Difference between revisions of "2006 AMC 8"

Revision as of 06:30, 29 November 2010

Problems

1. Mindy made three purchases for <dollar/>1.98, <dollar/>5.04, and <dollar/>9.89. What was her total, to the nearest dollar?

A) <dollar/>10

B) <dollar/>15

C) <dollar/>16

D) <dollar/>17

E) <dollar/>18

2. On the AMC 8 contest Billy answers 13 questions correctly, answers 7 incorrectly and doesn't answer the last 5. What is his score?

A) 1

B) 6

C) 13

D) 19

E) 26

3. Elisa swims laps in the pool. When she first started, she completed 10 laps in 25 minutes. Now she can finish 12 laps in 24 minutes. By how many minutes has she improved her lap time?

A) $\frac{1}{2}$

B) $\frac{3}{4}$

C) 1

D) 2

E) 3

4. Initially, a spinner points west. Chenille moves it clockwise $2\frac{1}{4}$ revolutions and then counterclockwise $3\frac{3}{4}$ revolutions. In what direction does the spinner point after the two moves?

A) north

B) east

C) south

D) west

E) northwest

5. Points $A$ , $B$ , $C$ , and $D$ are midpoints of the sides of the larger square. If the larger square has area 60, what is the area of the smaller square?

$[asy] size(100); draw((0,0)--(2,0)--(2,2)--(0,2)--cycle,linewidth(1)); draw((0,1)--(1,2)--(2,1)--(1,0)--cycle); label("$A$", (1,2), N); label("$B$", (2,1), E); label("$C$", (1,0), S); label("$D$", (0,1), W);[/asy]$

A) 15

B) 20

C) 24

D) 30

E) 40

6. The letter T is formed by placing two $2\times4$ inch rectangles next to each other, as shown. What is the perimeter of the T, in inches?

$[asy] size(150); draw((0,6)--(4,6)--(4,4)--(3,4)--(3,0)--(1,0)--(1,4)--(0,4)--cycle, linewidth(1)); [/asy]$

A) 12

B) 16

C) 20

D) 22

E) 24

7. Circle $x$ has a radius of $\pi$ . Circle $Y$ has a circumference of $8\pi$ . Circle $Z$ has an area of $9\pi$ . List the circles in order from smallest to largest radius.

A) $X, Y, Z$

B) $Z, X, Y$

C) $Y, X, Z$

D) $Z, Y, X$

E) $X, Z, Y$

8. The table shows some of the results of a survey by radiostation KAMC. What percentage of the males surveyed listen to the station?

$\begin{tabular}{|c|c|c|c|}\hline & Listen & Don't Listen & Total\\ \hline Males & ? & 26 & ?\\ \hline Females & 58 & ? & 96\\ \hline Total & 136 & 64 & 200\\ \hline\end{tabular}$

A) 39

B) 48

C) 52

D) 55

E) 75

9. What is the product of $\frac{3}{2}\times\frac{4}{3}\times\frac{5}{4}\times\cdots\times\frac{2006}{2005}$ ?

A) 1

B) 1002

C) 1003

D) 2005

E) 2006

10. Jorge's teacher asks him to plot all the ordered pairs $(w,l)$ of positive integers for which $w$ is the width and $l$ is the length of a rectangle with area 12. What should his graph look like?

A) $[asy] size(75); draw((0,-1)--(0,13)); draw((-1,0)--(13,0)); dot((1,12)); dot((2,6)); dot((3,4)); dot((4,3)); dot((6,2)); dot((12,1)); label("$l$", (0,6), W); label("$w$", (6,0), S);[/asy]$

B) $[asy] size(75); draw((0,-1)--(0,13)); draw((-1,0)--(13,0)); dot((1,1)); dot((3,3)); dot((5,5)); dot((7,7)); dot((9,9)); dot((11,11)); label("$l$", (0,6), W); label("$w$", (6,0), S);[/asy]$

C) $[asy] size(75); draw((0,-1)--(0,13)); draw((-1,0)--(13,0)); dot((1,11)); dot((3,9)); dot((5,7)); dot((7,5)); dot((9,3)); dot((11,1)); label("$l$", (0,6), W); label("$w$", (6,0), S);[/asy]$

D) $[asy] size(75); draw((0,-1)--(0,13)); draw((-1,0)--(13,0)); dot((1,6)); dot((3,6)); dot((5,6)); dot((7,6)); dot((9,6)); dot((11,6)); label("$l$", (0,6), W); label("$w$", (6,0), S);[/asy]$

E) $[asy] size(75); draw((0,-1)--(0,13)); draw((-1,0)--(13,0)); dot((6,1)); dot((6,3)); dot((6,5)); dot((6,7)); dot((6,9)); dot((6,11)); label("$l$", (0,6), W); label("$w$", (6,0), S);[/asy]$

Solutions

1. (D) Mindy's total was approximately $2+5+10=17.$

2. (C) On the AMC 8 a student's score is the number of problems answered correctly. So Billy's score is 13. Because there is no penalty for guessing, if he wants to increase his score, he probably should fill in the last five answers.

3. (A) When Elisa started, she completed a lap in $\frac{25}{10}=2.5$ minutes. Now she can complete a lap in $\frac{24}{12}=2$ minutes. She has improved her lap time by $2.5-2=0.5$ or $\frac{1}{2}$ minute.

4. (B) Ignore the number of complete revolutions because they do not affect direction. One-fourth of the distance around the circle clockwise from west is north. Three-fourth of the distance counterclockwise around the circle from north is east. Chenille's spinner points east.

5. (D) The area of the smaller square is actually half of the area of the bigger square. You see, those small triangles that you see between two adjacent letters and the corner between them can be flipped in to exactly cover the smaller square.

6. (C) The perimeter of the T is simply the perimeter of each rectangle minus the parts on the border of a rectangle that are no longer on the perimeter. The perimeter of each of 12, and the width of the lower one is two, (the top width of the lower one is no longer on the perimeter), but notice also that part of the bottom length of the top rectangle is covered. It must also have length two, so $12\times2-2-2=20$

7. (B) The radius of $X$ is given: $\pi$ . The radius of $Y$ can be found with the equation $C=2\pi r$ : Plugging in $8\pi$ for the circumference, we get a radius of $4$ . The radius of $Z$ can be found with $A=\pi r^2$ : Plugging in $9\pi$ for the circumference, we get $\sqrt{9}=3$ for the radius. Since $pi$ is between 3 and 4, the order of the radii is $Z, X, Y$ .

8. (E) The number of males that listen is $136-58=78$ . The total number of males is $78+26=104$ . $\frac{78}{104}$ can be reduced to $\frac{3}{4}$ , or $75\%$ .

9. (C) Each integer between 3 and 2005 inclusive are in the numerator and denominator. These may cancel, leaving us with $\frac{2006}{2}=1003$ .

10. (A) We see that $l$ should decrease as $w$ increases, leaving us with only choices $A$ and $C$ . But $C$ is a straight line. That must be wrong, because it would imply that the perimeter would remain constant, and the area of a rectangle changes as it becomes more oblong or square while keeping the same perimeter. We know that the area remains the same, so the answer must be $A$ .

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Difference between revisions of "2006 AMC 8"

Revision as of 06:30, 29 November 2010

@@ Line 1: / Line 1: @@
 '''Problems'''
-. Mindy made three purchases for <dollar/>1.98, <dollar/>5.04, and <dollar/>9.89.  What was her total, to the nearest dollar?
+.
+Mindy made three purchases for <dollar/>1.98, <dollar/>5.04, and <dollar/>9.89.  What was her total, to the nearest dollar?
 A) <dollar/>10
@@ Line 51: / Line 52: @@
 E) northwest
+. Points <math> A </math>, <math> B </math>, <math> C </math>, and <math> D </math> are midpoints of the sides of the larger square.  If the larger square has area 60, what is the area of the smaller square?
+<asy>
+size(100);
+draw((0,0)--(2,0)--(2,2)--(0,2)--cycle,linewidth(1));
+draw((0,1)--(1,2)--(2,1)--(1,0)--cycle);
+label("$A$", (1,2), N);
+label("$B$", (2,1), E);
+label("$C$", (1,0), S);
+label("$D$", (0,1), W);</asy>
+A) 15
+B) 20
+C) 24
+D) 30
+E) 40
+. The letter T is formed by placing two <math> 2\times4 </math> inch rectangles next to each other, as shown. What is the perimeter of the T, in inches?
+<asy>
+size(150);
+draw((0,6)--(4,6)--(4,4)--(3,4)--(3,0)--(1,0)--(1,4)--(0,4)--cycle, linewidth(1));
+</asy>
+A) 12
+B) 16
+C) 20
+D) 22
+E) 24
+. Circle <math> x </math> has a radius of <math> \pi </math>.  Circle <math> Y </math> has a circumference of <math> 8\pi </math>.  Circle <math> Z </math> has an area of <math> 9\pi </math>.  List the circles in order from smallest to largest radius.
+A) <math> X, Y, Z </math>
+B) <math> Z, X, Y </math>
+C) <math> Y, X, Z </math>
+D) <math> Z, Y, X </math>
+E) <math> X, Z, Y </math>
+. The table shows some of the results of a survey by radiostation KAMC. What percentage of the males surveyed listen to the station?
+<math> \begin{tabular}{|c|c|c|c|}\hline & Listen & Don't Listen & Total\\ \hline Males & ? & 26 & ?\\ \hline Females & 58 & ? & 96\\ \hline Total & 136 & 64 & 200\\ \hline\end{tabular} </math>
+A) 39
+B) 48
+C) 52
+D) 55
+E) 75
+. What is the product of <math> \frac{3}{2}\times\frac{4}{3}\times\frac{5}{4}\times\cdots\times\frac{2006}{2005} </math>?
+A) 1
+B) 1002
+C) 1003
+D) 2005
+E) 2006
+. Jorge's teacher asks him to plot all the ordered pairs <math> (w,l) </math> of positive integers for which <math> w </math> is the width and <math> l </math> is the length of a rectangle with area 12. What should his graph look like?
+A) <asy>
+size(75);
+draw((0,-1)--(0,13));
+draw((-1,0)--(13,0));
+dot((1,12));
+dot((2,6));
+dot((3,4));
+dot((4,3));
+dot((6,2));
+dot((12,1));
+label("$l$", (0,6), W);
+label("$w$", (6,0), S);</asy>
+B) <asy>
+size(75);
+draw((0,-1)--(0,13));
+draw((-1,0)--(13,0));
+dot((1,1));
+dot((3,3));
+dot((5,5));
+dot((7,7));
+dot((9,9));
+dot((11,11));
+label("$l$", (0,6), W);
+label("$w$", (6,0), S);</asy>
+C) <asy>
+size(75);
+draw((0,-1)--(0,13));
+draw((-1,0)--(13,0));
+dot((1,11));
+dot((3,9));
+dot((5,7));
+dot((7,5));
+dot((9,3));
+dot((11,1));
+label("$l$", (0,6), W);
+label("$w$", (6,0), S);</asy>
+D) <asy>
+size(75);
+draw((0,-1)--(0,13));
+draw((-1,0)--(13,0));
+dot((1,6));
+dot((3,6));
+dot((5,6));
+dot((7,6));
+dot((9,6));
+dot((11,6));
+label("$l$", (0,6), W);
+label("$w$", (6,0), S);</asy>
+E) <asy>
+size(75);
+draw((0,-1)--(0,13));
+draw((-1,0)--(13,0));
+dot((6,1));
+dot((6,3));
+dot((6,5));
+dot((6,7));
+dot((6,9));
+dot((6,11));
+label("$l$", (0,6), W);
+label("$w$", (6,0), S);</asy>
 '''Solutions'''
 . (D) Mindy's total was approximately <math>2+5+10=17.</math>
 . (C) On the AMC 8 a student's score is the number of problems answered correctly.  So Billy's score is 13.  Because there is no penalty for guessing, if he wants to increase his score, he probably should fill in the last five answers.
 . (A) When Elisa started, she completed a lap in <math>\frac{25}{10}=2.5</math> minutes.  Now she can complete a lap in <math>\frac{24}{12}=2</math> minutes.  She has improved her lap time by <math>2.5-2=0.5</math> or <math>\frac{1}{2}</math> minute.
 . (B) Ignore the number of complete revolutions because they do not affect direction.  One-fourth of the distance around the circle clockwise from west is north.  Three-fourth of the distance counterclockwise around the circle from north is east.  Chenille's spinner points east.
+. (D) The area of the smaller square is actually half of the area of the bigger square.  You see, those small triangles that you see between two adjacent letters and the corner between them can be flipped in to exactly cover the smaller square.
+. (C) The perimeter of the T is simply the perimeter of each rectangle minus the parts on the border of a rectangle that are no longer on the perimeter.  The perimeter of each of 12, and the width of the lower one is two, (the top width of the lower one is no longer on the perimeter), but notice also that part of the bottom length of the top rectangle is covered.  It must also have length two, so <math> 12\times2-2-2=20 </math>
+. (B) The radius of <math> X </math> is given: <math> \pi </math>.  The radius of <math> Y </math> can be found with the equation <math> C=2\pi r </math>: Plugging in <math> 8\pi </math> for the circumference, we get a radius of <math> 4 </math>.  The radius of <math> Z </math> can be found with <math> A=\pi r^2 </math>:  Plugging in <math> 9\pi </math> for the circumference, we get <math> \sqrt{9}=3 </math> for the radius.  Since <math> pi </math> is between 3 and 4, the order of the radii is <math> Z, X, Y </math>.
+. (E) The number of males that listen is <math> 136-58=78 </math>.  The total number of males is <math> 78+26=104 </math>.  <math>\frac{78}{104} </math> can be reduced to <math> \frac{3}{4} </math>, or <math> 75\% </math>.
+. (C) Each integer between 3 and 2005 inclusive are in the numerator and denominator.  These may cancel, leaving us with <math> \frac{2006}{2}=1003 </math>.
+. (A) We see that <math> l </math> should decrease as <math> w </math> increases, leaving us with only choices <math> A </math> and <math> C </math>.  But <math> C </math> is a straight line.  That must be wrong, because it would imply that the perimeter would remain constant, and the area of a rectangle changes as it becomes more oblong or square while keeping the same perimeter.  We know that the area remains the same, so the answer must be <math> A </math>.
 Copyright 2006, The Mathematical Association of America