30. Find a positive number for which the sum of its reciprocal and four times its square is the smallest possible.

Your company can manufacture x hundred grade A tires and y hundred grade B tires a day, where 0 ≤ x ≤ 4 and y = (40 - 10x)/(5-x). Your profit on a grade A tire is twice your profit on a grade B tire. What is the most profitable number of each kind to make?

Particle motion The positions of two particles on the s-axis are s₁ = cos t and s₂ = cos (t + π/4) .

a. What is the farthest apart the particles ever get?

The ladder problem What is the approximate length (in feet) of the longest ladder you can carry horizontally around the corner of the corridor shown here? Round your answer down to the nearest foot.

Particle motion The positions of two particles on the s-axis are s₁ = cos t and s₂ = cos (t + π/4) .

b. When do the particles collide?

32. Answer Exercise 31 if one piece is bent into a square and the other into a circle.

35. Determine the dimensions of the rectangle of largest area that can be inscribed in the right triangle shown in the accompanying figure.

6. You are planning to close off a corner of the first quadrant with a line segment 20 units long running from (a, 0) to (0,b). Show that the area of the triangle enclosed by the segment is largest when a = b.

10. Catching rainwater A 1125 ft^3 open-top rectangular tank with a square base x ft on a side and y ft deep is to be built with its top flush with the ground to catch runoff water. The costs associated with the tank involve not only the material from which the tank is made but also an excavation charge proportional to the product xy.

a. If the total cost is c=5(x^2+4xy) + 10xy, what values of x and y will minimize it?

b. Give a possible scenario for the cost function in part (a).