Math Mastery

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Part One

Algebra 2 Word Problems

⚡

Quick Memory Point — Algebra 2

QUADRATIC: ax² + bx + c = 0 → use FACTOR first, then FORMULA if stuck
SYSTEMS: SUBSTITUTE or ELIMINATE — pick the easier variable
EXPONENTIAL: SAME BASE → set exponents equal / DIFFERENT BASE → use log
KEY WORDS: "combined" → add | "difference" → subtract | "product" → multiply

Quadratic

★★☆☆☆

A rectangular garden has a length that is 3 meters more than twice its width. If the area of the garden is $54 \text{ m}^2$, what is the width of the garden?

💡 Hint

Let width = w. Then length = 2w + 3. Area = w(2w + 3) = 54. Expand and solve the quadratic.

A w = 4 meters

B w = 4.5 meters ✓

C w = 6 meters

D w = 3 meters

📖 Explanation

Let width = $w$. Then length = $2w + 3$.
Area: $w(2w + 3) = 54$ → $2w^2 + 3w - 54 = 0$
Factor: $(2w - 9)(w + 6) = 0$ → $w = \frac{9}{2} = 4.5$ (reject $w = -6$)
✦ KEY: Always reject negative solutions for length/width problems.

Systems

★★★☆☆

A coffee shop sells lattes for \$4 and cappuccinos for \$5. One morning, they sold 80 drinks total and made \$370. How many lattes were sold?

💡 Hint

Set up two equations: x + y = 80 and 4x + 5y = 370. Use substitution.

A 25 lattes

B 50 lattes ✓

C 30 lattes

D 40 lattes

📖 Explanation

Let $x$ = lattes, $y$ = cappuccinos.
$x + y = 80$ → $y = 80 - x$
$4x + 5(80 - x) = 370$ → $4x + 400 - 5x = 370$ → $-x = -30$ → $x = 50$
✦ KEY: SUBSTITUTION — solve for one variable first, then plug in.

Exponential

★★★☆☆

A bacteria population doubles every 3 hours. If you start with 200 bacteria, after how many hours will there be 1,600 bacteria?

💡 Hint

Model: P = 200 · 2^(t/3). Set equal to 1600. What power of 2 equals 8?

A 6 hours

B 9 hours ✓

C 12 hours

D 15 hours

📖 Explanation

$200 \cdot 2^{t/3} = 1600$ → $2^{t/3} = 8 = 2^3$ → $\frac{t}{3} = 3$ → $t = 9$ hours
✦ KEY: SAME BASE trick — when bases match, just set exponents equal!

Quadratic

★★★★☆

A ball is thrown upward. Its height (in feet) is given by $h(t) = -16t^2 + 64t + 5$. At what time does the ball reach its maximum height? (Tricky: don't solve h = 0!)

💡 Hint

Vertex of a parabola: t = -b/(2a). The maximum occurs at the vertex, NOT when h = 0.

A t = 1 second

B t = 2 seconds ✓

C t = 4 seconds

D t = 3 seconds

📖 Explanation

The vertex gives the maximum. Use $t = \frac{-b}{2a} = \frac{-64}{2(-16)} = \frac{-64}{-32} = 2$
⚠️ TRAP: Many students solve $h = 0$ to find max height — that gives you when the ball lands, not when it peaks!
✦ KEY: VERTEX FORMULA $t = \frac{-b}{2a}$ → always for max/min of quadratic.

Rational

★★★★☆

Pipe A fills a tank in 4 hours, Pipe B fills it in 6 hours. If both pipes are open together, how long does it take to fill the tank?

💡 Hint

Rate A = 1/4 per hour, Rate B = 1/6 per hour. Combined: 1/4 + 1/6 = 1/t. Find t.

A 2 hours

B 2.4 hours ✓

C 3 hours

D 5 hours

📖 Explanation

$\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}$ tanks per hour
Time = $\frac{12}{5} = 2.4$ hours
✦ KEY: WORK RATE — "rate + rate = combined rate", then flip it for time.

Logarithm

★★★★☆

An investment grows according to $A = P \cdot e^{0.06t}$. How long (in years) does it take for an investment to double? Round to the nearest tenth.

💡 Hint

Set A = 2P. Then 2P = P·e^(0.06t). Divide by P, take ln of both sides.

A 10.2 years

B 11.6 years ✓

C 13.0 years

D 8.5 years

📖 Explanation

$2 = e^{0.06t}$ → $\ln 2 = 0.06t$ → $t = \frac{\ln 2}{0.06} = \frac{0.693}{0.06} \approx 11.6$ years
✦ KEY: LN BOTH SIDES — when base is $e$, use $\ln$. When base is 10, use $\log$.

Inequalities

★★★☆☆

A taxi charges \$3 base fee plus \$2 per mile. Another taxi charges \$5 base fee plus \$1.50 per mile. After how many miles is the second taxi cheaper?

💡 Hint

Set up: 3 + 2m > 5 + 1.5m. Solve for m. The inequality flips when you divide by a negative!

A After 2 miles

B After 4 miles ✓

C After 6 miles

D After 3 miles

📖 Explanation

Second cheaper when: $5 + 1.5m < 3 + 2m$ → $5 - 3 < 2m - 1.5m$ → $2 < 0.5m$ → $m > 4$
✦ KEY: INEQUALITY FLIP — only flip the sign when multiplying/dividing by a negative number.

Sequences

★★★☆☆

A theater has 20 seats in the first row, and each row has 3 more seats than the row before. If there are 15 rows, what is the total number of seats?

💡 Hint

Arithmetic sequence: a₁ = 20, d = 3, n = 15. Sum = n/2 · (2a₁ + (n-1)d)

A 450 seats

B 615 seats ✓

C 525 seats

D 700 seats

📖 Explanation

$S_n = \frac{n}{2}(2a_1 + (n-1)d) = \frac{15}{2}(2 \cdot 20 + 14 \cdot 3) = \frac{15}{2}(40 + 42) = \frac{15}{2} \cdot 82 = 615$
✦ KEY: ARITHMETIC SUM = $\frac{n}{2}(\text{first} + \text{last})$ — add first and last, multiply by half the count.

Radical

★★★★☆

The speed $v$ (in m/s) of a roller coaster at the bottom of a hill is $v = \sqrt{2gh}$, where $g = 10$ and $h$ is height in meters. If the speed is 20 m/s, what is the height?

💡 Hint

Plug in v=20 and g=10. Then square both sides to eliminate the square root.

A 10 meters

B 20 meters ✓

C 40 meters

D 5 meters

📖 Explanation

$20 = \sqrt{2 \cdot 10 \cdot h} = \sqrt{20h}$ → Square both: $400 = 20h$ → $h = 20$ meters
✦ KEY: ISOLATE ROOT first, then SQUARE BOTH SIDES. Always check your answer back in the original!

A10

Polynomial

★★★★★

A box is made from a 12×12 cm square sheet by cutting equal squares of side $x$ from each corner. The volume is 100 cm³. Which value of $x$ (approximately) satisfies this? (Tricky — don't forget the domain!)

💡 Hint

V = x(12 - 2x)². Domain: 0 < x < 6. Expand and check each option.

A x ≈ 1.1 cm ✓

B x ≈ 7 cm

C x ≈ 5.9 cm

D x ≈ 3 cm

📖 Explanation

$V = x(12-2x)^2 = 100$
Test $x = 1.1$: $1.1 \cdot (9.8)^2 = 1.1 \cdot 96.04 \approx 105.6$ ✓ (closest)
⚠️ TRAP: Options B and C are outside the domain (x must be less than 6)!
✦ KEY: DOMAIN FIRST — always set up valid range before solving.

△

Part Two

Geometry Word Problems

📐

Quick Memory Point — Geometry

TRIANGLES: SOH-CAH-TOA (sin=opp/hyp, cos=adj/hyp, tan=opp/adj)
SIMILAR: CORRESPONDING sides are PROPORTIONAL — set up ratios carefully
CIRCLES: Inscribed angle = HALF the arc | Central angle = arc
3D: LATERAL + BASE → Surface Area | $\frac{1}{3} \cdot base \cdot height$ → Pyramid/Cone

Similar △

★★☆☆☆

A 6-foot tall person casts a 4-foot shadow. At the same time, a nearby tree casts a 30-foot shadow. How tall is the tree?

💡 Hint

Similar triangles: person height / person shadow = tree height / tree shadow. Set up the proportion.

A 40 feet

B 45 feet ✓

C 50 feet

D 20 feet

📖 Explanation

$\frac{6}{4} = \frac{h}{30}$ → $h = \frac{6 \times 30}{4} = \frac{180}{4} = 45$ feet
✦ KEY: PROPORTION — always match same type on same side: height/shadow = height/shadow.

Pythagorean

★★☆☆☆

A ladder is leaning against a wall. The base of the ladder is 5 feet from the wall, and the ladder reaches 12 feet up the wall. How long is the ladder?

💡 Hint

This is a right triangle. Use a² + b² = c². Which are legs and which is the hypotenuse?

A 11 feet

B 13 feet ✓

C 15 feet

D 17 feet

📖 Explanation

$c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$ feet
✦ KEY: PYTHAGOREAN TRIPLES to memorize: (3,4,5), (5,12,13), (8,15,17), (7,24,25)

Circles

★★★☆☆

A circular sprinkler waters a circular lawn of radius 7 meters. What is the area of lawn watered? (Use $\pi \approx 3.14$)

TRICKY: The sprinkler is at the center. A separate fence runs through the diameter. What is the area of the semicircle?

💡 Hint

Semicircle area = (1/2)πr². Read carefully — the question asks for the SEMICIRCLE, not the full circle.

A 153.86 m²

B 76.93 m² ✓

C 43.96 m²

D 49 m²

📖 Explanation

Full circle: $\pi r^2 = 3.14 \times 49 = 153.86$ m²
Semicircle: $\frac{153.86}{2} = 76.93$ m²
⚠️ TRAP: Option A is the full circle. Always re-read what shape is being asked for!

Trigonometry

★★★★☆

From the top of a 50-meter cliff, the angle of depression to a boat is 30°. How far is the boat from the base of the cliff?

💡 Hint

Angle of depression = angle of elevation from the boat. Draw a diagram! tan(30°) = opposite/adjacent = 50/d.

A 25 meters

B 86.6 meters ✓

C 50 meters

D 100 meters

📖 Explanation

$\tan 30° = \frac{50}{d}$ → $d = \frac{50}{\tan 30°} = \frac{50}{0.577} \approx 86.6$ m
✦ KEY: DEPRESSION = ELEVATION (alternate interior angles). Always draw the diagram with the horizontal line!

3D Solids

★★★☆☆

A cylindrical water tank has a radius of 3 meters and a height of 10 meters. How many liters of water can it hold? (1 m³ = 1000 L, use $\pi \approx 3.14$)

💡 Hint

Volume of cylinder = πr²h. Calculate in m³ first, then convert to liters.

A 94,200 L

B 282,600 L ✓

C 188,400 L

D 565,200 L

📖 Explanation

$V = \pi r^2 h = 3.14 \times 9 \times 10 = 282.6 \text{ m}^3 = 282{,}600 \text{ L}$
✦ KEY: CYLINDER = $\pi r^2 h$. Don't confuse radius and diameter! r = 3 (not 6).

Coordinate

★★★☆☆

Point A is at $(1, 2)$ and point B is at $(7, 10)$. What is the midpoint of segment AB?

TRICKY: The answer looks like it could be the distance formula — don't mix them up!

💡 Hint

Midpoint = average of x-values, average of y-values. NOT the distance formula.

A (3, 4)

B (4, 6) ✓

C (6, 8)

D (5, 7)

📖 Explanation

Midpoint $= \left(\frac{1+7}{2}, \frac{2+10}{2}\right) = (4, 6)$
⚠️ TRAP: Midpoint = AVERAGE (add and divide by 2). Distance = $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Don't confuse them!

Angles

★★★★☆

In a circle, an inscribed angle intercepts an arc of 140°. What is the measure of the inscribed angle?

COMMON MISTAKE: Students confuse inscribed angles with central angles.

💡 Hint

Inscribed Angle Theorem: inscribed angle = HALF the intercepted arc. Central angle = arc (they are equal).

A 140°

B 70° ✓

C 280°

D 35°

📖 Explanation

Inscribed angle $= \frac{1}{2} \times \text{arc} = \frac{140°}{2} = 70°$
⚠️ TRAP: Option A (140°) is what you get if you forget to halve it — that's the central angle!
✦ KEY: INSCRIBED = HALF. Central = Same.

Surface Area

★★★★☆

A square pyramid has a base side of 6 cm and a slant height of 5 cm. What is the total surface area?

💡 Hint

SA = base area + lateral area. Base = s². Lateral = 4 triangles, each = (1/2) × base × slant height.

A 60 cm²

B 96 cm² ✓

C 120 cm²

D 30 cm²

📖 Explanation

Base = $6^2 = 36$ cm²
4 triangles = $4 \times \frac{1}{2} \times 6 \times 5 = 60$ cm²
Total = $36 + 60 = 96$ cm²
✦ KEY: PYRAMID SA = BASE + LATERAL. Use slant height for triangles (not the actual height of the pyramid!)

Transformations

★★★☆☆

Triangle ABC has vertices $A(2, 1)$, $B(4, 1)$, $C(3, 4)$. After a reflection over the y-axis, what are the new coordinates of point A?

💡 Hint

Reflection over y-axis: negate the x-coordinate, keep y the same. (x,y) → (-x, y)

A (2, -1)

B (-2, 1) ✓

C (-2, -1)

D (1, 2)

📖 Explanation

Reflection rules: Over y-axis → $(x, y)$ becomes $(-x, y)$. So $(2,1)$ → $(-2,1)$
✦ KEY: REFLECTION CHEAT SHEET:
y-axis: flip x → $(-x, y)$ | x-axis: flip y → $(x, -y)$ | origin: flip both → $(-x, -y)$

G10

Proof Logic

★★★★★

Two triangles share the same base. Triangle 1 has base 8 cm and height 6 cm. Triangle 2 has base 8 cm and height 9 cm. What is the ratio of Area 1 to Area 2?

TRICKY: You don't need to calculate the actual areas!

💡 Hint

Both triangles have the same base, so the ratio of areas = ratio of heights. Think about WHY!

A 1 : 1

B 2 : 3 ✓

C 3 : 4

D 4 : 9

📖 Explanation

$A_1 = \frac{1}{2}(8)(6) = 24$, $A_2 = \frac{1}{2}(8)(9) = 36$
Ratio = $24 : 36 = 2 : 3$
✦ SHORTCUT: Same base → ratio of areas = ratio of heights = $6:9 = 2:3$. No full calculation needed!

🎉 Quiz Complete!