2024년 6월 1일 토요일

수학공부와 작업기억

 오랜 세월 동안 유럽에서는 수학교육을 언어교육과 함께 기초소양교육의 양대 기둥으로 삼아왔다. 왜 그렇게 수학을 중시하고, 왜 누구나 어려운 수학을 공부해야 할까. 학생들은 대개 그저 좋은 학교에 들어가기 위해 공부를 할 뿐이다. 성인들 중에는 “기본적인 개념만 이해하면 됐지 왜 어렵게 배배 꼰 문제들까지 풀어야 하죠?”라든가 “어렵게 배웠던 수학을 고등학교를 졸업한 후에는 써먹은 적이 없어요”라고 하는 분들이 많다.

수학공부를 중시하는 것은 ‘수학적 지식’을 나중에 잘 써먹게 하기 위해서가 아니다. 그보다는 사고력을 키우는 것이 주요 목적이라 할 수 있다. 흔히 수학공부는 논리적 사고력과 문제 해결력을 키워준다고 말한다. 이런 추상적인 능력 외에도 학생들이 수학공부를 통해 얻을 수 있는 이점에는 여러 가지가 있다. 그중에 사람들이 놓치기 쉬운 다음과 같은 두 가지 이점에 대해 이야기해 보고자 한다.

첫번째, 학생들은 수학공부를 통해 좋은 학습 태도를 기를 수 있다. 수학 문제를 풀면서 책상머리에 오랫동안 붙어 앉아있는 습관을 기를 수 있고, 깊게 사고하는 훈련을 할 수 있다. 학습도 습관이다. 학습하는 습관이 몸에 잘 붙게 하는 데에는 수학공부만 한 게 없다. 수학 문제 푸는 데에 빠지면 몇 시간은 금방 간다. 그래서 수학을 잘하는 학생들이 다른 과목도 잘하는 이유는 (그들의 머리보다는) 그들이 수학공부를 통해 얻은 학습 태도 때문일 가능성이 높다. 좋은 학습 태도는 공부할 때만이 아니라 성인이 되어 일을 할 때도 필요하다. 한 보험회사 임원이 나에게 이런 말을 한 적이 있다. “수학공부를 많이 한 사원은 수학적 사고력 그 자체보다는 문제를 대할 때의 태도가 다릅니다.”

작업기억, 업무수행의 중요 요소

두번째 이점은 수학적 사고법이다. 그중 핵심이 되는 것이 바로 ‘작업기억(working memory)’이다. 작업기억은 작업 중에 얻은 정보를 일시적으로 유지하면서 학습, 이해, 판단 등을 계획하고 수행하는 능력을 말한다. 최근에는 많은 심리학자·교육학자들이 이것에 주목하고 있고, 이것이 학습 능력에 미치는 영향이 IQ(지능지수)보다 더 크다는 연구 결과가 많이 나온다. 단기기억과 다른 점은 단기기억은 정보를 가공 없이 그대로 기억하고 유지하는 것이고 작업기억은 ‘정보의 조작’이 수반되는 것이다.

수학에서 ‘17×21’과 같은 계산을 할 때도 몇 번 곱셈을 한 후에 그 값을 더하게 되는데 각 단계에서 얻은 값을 기억했다가 그것을 다음 단계에 적용하는 것이 작업기억의 간단한 예이다. 모든 논리적·수학적 사고는 두 가지 과정의 반복으로 이루어진다. 하나는 추상적인 개념이나 정보를 머릿속에 잘 넣는 과정이고 또 다른 과정은 그것을 잠시 기억하고 있다가 그 바로 다음에 적용하는 것이다. 이 두 번째 과정이 바로 작업기억이다. 예를 들어 √2가 무리수임을 귀류법을 써서 증명하는 과정을 보면, 우선 ‘무리수’와 ‘귀류법’이라는 개념을 머릿속에 넣어야 한다. 여기에서 귀류법이란 “결론이 참이 아니라면 모순이다”라는 것을 통해 결론이 참임을 증명하는 증명법이다. 그러면 증명은 “√2가 무리수가 아닌 유리수라 하자”로 시작해야 한다. 그리고 √2=q/p라 놓은 후 그다음 단계들로 접어드는데 이 단계들에서 필요한 것이 바로 작업기억이다.

작업기억은 학생들의 학업에서만 중요한 것이 아니라 성인이 된 후에도 자신이 하고 있는 일과 연관하여 새로운 지식이나 개념을 익히고 적용하는 과정에서도 중요하다. 한 회사에 새로 입사한 사원이 회사의 주요 업무를 파악하고 실행하는 과정, 기존의 사원들이 새로운 프로젝트를 기획하고 수행하는 과정, 중요한 사안에 대하여 판단하고 결정하는 과정 등에서 작업기억은 중요한 요소로 작용할 수 있다

경향신문사


작성자: 시간: 댓글 없음:

2024년 5월 17일 금요일

2024 AMC 8 Problems 문제

 

Problem 1

What is the ones digit of\[222,222-22,222-2,222-222-22-2?\]$\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 6\qquad\textbf{(E) } 8$

Solution

Problem 2

What is the value of this expression in decimal form?\[\frac{44}{11} + \frac{110}{44} + \frac{44}{1100}\]

$\textbf{(A) } 6.4\qquad\textbf{(B) } 6.504\qquad\textbf{(C) } 6.54\qquad\textbf{(D) } 6.9\qquad\textbf{(E) } 6.94$

Solution

Problem 3

Four squares of side lengths $4$, $7$, $9$, and $10$ units are arranged in increasing size order so that their left edges and bottom edges align. The squares alternate in the color pattern white-gray-white-gray, respectively, as shown in the figure. What is the area of the visible gray region in square units?

[asy] size(150); filldraw((0,0)--(10,0)--(10,10)--(0,10)--cycle,gray(0.7),linewidth(1)); filldraw((0,0)--(9,0)--(9,9)--(0,9)--cycle,white,linewidth(1)); filldraw((0,0)--(7,0)--(7,7)--(0,7)--cycle,gray(0.7),linewidth(1)); filldraw((0,0)--(4,0)--(4,4)--(0,4)--cycle,white,linewidth(1)); draw((11,0)--(11,4),linewidth(1)); draw((11,6)--(11,10),linewidth(1)); label("$10$",(11,5),fontsize(14pt)); draw((10.75,0)--(11.25,0),linewidth(1)); draw((10.75,10)--(11.25,10),linewidth(1)); draw((0,11)--(4,11),linewidth(1)); draw((6,11)--(9,11),linewidth(1)); draw((0,11.25)--(0,10.75),linewidth(1)); draw((9,11.25)--(9,10.75),linewidth(1)); label("$9$",(5,11),fontsize(14pt)); draw((-1,0)--(-1,1),linewidth(1)); draw((-1,3)--(-1,7),linewidth(1)); draw((-1.25,0)--(-0.75,0),linewidth(1)); draw((-1.25,7)--(-0.75,7),linewidth(1)); label("$7$",(-1,2),fontsize(14pt)); draw((0,-1)--(1,-1),linewidth(1)); draw((3,-1)--(4,-1),linewidth(1)); draw((0,-1.25)--(0,-.75),linewidth(1)); draw((4,-1.25)--(4,-.75),linewidth(1)); label("$4$",(2,-1),fontsize(14pt)); [/asy]

$\textbf{(A)}\ 42 \qquad \textbf{(B)}\ 45 \qquad \textbf{(C)}\ 49 \qquad \textbf{(D)}\ 50 \qquad \textbf{(E)}\ 52$

Solution

Problem 4

When Yunji added all the integers from $1$ to $9$, she mistakenly left out a number. Her incorrect sum turned out to be a square number. What number did Yunji leave out?

$\textbf{(A) } 5\qquad\textbf{(B) } 6\qquad\textbf{(C) } 7\qquad\textbf{(D) } 8\qquad\textbf{(E) } 9$


Solution

Problem 5

Aaliyah rolls two standard 6-sided dice. She notices that the product of the two numbers rolled is a multiple of $6$. Which of the following integers cannot be the sum of the two numbers?

$\textbf{(A) } 5\qquad\textbf{(B) } 6\qquad\textbf{(C) } 7\qquad\textbf{(D) } 8\qquad\textbf{(E) } 9$

Solution

Problem 6

Sergei skated around an ice rink, gliding along different paths. The gray lines in the figures below show four of the paths labeled $P$, $Q$, $R$, and $S.$ What is the sorted order of the four paths from shortest to longest?

2024 AMC 8-Problem 6.png

$\textbf{(A)}\ P,Q,R,S \qquad \textbf{(B)}\ P,R,S,Q \qquad \textbf{(C)}\ Q,S,P,R \qquad \textbf{(D)}\ R,P,S,Q \qquad \textbf{(E)}\ R,S,P,Q$

Solution

Problem 7

A $3\times 7$ rectangle is covered without overlap by 3 shapes of tiles: $2\times 2$, $1\times 4$, and $1\times 1$, shown below. What is the minimum possible number of $1\times 1$ tiles used?

caption

$\textbf{(A) } 1\qquad\textbf{(B) } 2\qquad\textbf{(C) } 3\qquad\textbf{(D) } 4\qquad\textbf{(E) } 5$

Solution

Problem 8

On Monday Taye has $$2$. Every day, he either gains $$3$ or doubles the amount of money he had on the previous day. How many different dollar amounts could Taye have on Thursday, $3$ days later?

$\textbf{(A) } 3\qquad\textbf{(B) } 4\qquad\textbf{(C) } 5\qquad\textbf{(D) } 6\qquad\textbf{(E) } 7$

Solution

Problem 9

All the marbles in Maria's collection are red, green, or blue. Maria has half as many red marbles as green marbles and twice as many blue marbles as green marbles. Which of the following could be the total number of marbles in Maria's collection?

$\textbf{(A) } 24\qquad\textbf{(B) } 25\qquad\textbf{(C) } 26\qquad\textbf{(D) } 27\qquad\textbf{(E) } 28$

Solution

Problem 10

In January 1980 the Moana Loa Observation recorded carbon dioxide $(CO_2)$ levels of 338 ppm (parts per million). Over the years the average $CO_2$ reading has increased by about 1.515 ppm each year. What is the expected $CO_2$ level in ppm in January 2030? Round your answer to the nearest integer.

$\textbf{(A)}\ 399 \qquad \textbf{(B)}\ 414 \qquad \textbf{(C)}\ 420 \qquad \textbf{(D)}\ 444 \qquad \textbf{(E)}\ 459$

Solution

Problem 11

The coordinates of $\triangle ABC$ are $A(5,7)$, $B(11,7)$, and $C(3,y)$, with $y>7$. The area of $\triangle ABC$ is 12. What is the value of $y$?

[asy] // Diagram inaccurate to prevent measuring with ruler. size(10cm); draw((3,10)--(11,7)--(5,7)--(3,10)); dot((5,7)); label("$A(5,7)$",(5,7),S); dot((11,7)); label("$B(11,7)$",(11,7),S); dot((3,10)); label("$C(3,y)$",(3,10),NW); // Problem 11: put on here by Andrei.martynau [/asy]

$\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad \textbf{(E) }12$

Solution

Problem 12

Rohan keeps a total of $90$ guppies in $4$ fish tanks.

  • There is $1$ more guppy in the $2$nd tank than the $1$st tank.
  • There are $2$ more guppies in the 3rd tank than the $2$nd tank.
  • There are $3$ more guppies in the 4th tank than the $3$rd tank.

How many guppies are in the $4$th tank?

$\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 21 \qquad \textbf{(C)}\ 23 \qquad \textbf{(D)}\ 24 \qquad \textbf{(E)}\ 26$

Solution

Problem 13

Buzz Bunny is hopping up and down a set of stairs, one step at a time. In how many ways can Buzz start on the ground, make a sequence of $6$ hops, and end up back on the ground? (For example, one sequence of hops is up-up-down-down-up-down.)

[asy] /* AMC8 P13 2024, revised by Teacher David */ /** * This Geometry Artwork/Graph is designed using GeoSketch v1.0, * a free software tool created by Tina Yan, William Zhong, and * Teacher David. * * For more information, please refer to * https://geosketch.org (under construction) */ defaultpen(linewidth(1pt)); unitsize(0.3pt); import graph; /** * Define a quadratic bezier curve function. */ typedef pair quad_bezier(real t); quad_bezier fungen (pair a, pair b, pair c) { return new pair (real t) { real x = (1-t)*(1-t)*a.x + 2*(1-t)*t*b.x + t*t*c.x; real y = (1-t)*(1-t)*a.y + 2*(1-t)*t*b.y + t*t*c.y; return (x,y); }; } quad_bezier qb0 = fungen((293,243),(237,276),(239,310)); draw(graph(qb0, 0,1)); quad_bezier qb1 = fungen((239,310),(274,301),(295,254)); draw(graph(qb1, 0,1)); quad_bezier qb2 = fungen((266,294),(260,309),(266,323)); draw(graph(qb2, 0,1)); quad_bezier qb3 = fungen((266,323),(294,311),(302,257)); draw(graph(qb3, 0,1)); quad_bezier qb4 = fungen((333,258),(341,249),(348,244)); draw(graph(qb4, 0,1)); quad_bezier qb5 = fungen((348,244),(355,241),(351,234)); draw(graph(qb5, 0,1)); quad_bezier qb6 = fungen((351,234),(348,226),(338,226)); draw(graph(qb6, 0,1)); quad_bezier qb7 = fungen((351,234),(350,219),(321,208)); draw(graph(qb7, 0,1)); quad_bezier qb8 = fungen((260,247),(135,293),(137,170)); draw(graph(qb8, 0,1)); quad_bezier qb9 = fungen((122,161),(132,147),(148,144)); draw(graph(qb9, 0,1)); quad_bezier qb10 = fungen((148,144),(176,155),(204,146)); draw(graph(qb10, 0,1)); quad_bezier qb11 = fungen((204,146),(216,141),(235,137)); draw(graph(qb11, 0,1)); quad_bezier qb12 = fungen((228,156),(208,160),(188,161)); draw(graph(qb12, 0,1)); quad_bezier qb13 = fungen((319,214),(313,174),(283,168)); draw(graph(qb13, 0,1)); quad_bezier qb14 = fungen((228,156),(242,158),(247,171)); draw(graph(qb14, 0,1)); quad_bezier qb15 = fungen((245,181),(250,158),(266,143)); draw(graph(qb15, 0,1)); quad_bezier qb16 = fungen((266,143),(287,134),(298,135)); draw(graph(qb16, 0,1)); quad_bezier qb17 = fungen((298,135),(309,143),(300,148)); draw(graph(qb17, 0,1)); quad_bezier qb18 = fungen((300,148),(272,150),(270,175)); draw(graph(qb18, 0,1)); quad_bezier qb19 = fungen((282,177),(274,158),(300,148)); draw(graph(qb19, 0,1)); draw(arc((317.8948497854077,245.25965665236052), 19.760615163024095, 143.54947493250435, 40.14574559948477)); draw(arc((282.65584415584414,295.7857142857143), 53.78971270402217, -78.91253214600312, -114.90992209204622)); draw(arc((127.7,168.5), 9.420191080864546, 9.162347045721713, 232.7651660184253)); draw(arc((229.125,145.625), 10.435815732370902, -55.73889710090544, 96.1886159632416)); draw(arc((186.26470588235293,181.5), 20.573313920580237, -85.1615330431756, 85.1615330431756)); filldraw(ellipse((314,235), 13.0, 10.04987562112089), rgb(254,255,255), black); filldraw(rotate(14.036243467926468,(315,251))*ellipse((315,235), 9.219544457292887, 8.246211251235321), rgb(0,0,0), black); pair o = (400,190); real len=80; real height=56; for (int i=0; i<4; ++i) { pair a = (i*len, i*height); path p = a -- a+(len,0) -- a+(len, height); draw(shift(o)*p); } path p = (0,0)--(0,-height)--(4*len,-height); draw(shift(o)*p); [/asy]


$\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 12$

Solution

Problem 14

The one-way routes connecting towns $A,M,C,X,Y,$ and $Z$ are shown in the figure below (not drawn to scale).The distances in kilometers along each route are marked. Traveling along these routes, what is the shortest distance from A to Z in kilometers?

2024-AMC8-q14.png

$\textbf{(A)}\ 28 \qquad \textbf{(B)}\ 29 \qquad \textbf{(C)}\ 30 \qquad \textbf{(D)}\ 31 \qquad \textbf{(E)}\ 32$

Solution

Problem 15

Let the letters $F$,$L$,$Y$,$B$,$U$,$G$ represent distinct digits. Suppose $\underline{F}~\underline{L}~\underline{Y}~\underline{F}~\underline{L}~\underline{Y}$ is the greatest number that satisfies the equation

\[8\cdot\underline{F}~\underline{L}~\underline{Y}~\underline{F}~\underline{L}~\underline{Y}=\underline{B}~\underline{U}~\underline{G}~\underline{B}~\underline{U}~\underline{G}.\]

What is the value of $\underline{F}~\underline{L}~\underline{Y}+\underline{B}~\underline{U}~\underline{G}$?

$\textbf{(A)}\ 1089 \qquad \textbf{(B)}\ 1098 \qquad \textbf{(C)}\ 1107 \qquad \textbf{(D)}\ 1116 \qquad \textbf{(E)}\ 1125$

Solution

Problem 16

Minh enters the numbers $1$ through $81$ into the cells of a $9 \times 9$ grid in some order. She calculates the product of the numbers in each row and column. What is the least number of rows and columns that could have a product divisible by $3$?

$\textbf{(A) } 8\qquad\textbf{(B) } 9\qquad\textbf{(C) } 10\qquad\textbf{(D) } 11\qquad\textbf{(E) } 12$

Solution

Problem 17

A chess king is said to attack all squares one step away from it (basically any square right next to it in any direction), horizontally, vertically, or diagonally. For instance, a king on the center square of a 3 x 3 grid attacks all 8 other squares, as shown below. Suppose a white king and a black king are placed on different squares of 3 x 3 grid so that they do not attack each other. In how many ways can this be done?

[asy] /* AMC8 P17 2024, revised by Teacher David */ unitsize(29pt); import math; add(grid(3,3)); pair [] a = {(0.5,0.5), (0.5, 1.5), (0.5, 2.5), (1.5, 2.5), (2.5,2.5), (2.5,1.5), (2.5,0.5), (1.5,0.5)}; for (int i=0; i<a.length; ++i) { pair x = (1.5,1.5) + 0.4*dir(225-45*i); draw(x -- a[i], arrow=EndArrow()); } label("$K$", (1.5,1.5)); [/asy]

$\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 27 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 32$

Solution

Problem 18

Three concentric circles centered at $O$ have radii of $1$, $2$, and $3$. Points $B$ and $C$ lie on the largest circle. The region between the two smaller circles is shaded, as is the portion of the region between the two larger circles bounded by central angle $BOC$, as shown in the figure below. Suppose the shaded and unshaded regions are equal in area. What is the measure of $\angle{BOC}$ in degrees?

[asy] size(100); import graph; draw(circle((0,0),3)); real radius = 3; real angleStart = -54; // starting angle of the sector real angleEnd = 54; // ending angle of the sector label("$O$",(0,0),W); pair O = (0, 0); filldraw(arc(O, radius, angleStart, angleEnd)--O--cycle, lightgray); filldraw(circle((0,0),2),lightgray); filldraw(circle((0,0),1),white); draw((1.763,2.427)--(0,0)--(1.763,-2.427)); label("$B$",(1.763,2.427),NE); label("$C$",(1.763,-2.427),SE); [/asy]

$\textbf{(A)}\ 108 \qquad \textbf{(B)}\ 120 \qquad \textbf{(C)}\ 135 \qquad \textbf{(D)}\ 144 \qquad \textbf{(E)}\ 150$

Solution

Problem 19

Jordan owns 15 pairs of sneakers. Three fifths of the pairs are red and the rest are white. Two thirds of the pairs are high-top and the rest are low-top. The red high-top sneakers make up a fraction of the collection. What is the least possible value of this fraction?

caption


$\textbf{(A) } 0\qquad\textbf{(B) } \dfrac{1}{5} \qquad\textbf{(C) } \dfrac{4}{15} \qquad\textbf{(D) } \dfrac{1}{3} \qquad\textbf{(E) } \dfrac{2}{5}$

Solution

Problem 20

Any three vertices of the cube $PQRSTUVW,$ shown in the figure below, can be connected to form a triangle. $($For example, vertices $P, Q,$ and $R$ can be connected to form $\triangle{PQR}.)$ How many of these triangles are equilateral and contain $P$ as a vertex?

[asy] unitsize(4); pair P,Q,R,S,T,U,V,W; P=(0,30); Q=(30,30); R=(40,40); S=(10,40); T=(10,10); U=(40,10); V=(30,0); W=(0,0); draw(W--V); draw(V--Q); draw(Q--P); draw(P--W); draw(T--U); draw(U--R); draw(R--S); draw(S--T); draw(W--T); draw(P--S); draw(V--U); draw(Q--R); dot(P); dot(Q); dot(R); dot(S); dot(T); dot(U); dot(V); dot(W); label("$P$",P,NW); label("$Q$",Q,NW); label("$R$",R,NE); label("$S$",S,N); label("$T$",T,NE); label("$U$",U,NE); label("$V$",V,SE); label("$W$",W,SW); [/asy]

$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }6$

Solution

Problem 21

A group of frogs (called an army) is living in a tree. A frog turns green when in the shade and turns yellow when in the sun. Initially, the ratio of green to yellow frogs was $3 : 1$. Then $3$ green frogs moved to the sunny side and $5$ yellow frogs moved to the shady side. Now the ratio is $4 : 1$. What is the difference between the number of green frogs and the number of yellow frogs now?

$\textbf{(A) } 10\qquad\textbf{(B) } 12\qquad\textbf{(C) } 16\qquad\textbf{(D) } 20\qquad\textbf{(E) } 24$

Solution

Problem 22

A roll of tape is $4$ inches in diameter and is wrapped around a ring that is $2$ inches in diameter. A cross section of the tape is shown in the figure below. The tape is $0.015$ inches thick. If the tape is completely unrolled, approximately how long would it be? Round your answer to the nearest $100$ inches.

[asy] /* AMC8 P22 2024, revised by Teacher David */ size(150); pair o = (0,0); real r1 = 1; real r2 = 2; filldraw(circle(o, r2), gray, linewidth(1pt)); filldraw(circle(o, r1), white, linewidth(1pt)); draw((-2,-2.6)--(-2,-2.4)); draw((2,-2.6)--(2,-2.4)); draw((-2,-2.5)--(2,-2.5), L=Label("4 in.")); draw((-1,0)--(1,0), L=Label("2 in.", align=(0,1)), arrow=Arrows()); draw((2,0)--(2,-1.3), linewidth(1pt)); [/asy]

$\textbf{(A) } 300\qquad\textbf{(B)} 600\qquad\textbf{(C) } 1200\qquad\textbf{(D) } 1500\qquad\textbf{(E) } 1800$

Solution

Problem 23

Rodrigo has a very large sheet of graph paper. First he draws a line segment connecting point $(0,4)$ to point $(2,0)$ and colors the $4$ cells whose interiors intersect the segment, as shown below. Next Rodrigo draws a line segment connecting point $(2000,3000)$ to point $(5000,8000)$. How many cells will he color this time?

[asy] filldraw((0,4)--(1,4)--(1,3)--(0,3)--cycle, gray(.75), gray(.5)+linewidth(1)); filldraw((0,3)--(1,3)--(1,2)--(0,2)--cycle, gray(.75), gray(.5)+linewidth(1)); filldraw((1,2)--(2,2)--(2,1)--(1,1)--cycle, gray(.75), gray(.5)+linewidth(1)); filldraw((1,1)--(2,1)--(2,0)--(1,0)--cycle, gray(.75), gray(.5)+linewidth(1)); draw((-1,5)--(-1,-1),gray(.9)); draw((0,5)--(0,-1),gray(.9)); draw((1,5)--(1,-1),gray(.9)); draw((2,5)--(2,-1),gray(.9)); draw((3,5)--(3,-1),gray(.9)); draw((4,5)--(4,-1),gray(.9)); draw((5,5)--(5,-1),gray(.9)); draw((-1,5)--(5, 5),gray(.9)); draw((-1,4)--(5,4),gray(.9)); draw((-1,3)--(5,3),gray(.9)); draw((-1,2)--(5,2),gray(.9)); draw((-1,1)--(5,1),gray(.9)); draw((-1,0)--(5,0),gray(.9)); draw((-1,-1)--(5,-1),gray(.9)); dot((0,4)); label("$(0,4)$",(0,4),NW); dot((2,0)); label("$(2,0)$",(2,0),SE); draw((0,4)--(2,0)); draw((-1,0) -- (5,0), arrow=Arrow); draw((0,-1) -- (0,5), arrow=Arrow); [/asy]

$\textbf{(A) }6000\qquad\textbf{(B) }6500\qquad\textbf{(C) }7000\qquad\textbf{(D) }7500\qquad\textbf{(E) }8000$

Solution

Problem 24

Jean has made a piece of stained glass art in the shape of two mountains, as shown in the figure below. One mountain peak is $8$ feet high while the other peak is $12$ feet high. Each peak forms a $90^\circ$ angle, and the straight sides form a $45^\circ$ angle with the ground. The artwork has an area of $183$ square feet. The sides of the mountain meet at an intersection point near the center of the artwork, $h$ feet above the ground. What is the value of $h?$

[asy] unitsize(.3cm); filldraw((0,0)--(8,8)--(11,5)--(18,12)--(30,0)--cycle,gray(0.7),linewidth(1)); draw((-1,0)--(-1,8),linewidth(.75)); draw((-1.4,0)--(-.6,0),linewidth(.75)); draw((-1.4,8)--(-.6,8),linewidth(.75)); label("$8$",(-1,4),W); label("$12$",(31,6),E); draw((-1,8)--(8,8),dashed); draw((31,0)--(31,12),linewidth(.75)); draw((30.6,0)--(31.4,0),linewidth(.75)); draw((30.6,12)--(31.4,12),linewidth(.75)); draw((31,12)--(18,12),dashed); label("$45^{\circ}$",(.75,0),NE,fontsize(10pt)); label("$45^{\circ}$",(29.25,0),NW,fontsize(10pt)); draw((8,8)--(7.5,7.5)--(8,7)--(8.5,7.5)--cycle); draw((18,12)--(17.5,11.5)--(18,11)--(18.5,11.5)--cycle); draw((11,5)--(11,0),dashed); label("$h$",(11,2.5),E); [/asy]

$\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 4\sqrt{2} \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 5\sqrt{2}$

Solution

Problem 25

A small airplane has $4$ rows of seats with $3$ seats in each row. Eight passengers have boarded the plane and are distributed randomly among the seats. A married couple is next to board. What is the probability there will be $2$ adjacent seats in the same row for the couple?

2024 AMC 8-Problem 25.png

$\textbf{(A)}\ \dfrac{8}{15} \qquad \textbf{(B)}\ \dfrac{32}{55} \qquad \textbf{(C)}\ \dfrac{20}{33} \qquad \textbf{(D)}\ \dfrac{34}{55} \qquad \textbf{(E)}\ \dfrac{8}{11}$

작성자: 시간: 댓글 없음:
피드 구독하기: 글 (Atom)