Self-Study Workbook

Algebra 2
& Geometry

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📐 Algebra 2

Word Problems — Core Units

Quadratics · Systems · Exponentials · Logs · Sequences · Rationals

1
Quadratic · Vertex Form Tricky
⚡
Quick Recall
y = a(x − h)² + k → vertex = (h, k), axis = x = h
꼭짓점 (h, k) — 부호 반드시 확인! h 앞에 마이너스 붙음

A ball is thrown upward. Its height (in feet) after t seconds is given by h(t) = −16(t − 3)² + 150. What is the maximum height the ball reaches, and at what time does it occur?

💡 Worked Example (similar)
f(t) = −2(t − 4)² + 80
Vertex = (4, 80) → max height = 80 ft at t = 4 sec
📖 Explanation
In vertex form h(t) = a(t − h)² + k, the vertex is (h, k). Here h = 3, k = 150. Since a = −16 < 0, the parabola opens down, so the vertex is the maximum.
  • 1. Vertex form: a(t − 3)² + 150 → vertex (3, 150)
  • 2. a < 0 → opens down → vertex = maximum
  • 3. Max height = 150 ft at t = 3 sec
⚠️ Choice A (134) is wrong — don't substitute t = 0. The vertex k-value IS the max directly.
2
Systems of Equations Tricky
⚡
Quick Recall
SUBSTITUTION: isolate one variable → plug in → solve back
한 변수 분리 → 대입 → 역대입 순서 지키기

Two pumps fill a tank. Pump A alone takes 6 hours; Pump B alone takes 4 hours. If both pumps run together, how many hours does it take to fill the tank? Express your answer as a simplified fraction.

💡 Worked Example (similar)
Rates: A = 1/6, B = 1/4 per hour
Combined rate = 1/6 + 1/4 = 2/12 + 3/12 = 5/12
Time = 1 ÷ (5/12) = 12/5 hours
📖 Explanation
  • 1. Rate A = 1/6 tank/hr, Rate B = 1/4 tank/hr
  • 2. Combined rate = 1/6 + 1/4 = 2/12 + 3/12 = 5/12
  • 3. Time = 1 ÷ (5/12) = 12/5 hours = 2.4 hrs
⚠️ Never just add the hours (6 + 4 = 10). Add the rates, not the times!
3
Exponential Growth Tricky
⚡
Quick Recall
A(t) = A₀ · (1 + r)^t → growth; (1 − r)^t → decay
증가: +r, 감소: −r / r은 소수로 변환 필수 (% ÷ 100)

A town's population is 12,000 and grows at 5% per year. Which expression gives the population after t years, and what is the population after 3 years (rounded to nearest whole number)?

💡 Worked Example
A(t) = 12000 · (1.05)^t
A(3) = 12000 · (1.05)³ = 12000 · 1.157625 ≈ 13,892
📖 Explanation
  • 1. Growth formula: A(t) = A₀(1 + r)^t, r = 0.05
  • 2. A(3) = 12000 × (1.05)³ = 12000 × 1.157625
  • 3. = 13,891.5 ≈ 13,891
⚠️ Choice C uses (1.05)t (linear), not (1.05)^t (exponential) — huge difference long-term!
4
Logarithms Tricky
⚡
Quick Recall
log_b(x) = y ↔ b^y = x · log(AB) = logA + logB
로그 = 지수로 바꾸는 게 핵심. 덧셈 ↔ 곱셈 변환 암기

The loudness of a sound in decibels is given by L = 10 · log(I / I₀) where I₀ = 10⁻¹² W/m². A sound has intensity I = 10⁻⁴ W/m². What is its loudness in decibels?

💡 Key log rule
log(10^n) = n
log(10⁻⁴ / 10⁻¹²) = log(10⁸) = 8
📖 Explanation
  • 1. I / I₀ = 10⁻⁴ / 10⁻¹² = 10^(−4−(−12)) = 10⁸
  • 2. log(10⁸) = 8
  • 3. L = 10 × 8 = 80 dB
⚠️ Choice D forgets to multiply by 10. The formula has a ×10 factor — don't skip it!
5
Arithmetic Sequence Tricky
⚡
Quick Recall
aₙ = a₁ + (n−1)d · Sₙ = n/2 · (a₁ + aₙ)
n번째 항: 첫 항 + (n−1) × 공차 / 합: 항 수 × 평균

A theater has 20 rows. The first row has 15 seats, and each successive row has 3 more seats than the previous one. How many total seats are in the theater?

💡 Formula
a₁ = 15, d = 3, n = 20
a₂₀ = 15 + 19·3 = 72
S₂₀ = 20/2 · (15 + 72) = 10 · 87 = 870
📖 Explanation
  • 1. Last row: a₂₀ = 15 + (20−1)×3 = 15 + 57 = 72
  • 2. Sum = (20/2)(15 + 72) = 10 × 87 = 870
⚠️ (n−1) not n! Many students write a₂₀ = 15 + 20×3 = 75 → wrong!
6
Quadratic · Discriminant Tricky
⚡
Quick Recall
Discriminant Δ = b²−4ac → Δ>0: 2 real, Δ=0: 1 real, Δ<0: no real
판별식 양수=두근, 0=중근, 음수=허근

A company's profit in thousands of dollars is modeled by P(x) = −2x² + 12x − 20 where x is the number of items sold (in hundreds). For how many values of x does the company break even (profit = 0)?

💡 Discriminant Check
Δ = b² − 4ac = 144 − 4(−2)(−20) = 144 − 160 = −16
📖 Explanation
  • 1. a = −2, b = 12, c = −20
  • 2. Δ = 12² − 4(−2)(−20) = 144 − 160 = −16
  • 3. Δ < 0 → no real solutions → company never breaks even
⚠️ Watch signs! 4ac = 4(−2)(−20) = +160, so you subtract it: 144 − 160 = −16.
7
Geometric Sequence Tricky
⚡
Quick Recall
aₙ = a₁ · r^(n−1) · Sₙ = a₁(1−rⁿ)/(1−r), r≠1
등비수열: 첫 항 × 공비^(n-1) / 합 공식 분모에 (1−r)

A bacteria culture starts with 500 cells and doubles every hour. After 5 hours, how many total cells have been produced in all 5 hours combined (sum from hour 0 through hour 4 doubling cycles = first 5 terms)?

💡 Geo Sum Formula
a₁=500, r=2, n=5
S₅ = 500·(2⁵−1)/(2−1) = 500·31 = 15,500
📖 Explanation
  • 1. Terms: 500, 1000, 2000, 4000, 8000
  • 2. S₅ = 500(2⁵ − 1)/(2 − 1) = 500 × 31 = 15,500
⚠️ Choice A (16,000) is just the 5th term alone. The question asks for the SUM of all 5 terms!
8
Rational Equations Tricky
⚡
Quick Recall
Rational eq → multiply by LCD → check for excluded values!
분수 방정식: LCD 곱하기 → 반드시 분모=0 되는 값 제외 확인

Maria can paint a fence in 8 hours, and her brother can do it in 12 hours. They start together, but after 2 hours, Maria leaves. How many more hours does the brother need to finish the fence alone?

💡 Setup
Work done together in 2 hrs: 2(1/8 + 1/12) = 2(5/24) = 5/12
Remaining: 1 − 5/12 = 7/12
Brother's rate = 1/12, time = (7/12)÷(1/12) = 7 hrs
📖 Explanation
  • 1. Together rate: 1/8 + 1/12 = 3/24 + 2/24 = 5/24 per hr
  • 2. Work done in 2 hrs: 2 × 5/24 = 10/24 = 5/12
  • 3. Remaining work: 1 − 5/12 = 7/12
  • 4. Brother alone: (7/12) ÷ (1/12) = 7 hours
9
Exponential Decay · Half-Life Tricky
⚡
Quick Recall
A(t) = A₀ · (1/2)^(t/h) where h = half-life
반감기 공식: t를 반감기로 나눈 값이 지수

A radioactive substance has a half-life of 10 years. A sample starts with 640 grams. How many grams remain after 40 years?

💡 Step-by-step
A(40) = 640 · (1/2)^(40/10) = 640 · (1/2)⁴ = 640/16 = 40
📖 Explanation
  • 1. t/h = 40/10 = 4 half-lives
  • 2. 640 → 320 → 160 → 80 → 40 grams
  • 3. Formula: 640 × (1/2)⁴ = 640/16 = 40
⚠️ Choice A (80) = after 3 half-lives, not 4. Count carefully: 40 ÷ 10 = 4 cycles.
10
Systems · Linear + Quadratic Tricky
⚡
Quick Recall
Line meets parabola → substitute y = mx+b into y = ax²+bx+c
선 + 포물선 교점: y 대입 후 이차방정식 풀기

A ball is launched from ground level with height h = −t² + 6t (feet, t in seconds). A ledge is at height h = 5 feet. At what times does the ball pass the ledge?

💡 Set equal and solve
−t² + 6t = 5 → t² − 6t + 5 = 0
(t − 1)(t − 5) = 0 → t = 1 or t = 5
📖 Explanation
  • 1. Set −t² + 6t = 5
  • 2. Rearrange: t² − 6t + 5 = 0
  • 3. Factor: (t − 1)(t − 5) = 0 → t = 1 and t = 5
⚠️ D (t=0, t=6) is where h=0 (ground), not where h=5. Don't confuse "start/land" with "pass the ledge".
📏 Geometry

Word Problems — Core Units

Triangles · Circles · Volume · Similarity · Proofs · Coordinate Geometry

11
Pythagorean Theorem Tricky
⚡
Quick Recall
a² + b² = c² → c is always the LONGEST side (hypotenuse)
빗변(hypotenuse)은 직각의 맞은편 — 항상 가장 긴 변

A ladder 13 feet long leans against a wall. The base of the ladder is 5 feet from the wall. How high up the wall does the ladder reach?

💡 Setup
5² + h² = 13²
25 + h² = 169 → h² = 144 → h = 12
📖 Explanation
  • 1. a = 5 (base), c = 13 (ladder = hypotenuse)
  • 2. b² = 13² − 5² = 169 − 25 = 144
  • 3. b = √144 = 12 feet
⚠️ 5-12-13 is a Pythagorean triple — memorize it! Also know: 3-4-5, 8-15-17.
12
Similar Triangles · Scale Factor Tricky
⚡
Quick Recall
Similar △: sides proportional · Area ratio = (scale factor)²
닮음비 k → 넓이비 k², 부피비 k³

Two similar triangles have corresponding sides in a ratio of 3 : 5. The area of the smaller triangle is 27 cm². What is the area of the larger triangle?

💡 Area ratio
Scale factor = 3:5 → Area ratio = 3²:5² = 9:25
27/A = 9/25 → A = 27 × 25/9 = 75
📖 Explanation
  • 1. Side ratio = 3:5, so area ratio = 9:25
  • 2. 27/A = 9/25 → A = 27 × 25/9 = 3 × 25 = 75 cm²
⚠️ Choice A (45): multiplied by 5/3 instead of (5/3)² = 25/9. Areas scale as the square!
13
Circle · Arc Length Tricky
⚡
Quick Recall
Arc length = (θ/360) · 2πr · Sector area = (θ/360) · πr²
호의 길이/넓이: 각도 비율 × 전체 원 공식

A circular sprinkler covers a sector with a radius of 9 meters and a central angle of 120°. What is the area of the sector it waters? (Use π)

💡 Formula
A = (120/360) · π · 9² = (1/3) · 81π = 27π
📖 Explanation
  • 1. Full area = πr² = 81π
  • 2. Fraction = 120/360 = 1/3
  • 3. Sector area = (1/3)(81π) = 27π m²
⚠️ D (81π) is the full circle area. B (18π) = arc length formula applied wrongly. Always use πr².
14
Volume · Cylinder vs. Cone Tricky
⚡
Quick Recall
V_cyl = πr²h · V_cone = (1/3)πr²h · V_sphere = (4/3)πr³
원뿔 = 원기둥의 1/3 — 분수 빠뜨리지 않기!

An ice cream cone has a radius of 3 cm and height of 12 cm. A scoop of ice cream (sphere) of radius 3 cm sits on top. What is the total volume of ice cream (cone + sphere)? (Exact answer with π)

💡 Both formulas
Cone: (1/3)π(3²)(12) = 36π
Sphere: (4/3)π(3³) = 36π
Total = 72π cm³
📖 Explanation
  • 1. Cone = (1/3)π(9)(12) = 36π
  • 2. Sphere = (4/3)π(27) = 36π
  • 3. Total = 36π + 36π = 72π cm³
⚠️ A (108π): forgot the 1/3 in the cone formula. D (144π): forgot 1/3 in cone AND used wrong sphere.
15
Triangle · Exterior Angle Tricky
⚡
Quick Recall
Exterior angle = sum of the TWO non-adjacent interior angles
외각 = 나머지 두 내각의 합 (바로 옆 내각과는 보각)

In a triangle, two interior angles are 48° and 67°. What is the measure of the exterior angle at the third vertex?

💡 Exterior Angle Theorem
Exterior ∠ = 48° + 67° = 115°
📖 Explanation
  • 1. Third interior angle = 180 − 48 − 67 = 65°
  • 2. Exterior angle = 180 − 65° = 115°
  • 3. OR: shortcut — exterior = 48 + 67 = 115°
⚠️ A (65°) is the interior angle at that vertex, not the exterior. They're supplementary (add to 180°).
16
Coordinate Geometry · Distance Tricky
⚡
Quick Recall
d = √[(x₂−x₁)² + (y₂−y₁)²] · Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)
거리공식 = 피타고라스 응용. 각 좌표 차이의 제곱 합에 루트

A city planner places two fire stations at coordinates A(2, 5) and B(−4, 13) on a grid where each unit = 1 km. What is the straight-line distance between the stations?

💡 Apply formula
d = √[(−4−2)² + (13−5)²]
= √[36 + 64] = √100 = 10
📖 Explanation
  • 1. Δx = −4 − 2 = −6 → (−6)² = 36
  • 2. Δy = 13 − 5 = 8 → 8² = 64
  • 3. d = √(36 + 64) = √100 = 10 km
⚠️ B (14) = just added |−6| + |8| = 14 — that's Manhattan distance, not straight-line!
17
Special Right Triangles · 30-60-90 Tricky
⚡
Quick Recall
30-60-90: sides = x, x√3, 2x · 45-45-90: x, x, x√2
30-60-90: 짧은변×√3=긴변, 짧은변×2=빗변

A ramp makes a 30° angle with the ground. The horizontal length of the ramp is 20 feet. How tall is the ramp at its highest point, and how long is the ramp surface itself?

💡 30-60-90 sides
Horizontal (adj to 30°) = x√3 = 20 → x = 20/√3 = 20√3/3
Height (opp 30°) = x = 20√3/3
Hypotenuse = 2x = 40√3/3
📖 Explanation
  • 1. The 30° angle is at the base. Horizontal = adjacent to 30° = x√3
  • 2. x√3 = 20 → x = 20/√3 = 20√3/3
  • 3. Height (opposite 30°) = x = 20√3/3 ≈ 11.55 ft
  • 4. Ramp (hypotenuse) = 2x = 40√3/3 ≈ 23.09 ft
⚠️ D assumes the SHORT leg = 20, but here the LONG leg (horizontal) = 20. Know which side is given!
18
Circle · Inscribed Angle Tricky
⚡
Quick Recall
Inscribed angle = (1/2) × intercepted arc · Central angle = arc
원주각 = 호의 절반 / 중심각 = 호 / 반원의 원주각 = 90°

In a circle, an inscribed angle intercepts an arc of 140°. What is the measure of the inscribed angle? Also: if this angle were a central angle intercepting the same arc, what would it measure?

💡 Key relationship
Inscribed angle = arc / 2 = 140° / 2 = 70°
Central angle = arc = 140°
📖 Explanation
  • 1. Inscribed Angle Theorem: angle = (1/2) × arc = 70°
  • 2. Central angle = intercepted arc directly = 140°
⚠️ A flips them — the inscribed angle is HALF the arc, not double! The central angle equals the arc.
19
Surface Area · Composite Solid Tricky
⚡
Quick Recall
SA_cylinder = 2πr² + 2πrh · Lateral only = 2πrh
겉넓이: 두 원면 + 옆면 / 조합 도형은 겹치는 면 빼기!

A cylindrical can has radius 4 cm and height 10 cm. What is the total surface area of the can (including top and bottom)? (Exact in terms of π)

💡 Surface area breakdown
2 circles: 2π(4²) = 32π
Lateral: 2π(4)(10) = 80π
Total: 32π + 80π = 112π
📖 Explanation
  • 1. Two circular bases: 2 × π(4²) = 32π
  • 2. Lateral surface: 2π(4)(10) = 80π
  • 3. Total = 32π + 80π = 112π cm²
⚠️ A (80π) = lateral only. B (96π) = used radius instead of diameter somewhere. Always include BOTH circles!
20
Parallel Lines · Transversal Tricky
⚡
Quick Recall
Parallel lines cut by transversal: Alt. Int. = equal · Co-int. = 180°
엇각(alternate interior) = 같음 / 동측내각(co-interior) = 합이 180°

Two parallel lines are cut by a transversal. One angle formed is (3x + 20)° and its co-interior (same-side interior) angle is (2x + 10)°. Find the value of x and both angle measures.

💡 Co-interior angles are supplementary
(3x + 20) + (2x + 10) = 180
5x + 30 = 180 → 5x = 150 → x = 30
Angles: 110° and 70°
📖 Explanation
  • 1. Co-interior (same-side interior) angles are supplementary: sum = 180°
  • 2. (3x + 20) + (2x + 10) = 180 → 5x + 30 = 180
  • 3. 5x = 150 → x = 30
  • 4. Angles: 3(30)+20 = 110° and 2(30)+10 = 70°
  • 5. Check: 110 + 70 = 180 ✓
⚠️ C has x = 30 correct but wrong angle calculation — always substitute x back!
20 Problems Complete · Algebra 2 + Geometry · Self-Study Edition
Built for independent learners who think deeply.