Ma 310 Exam 2 Worksheet With Answers - University Of Kentucky (Page 2 of 4) in pdf

3. You are playing a two-person game that begins with a single pile of 15 pennies. A move

consists of removing 1, 2, or 3 pennies. The person who takes the very last penny wins.

What is the winning strategy? Why?

If you can reach the position 4, then you can win, because then your opponent can

only reach 3, 2, or 1, and you will win on the next move. So 4 is a goal position. If

you can reach the position 8, then you can win, because then in the next move your

opponent can only reach 7, 6, or 5, from which you can reach the goal position 4. So

8 is a goal position. If you can reach the position 12, then you can win, because then

in the next move your opponent can only reach 11, 10, or 9, from which you can reach

the goal position 8. So 12 is a goal position. Therefore, the ﬁrst player can win by

moving to position 12, and thereafter moving to positions 8, 4, and 0.

4. On a recent trip I drove from city A to city B. The trip took 3 hours and I averaged

50 miles/hour. I then drove on the same route from city B back to city A. My overall

average for the entire round trip from A to B and back to A was 60 miles/hour. What

was my average rate for the portion of the trip from B to A. Explain your answer.

Warning: the answer is NOT 70 miles/hour.

On the ﬁrst leg, the average rate is r

= 50 mi/hr, the time is t

= 3 hr, and so the

× t

distance is d

= r

= 150 mi. On the round trip, the average rate is r = 60 mi/hr

and the distance is d = 2d

= 300 mi. So the time is t = 300/60 = 5 hr. Therefore, on

= t − t

= 5 − 3 = 2 hr, the distance is d

the second leg, the time is t

= d

= 150 mi,

and so the average rate is r

= 150/2 = 75 mi/hr.

5. 120 small unpainted cubes are assembled into a large 4 × 5 × 6 rectangular prism

(“box”). All six faces of this large box are then painted.

(a) How many of the small cubes remain unpainted? Why? These cubes consist of

the inner core of 2 × 3 × 4 = 24 cubes.

(b) How many of the small cubes end up with paint on exactly one of their faces?

Why? These cubes consist of the cubes on each of the six faces, not counting the

cubes on the edges or corners. For each of the two 4 × 5 faces there are 2 × 3 = 6

cubes. For each of the two 4 × 6 faces there are 2 × 4 = 8 cubes. For each of the

two 5 × 6 faces there are 3 × 4 = 12 cubes. The grand total is 52 cubes.

Why? These cubes consist of the cubes on the 12 edges, not counting the 8

corner cubes. The four edges of length 4 each contribute 2 cubes. The four edges

of length 5 each contribute 3 cubes. The four edges of length 6 each contribute 4

cubes. The grand total is 36 cubes.

(d) How many of the small cubes end up with paint on exactly three of their faces?

Why? These are the 8 corner cubes.

6. Two waitresses, Robin and Jen, and their sons, Nicholas, Dustin, and Miles, just

started collecting state quarters. To start their collection, they each acquired one

quarter representing a diﬀerent state: Rhode Island, Delaware, Massachusetts, New

York, and Pennsylvania. Use the clues to match the names and the states. You do not

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