Self-Study Worksheet ยท Interactive Edition

Algebra 2 & Geometry

20 carefully selected word problems โ€” the ones students miss most. Pick an answer, get instant feedback.

Part 01

Algebra 2 โ€” Word Problems

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0 / 10
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Q 01 โš  Most Missed
A ball is thrown upward from a height of 5 feet with an initial velocity of 40 ft/s. Its height in feet after \(t\) seconds is modeled by
\[ h(t) = -16t^2 + 40t + 5 \] How long does it take the ball to reach its maximum height?
๐Ÿ“Œ KEY VERTEX-X = โˆ’b รท 2a โ†’ gives MAX time
\(t_{max} = \dfrac{-b}{2a} = \dfrac{-40}{2(-16)}\)
Q 02 โšก Tricky
A bank account starts with $2,000 and earns 6% annual interest compounded monthly. Which expression gives the value after 3 years?

Watch out: "compounded monthly" changes both the rate and the number of periods!
๐Ÿ“Œ KEY COMPOUND = P(1 + r/n)^(nt) ยท divide rate, multiply time
Q 03 โš  Most Missed
The number of bacteria in a culture triples every 4 hours. If there are 500 bacteria initially, how many are there after 12 hours?

Hint: Think carefully about how many "tripling periods" fit into 12 hours.
๐Ÿ“Œ KEY EXPO-GROWTH = initial ร— rate^(t รท period)
Q 04 โšก Tricky
A rocket is launched and its height (in meters) is modeled by \(h = -5t^2 + 30t\). A sensor detects the rocket only when \(h \geq 40\) meters. For how many seconds is the rocket detected?
๐Ÿ“Œ KEY INEQUALITY-QUAD โ†’ solve =, take BETWEEN roots (upward โˆฉ)
Q 05 โš  Most Missed
A car's value depreciates by 15% each year. If it was purchased for $24,000, write the function \(V(t)\) that gives its value after \(t\) years. What is the value after 5 years? (Round to the nearest dollar.)
๐Ÿ“Œ KEY DECAY = P ร— (1 โˆ’ r)^t ยท subtract, not add!
Q 06 โšก Tricky
The loudness \(L\) (in decibels) of a sound is \(L = 10\log\!\left(\dfrac{I}{I_0}\right)\), where \(I_0 = 10^{-12}\). A sound has intensity \(I = 10^{-4}\) W/mยฒ. What is its loudness?
๐Ÿ“Œ KEY LOG-DIVISION โ†’ log(a/b) = log a โˆ’ log b
Q 07 โš  Most Missed
A rectangular garden has a perimeter of 60 feet. The length is 3 times the width. What is the area of the garden in square feet?
๐Ÿ“Œ KEY SYSTEM-SETUP โ†’ label variables, write 2 equations, substitute
Q 08 โšก Tricky
A population of deer is modeled by \(P(t) = 300 \cdot e^{0.05t}\). After how many years will the population first exceed 600? (Use \(\ln 2 \approx 0.693\))
๐Ÿ“Œ KEY DOUBLE-TIME โ†’ t = ln(2) รท rate (for continuous growth)
Q 09 โš  Most Missed
Two pipes fill a tank together in 6 hours. Pipe A alone takes 10 hours. How many hours does Pipe B alone take?
๐Ÿ“Œ KEY RATE-ADD โ†’ 1/A + 1/B = 1/together (work = rate ร— time)
Q 10 โšก Tricky
A sequence of seats in a theater follows an arithmetic pattern. Row 1 has 15 seats, Row 2 has 18, Row 3 has 21, and so on. The theater has 20 rows total. How many seats are in the entire theater?
๐Ÿ“Œ KEY ARITH-SUM = n/2 ร— (first + last) ยท find last term first!
Part 02

Geometry โ€” Word Problems

Score
0 / 10
Progress
G 01 โš  Most Missed
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?
๐Ÿ“Œ KEY PYTHAGORAS โ†’ aยฒ + bยฒ = cยฒ ยท c is ALWAYS the hypotenuse
G 02 โšก Tricky
A circular pizza has a diameter of 16 inches. It is cut into 8 equal slices. What is the arc length of the crust of ONE slice? (Use \(\pi \approx 3.14\))
๐Ÿ“Œ KEY ARC-LENGTH = rฮธ ยท convert degrees โ†’ radians first!
G 03 โš  Most Missed
A cone-shaped paper cup has a radius of 3 cm and a height of 9 cm. Water is filled to the halfway point (height = 4.5 cm). What fraction of the cup's total volume is filled?

Remember: the water forms a smaller, similar cone.
๐Ÿ“Œ KEY SIMILAR-CONE โ†’ volume ratio = (scale factor)ยณ not ยฝ!
G 04 โšก Tricky
From the top of a 50-meter lighthouse, the angle of depression to a boat is 30ยฐ. How far is the boat from the base of the lighthouse?
๐Ÿ“Œ KEY DEPRESSION = ELEVATION (alternate interior angles) ยท use TAN
\(\tan(30ยฐ) = \dfrac{50}{d} \Rightarrow d = \dfrac{50}{\tan 30ยฐ}\)
G 05 โš  Most Missed
Two triangles are similar. The sides of the smaller triangle are 4, 6, and 8. The longest side of the larger triangle is 20. What is the perimeter of the larger triangle?
๐Ÿ“Œ KEY SIMILAR โ†’ find scale factor first, then multiply ALL sides
G 06 โšก Tricky
A sphere and a cylinder have the same radius \(r = 6\) cm. The cylinder has height \(h = 8\) cm. What is the ratio of the sphere's volume to the cylinder's volume? Simplify your answer.
๐Ÿ“Œ KEY SPHERE = (4/3)ฯ€rยณ ยท CYLINDER = ฯ€rยฒh ยท divide, cancel ฯ€ and rยฒ
G 07 โš  Most Missed
The vertices of a triangle are at \(A(0,0)\), \(B(6,0)\), and \(C(3,4)\). What is the area of the triangle?
๐Ÿ“Œ KEY COORD-AREA = ยฝ ร— base ร— height ยท draw it to spot the base!
G 08 โšก Tricky
A regular hexagon has a side length of 6 cm. What is its area?

Note: A regular hexagon can be divided into 6 equilateral triangles.
๐Ÿ“Œ KEY HEX-AREA = 6 ร— (โˆš3/4)sยฒ ยท equilateral triangle formula ร— 6
G 09 โš  Most Missed
A chord is 8 cm long and is 3 cm from the center of a circle. What is the radius of the circle?
๐Ÿ“Œ KEY CHORD-RADIUS โ†’ perpendicular from center bisects chord โ†’ Pythagoras!
G 10 โšก Tricky
A rectangular prism has a length of 10, width of 6, and height of 8. What is the length of the space diagonal (the line from one corner to the opposite corner)?
๐Ÿ“Œ KEY 3D-DIAGONAL = โˆš(lยฒ + wยฒ + hยฒ) ยท Pythagoras in 3 steps!