Warm-Up If last month was July, then what month will it be 22 months from now?

Transcription

1 Warm-Up. If last month was July, then what month will it be months from now?. The maximum slope for handicap ramps was changed from to. What is the positive difference of these slopes? Express your answer as a common fraction.. How many natural-number factors does N have if = " 4. A building modeled after the Chicago Sears Tower consists of nine square towers arranged in a three by three grid. They have congruent bases, and the heights, in feet, are indicated in the grid to the right. Which of the following is a side view of the building from some direction? 5. Let I [ = [ and J[ = [ + What is the value of I+ J " 6. An oil tanker containing 08,000 gallons of oil releases one third of its remaining volume every two hours. How many gallons have been released after the first six hours? 7. What is the largest perfect square less than 5 that is a multiple of 9? 8. Merina s annual salary is a whole number of dollars between $4,400 and $4,500. The digits in the hundreds, tens and units places are in strictly ascending order. How many distinct possibilities exist for her annual salary? 9. The area of a circular plate is π cm. What is the number of centimeters in the radius of the plate? Express your answer in simplest radical form. 0. Three standard dice are tossed. What is the probability that the sum of the numbers on the tops of the three dice is 7 or greater? Express your answer as a common fraction. MATHCOUNTS

2 Los Primeros MATHCOUNTS Quiz 8: Warm-Up November 7, 00. If last month was July, then what month will it be months from now? It must now be August. In 4 months it will again be August. In months it will be two months prior to August, i.e., it will be June.. The maximum slope for handicap ramps was changed from 0 to.whatis the positive difference of these slopes? Express your answer as a common fraction. 0 = = 60. How many natural-number factors does N have if N = 5? Counting factors is a very common problem in MATHCOUNTS, so it is important to master this technique. For this problem, N = 60 is already in prime factorization form, which is important when solving a problem like this. Once we ve written the N in this fashion, it is easy to see that any factor of N must be of the form l m 5 n where the integers l, m, andn must satisfy 0 l 4 choices for l 0 m choices for m 0 n choices for n. The values for l, m, andn can each be chosen independently of the others. Thus, by the fundamental principle of counting, 5 has 4 = 4 factors, or 4 ways in which the, and 5 can be used to form a factor of the original value. In general any natural number M can be written in prime factorization form as M = P m P m P m P m n n where P, P,...P n are the prime factors of M and m, m,...,m n are the integer exponents, each greater than 0. The number of factors of M (including and M) is (m + )(m + )(m + ) (m n + ).

3 4. A building modeled after the Chicago Sears Tower consists of nine square towers arranged in a three by three grid. They have congruent bases, and the heights, in feet, are indicated in the grid to the right Which of the following is a side view of the building from some direction? We consider four possible observers located at the positions labeled front, rear, left, and right, as defined below: back left right front These observers would see the following views of the buildings: left back right front We see that the front view is the same as the solution labeled D. 5. Let f (x) = x andg(x) = x +. What is the value of f ( + g())? Since g() = + =, then f ( + g()) = f ( + ) = f (4) = (4) = An oil tanker containing 08,000 gallons of oil releases one third of its remaining volume every two hours. How many gallons have been released after the first six hours? After hours, the tanker has released = 6000 gallons with = 7000 gallons remaining. After 4 hours, another 7000 = 4000 gallons have been released, with 7000 = gallons remaining. After 6 hours, another = 6000 gallons have been released, with = 000 gallons remaining. The total amount of oil released is = 76,000 gallons. Another way to look at the problem is to realize that after every two hours the remaining volume is reduced by a factor of so that after six hours, the fraction

4 remaining is (, ( ) and the fraction released is. ) It follows that the total amount released is [ ( ) ] [ Total oil released = = 8 ] = = 76,000 gal 7 7. What is the largest perfect square less than 5 that is a multiple of 9? Any perfect square that is a multiple of 9 must have 9 = as a factor. Note that 5 = 5 = 5. The next smaller perfect square containing 9 as a factor must be 4 = = Merinaí s annual salary is a whole number of dollars between $4,400 and $4,500. The digits in the hundreds, tens and units places are in strictly ascending order. How many distinct possibilities exist for her annual salary? Her salary must start with the digits 44. Since the hundreds, tens and units digits are in strict increasing order, we can easily list the possibilities for the remaining two digits: = T 4 = = 0 possibilities 9. The area of a circular plate is 00π cm. What is the number of centimeters in the radius of the plate? Express your answer in simplest radical form. The radius r and area A of a circle are related by A = πr or r = A/π so that r = A π = 00π π = 00 = 00 = Three standard dice are tossed. What is the probability that the sum of the numbers on the tops of the three dice is 7 or greater? Express your answer as a common fraction. We can get 7 or greater for the sum only if the sum is 7 or 8. For the sum to be 8 would mean that we rolled 6 on each die. The probability of obtaining 8 for the sum is therefore (/6) = /6. To obtain 7 for the sum, we must have

5 rolled two 6 s and one 5. There are three ways in which this could be done: (6,6,5), (6,5,6), and (5,6,6). Each of these three outcomes also has probability /6. The probability of rolling a sum of 7 or greater is therefore equal to the sum of the individual outcome probabilities: 4 6 = 54. 4