Worksheet on topic 1 - area and volume problems - Math 165 9/10
1. (Putnam A1, 1988) Let R be the region consisting of pairs (x, y) of the cartesian plane
satisfying both |x| − |y| ≤ 1 and |y| ≤ 1. Sketch the region R and find its area.
2. (VTRMC 1, 1989) A square of side a is inscribed in a triangle of base b and height h.
Prove that the area of the square cannot exceed one-half of the area of the triangle.
3. (Putnam A1, 1993) The horizontal line y = c intersects the curve y = 2x − 3x3 in the
first quadrant as in the figure. Find c so that the areas of the two shaded regions are equal.
4. (Putnam B1, 1989) A dart, thrown at random, hits a square target. Assuming that any
two parts of the target of equal area are equally likely to be hit, find the probability√that the
point hit is nearer the center than to any edge. Express your answer in the form (a b + c)/d
where a, b, c and d are positive integers.
5. (VTRMC 5, 1999) A rectangular box has sides of length 3, 4 and 5. Find the volume of
the region consisting of all points that are within distaince 1 of at least one of the sides.
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6. (VTRMC 1, 2001, also VTRMC 3, 1998) Three infinitely long circular cylinders each
with unit radius have their axes along the x, y and z-axes. Determine the volume of the
region common to all three cylinders. (Thus one needs to determine the volume common to
{y 2 + z 2 ≤ 1}, {z 2 + x2 ≤ 1}, {x2 + y 2 ≤ 1}.)
7. (Putnam A1, 2006) Find the volume of the region of points (x, y, z) such that (x2 + y 2 +
z 2 + 8)2 ≤ 36(x2 + y 2 ).
8. (Putnam B3, 1993) Two real numbers x and y are chosen at random in the integral (0, 1)
with respect to the uniform distribution. What is the probability that the closest integer to
x/y is even? Express the answer in the form r + sπ, where r and s are rational numbers.
9. (Putnam A2, 1998) Let s be any arc of the unit circle lying entirely in the first quadrant.
Let A be the area of the region lying below s and above the x-axis and let B be the area of
the region lying to the right of the y-axis and to the left of s. Prove that A + B depends
only on the arc length, and not on the position, of s.
10. (VTRMC 6, 1995) A straight rod of length 4 inches has ends which are allowed to slide
along the perimeter of a square whose sides each have length 12 inches. A paint brush is
attached to the rod so that it can slide between the two ends of the rod. Determine the total
possible area of the square which can be painted by the brush.
11. (Putnam B3, 1998) Let H be the unit hemisphere {(x, y, z) : x2 + y 2 + z 2 = 1, z ≥ 0}, C
the unit circle {(x, y, 0) : x2 +y 2 = 1}, and P the regular pentagon inscribed in C. Determine
the surface area of the portion of H lying over the planar region inside P , and write your
answer in the form A sin(α) + B cos(β), where A, B, α and β are real numbers.
12. (Putnam A4, 2001) Triangle ABC has area 1. Points E, F and G lie, respectively, on
sides BC, CA, AB such that AE bisects BF at point R, BF bisects CG at point S, and
CG bisects AE at point T . Find the area of the triangle RST .