhour in the woods. Find the path which will minimize the time it takes for him to getto the cabin.
27. A park ranger needs to get to a fire observation tower which is one mile from a longstraight road in a dense forest. The point on the road closest to the observation toweris 10 miles down the road on which the park ranger is standing. Knowing that he canwalk at 4 miles per hour on the road but only one mile per hour in the forest, howfar down the road should he walk before entering the forest, in order to minimize thetravel time?
28. A refinery is on a straight shore line. Oil needs to flow from a mooring place foroil tankers to this refinery. Suppose the mooring place is two miles off shore from apoint on the shore 8 miles away from the refinery which is also on the shore and thatit costs five times as much to lay pipe under water than above the ground. Describethe most economical route for a pipeline from the mooring place to the refinery.
29. Two hallways, one 5 feet wide and the other 6 feet wide meet. It is desired to carrya ladder horizontally around the corner. What is the longest ladder which can becarried in this way? Hint: Consider a line through the inside corner which extendsto the opposite walls. The shortest such line will be the length of the longest ladder.
30. A window is to be constructed for the wall of a church which is to consist of arectangle of height b surmounted by a half circle of radius a. Suppose the totalperimeter of the window is to be no more than 4π + 8 feet. Find the dimensions ofthe window which will admit the most light.
31. ∗ A parabola opens down. The vertex is at the point (0,a) and the parabola interceptsthe x axis at the points (−b,0) and (b,0) . A tangent line to the parabola is drawnin the first quadrant which has the property that the triangle formed by this tangentline and the x and y axes has smallest possible area. Find a relationship between aand b such that the normal line to the point of tangency passes through (0,0) . Alsodetermine what kind of triangle this is.
32. Show that for r a rational number and y = xr, it must be the case that if this functionis differentiable, then y′ = rxr−1. This was shown in more generality, but use thechain rule to verify this directly.
33. Let
f (x) =
1 if x ∈Q
0 if x /∈Q
Now let g(x) = x2 f (x) . Find where g is continuous and differentiable if anywhere.
34. Use induction to show that for u,v smooth functions,
dn
dxn (uv) =n
∑k=0
(nk
)u(n−k)v(k)
Here v(k) denotes the kth derivative of v.