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1. Mechanical & Automotive Engineering

2. Civil & Structural Engineering

3. Nuclear, Chemical & Energy Engineering

PROMPT 1 — Mechanical & Automotive Engineering

"Altitude Apex: Engineering a Winning Lap"

Scenario

Your team is the race engineering crew for a mid-tier team at the Sierra Alta International Raceway, a fictional road course. Your car currently qualifies mid-pack. Management has given you a development budget and one week to redesign the car's setup before the race.

Track Specifications (Parameter: Value)

Lap length: 3.4 miles (5,470 m)

Turns: 14 total: 5 hairpins, radius 25–30 m (flat) · 6 medium sweepers, radius 65–90 m (flat) · 3 high-speed corners, radius 150–180 m, banked at 8°

Longest straight: 1,100 m (the "back straight"), feeding directly into one of the 8°-banked high-speed corners

Elevation: 7,200 ft (2,195 m) above sea level

Climate: Race-day ambient 88°F (31°C), low humidity. On the back straight, wind blows at a steady 20 mph along a bearing 30° off the car's direction of travel (partly head/tailwind, partly crosswind)

Track surface: Dry asphalt, tire–track friction coefficient μ = 1.05

Baseline Car (Parameter: Value)

Engine: Naturally aspirated V8, rated 450 hp at sea level

Mass (m), with driver: 1,350 kg

Wheelbase: 2.7 m

Drag: Cd·A: 0.90 m²

Downforce reference: 1,200 lb at 150 mph, measured at sea-level test conditions (ρ₀ = 1.225 kg/m³)

Rolling resistance coefficient (Crr): 0.015

Baseline lap time (given): 2:13.400

Target: Break 2:03.000 — at least a 10.4-second improvement

Development budget: $150,000

Given Physics — use these, you don't need to re-derive them

Air density at altitude: ρ = ρ₀·e^(−h/H), where H ≈ 8,500 m (standard atmospheric scale height). Compute this ratio yourself — it's your first real number.
Naturally aspirated engine power scales approximately proportionally with air density: P(altitude) ≈ P(sea level) × [ρ(altitude)/ρ₀]. (Physically: power depends on the mass of air the engine can ingest per cycle.)
Aerodynamic drag and downforce both follow F = ½·ρ·C·v², where C is the relevant coefficient×area product.
Banked-curve maximum speed (standard result, given): v_max = √[r·g·(tanθ + μ)/(1 − μ·tanθ)]

Upgrade & Fuel Menu (any combination, within budget)

Turbocharger kit

+120 hp; +18 kg

Cost: $45,000

Carbon-fiber panels

−40 kg per stage, up to 2 stages

Cost: $30,000/stage

Adjustable aero package

±10% downforce per step, up to ±20%; each 10% of downforce also shifts drag (Cd·A) by about 4% in the same direction

Cost: $12,000/step

Sticky race tire compound

μ: 1.05 → 1.20

Cost: $18,000

Switch to E85 ethanol

~30% lower energy density by volume than gasoline; higher octane permits more aggressive ignition timing; research typical net power and emissions effect

Cost: $8,000

Switch to biodiesel (B20)

Slightly lower power, better low-end torque, lower particulate emissions

Cost: $6,000

Suspension/gearing retune

No direct hp/weight change; justify a modest, cited corner-exit acceleration improvement

Cost: $10,000

Core Requirements

Altitude penalty. Compute the density ratio at 2,195 m, and the car's actual horsepower at altitude before any upgrades.
Back-calculate your downforce coefficient. Using the 1,200 lb @ 150 mph reference (at ρ₀), find the effective Cl·A your car generates. You'll need it for every corner-speed calculation that follows.
Top speed. At speed v, total resisting force is drag plus rolling resistance: F = ½·ρ·(Cd·A)·v² + Crr·m·g. Setting Power = F·v gives a cubic equation in v (a v³ term and a v term, no v²). Solve it — graphically, by iteration/spreadsheet goal-seek, or by any method that gets you a defensible root — for both the baseline car and your upgraded car.
Flat corners (hairpins & sweepers). At the limit, required centripetal force equals available friction: m·v²/r = μ·(m·g + F_down(v)), where F_down(v) = ½·ρ·(Cl·A)·v². Downforce depends on the very speed you're solving for — isolate v² algebraically (it's linear in v² once you collect terms) and solve in closed form. Do this for a representative hairpin and a representative sweeper, baseline and upgraded.
Banked corners. Apply the given banked-curve formula to your 3 high-speed corners, baseline and upgraded.
Traction ellipse check. At the exit of your 3 banked corners — where the car is simultaneously still cornering and accelerating hard onto the next straight — grip is shared between lateral and longitudinal demands: (F_lateral/F_max)² + (F_longitudinal/F_max)² ≤ 1, with F_max = μ(m·g + F_down). Confirm your assumed corner-exit acceleration doesn't violate this constraint.
Crosswind check. Resolve the 20 mph / 30°-off-axis wind on the back straight into headwind and crosswind components. Use the headwind component to adjust air-relative speed in your drag calculation (drag depends on speed through the air, not over the ground). Using a side-profile area of 2.6 m² and a side-force coefficient of 1.1, estimate the lateral force from the crosswind component and compare it to available tire grip. State — with numbers — whether this is significant or safely negligible for your setup.
Spend the budget. Choose upgrades, recompute everything above with your new numbers, and report your new estimated lap time.
Address the trade-off explicitly. More downforce isn't free — it costs drag. Show, with your own numbers, whether more or less downforce than baseline actually helps on this specific track (long straights and high-speed corners both matter — don't assume the answer, calculate it).

PROMPT 2 — Civil & Structural Engineering

"Spanning the Cascade Gorge"

Scenario

A regional transportation authority needs a new crossing over the Cascade Gorge, connecting two communities currently separated by a 40-minute detour. You are the lead engineering consultant: recommend a bridge type, justify it against the site, and produce a design that fits the budget.

Site Specifications (Parameter: Value)

Clear span required (L): 480 m

Gorge depth: 135 m, deck to river

River behavior: Normal flow ~1.2 m/s; spring flood (April–May) raises water level 8 m, flow to 4 m/s

North rim geology: Granite, safe bearing capacity 5 MPa

South rim geology: Sandstone, safe bearing capacity 2 MPa

Seismic zone: Moderate; design peak ground acceleration 0.25g (discuss qualitatively — full seismic analysis is not required)

Wind: Sustained 45 mph; gusts to 65 mph, oriented mostly perpendicular to the span. Dynamic pressure at 65 mph gust = 517 Pa (already computed for you via q = ½ρv²)

Environmental constraint: Protected raptor nesting on the north cliff face — no construction within 50 m of the nest zone, March–July

Required deck: 2 vehicle lanes + 1 pedestrian/bicycle lane

Design live load: 9.3 kN/m per lane (uniform), plus one 320 kN design-truck load positioned at the worst-case point for a simple check

Budget & Material Costs

Total budget: $85,000,000

Structural steel (fabricated, erected): $3,200 / metric ton

Prestressed concrete (cast, placed): $650 / m³

High-strength cable (suspension/stay): $6,800 / metric ton

Stone/masonry: $410 / m³

Foundation excavation, granite: $1,900 / m³

Foundation excavation, sandstone (extra reinforcement needed): $2,700 / m³

Environmental mitigation / permitting: $2,000,000 (fixed)

Engineering, inspection, contingency: 15% of construction subtotal

Starting Formulas by Bridge Type

Pick the section relevant to your choice.

Suspension: main cable approximated as a parabola with sag d below the towers. For a uniform load w per unit length, horizontal tension component H = wL²/(8d) (constant along the cable). Maximum tension, at the towers, combines the horizontal and vertical components: T_max = √(H² + (wL/2)²).

Cable-stayed: each stay, anchored at angle θ from the deck, carries a share of the load. For a stay responsible for tributary load W: T = W / sin(θ).

Arch: for a parabolic arch of span L and rise h under uniform load w, horizontal thrust at the abutments: H = wL²/(8h) — the same form as the suspension cable, except the arch is in compression where the cable is in tension. (A hanging chain and an inverted arch carry load the same way; this is why chain-model studies were historically used to design arches.)

Truss: member forces via method of joints/sections (ΣFx = 0, ΣFy = 0 at each pin). Note: simple/continuous truss spans rarely exceed roughly 300–550 m economically — research whether truss is a realistic candidate at 480 m before investing significant report space in it.

Core Requirements

Choose and justify a bridge type, explicitly addressing whether it's span-feasible at 480 m (research typical economical span ranges for each type).
Load. Estimate total dead load from your own material choices, add the given live load, and combine into a design load per unit length, w.
Size your primary member. Using the formula for your chosen type, find the peak internal force, then size that member's cross-section using a researched, cited allowable stress for your material and a factor of safety ≥ 2.
Foundation design — check both failure modes, at both rims:
- Bearing: footing area must satisfy (vertical load ÷ footing area) ≤ that rim's safe bearing capacity.
- Overturning: wind force (517 Pa × your structure's exposed frontal area) creates an overturning moment about the base of your tower/pier; your structure's weight creates a stabilizing moment. Require stabilizing moment ≥ 1.5 × overturning moment. Report final footing dimensions for both rims, and state which failure mode governed each.
Identify and mitigate at least two of: aeroelastic flutter, scour, seismic resonance, thermal expansion, differential settlement.
Construction sequencing that respects the March–July restriction near the north cliff.
Full budget: materials + a researched, cited labor/equipment multiplier + fixed costs + 15% contingency, within $85,000,000.
Diagram: a labeled elevation view of the gorge and your bridge, roughly to scale.

PROMPT 3 — Nuclear, Chemical & Energy Engineering

"Powering Meridian Falls: A Fusion Feasibility Study"

Scenario

The town of Meridian Falls (population 15,000; average electrical demand 15 MW, peak 22 MW) wants to replace its aging coal plant. A national lab is offering a small, already-proven, mass-produced deuterium-tritium (D-T) fusion power module — this is not a first-of-a-kind research prototype, it's an established commercial design being deployed for the first time in your town. You're producing the engineering and economic feasibility study.

Given Physics & Chemistry

D-T reaction: deuterium + tritium → helium-4 + neutron, releasing 17.6 MeV per reaction. 1 MeV = 1.602 × 10⁻¹³ J.
Lawson-type ignition criterion (simplified, given): at plasma temperature 15 keV, the reactor needs n·τ_E ≥ 2 × 10²⁰ s/m³, where n = plasma density (particles/m³) and τ_E = energy confinement time (s).
Tritium breeding: Li-6 + neutron → He-4 + tritium + 4.8 MeV. Each fusion reaction's neutron breeds one tritium atom (1:1, this simplified design assumption). Natural lithium is ~7.5% Li-6 / ~92.5% Li-7; this reactor's blanket uses lithium enriched to 30% Li-6.
Conversion: thermal → electric at 35% efficiency (steam cycle). 15% of gross electrical output must be recirculated to run magnets, heating, and cryogenics — only the remaining 85% is net power to the town.

Reactor Geometry & Costs (given) (Component: Spec / Cost)

Vacuum vessel + magnet assembly: Outer radius 5.0 m (fixed, given — not something you derive)

Superconducting magnet system: $340,000,000 (fixed)

Vacuum vessel & first wall: $120,000,000 (fixed)

Cryogenic cooling system: $95,000,000 (fixed)

Tritium-breeding blanket: Shell surrounding the vessel, thickness 0.8 m; average density 1,900 kg/m³; lithium is 12% of blanket mass by mass; installed cost $92,000 / m³

Radiation shielding: Surrounds the blanket; each meter of thickness reduces neutron flux by a factor of 10 (given). Site boundary is 30 m from reactor center and must see flux reduced by at least a factor of 4 × 10¹¹ from the unshielded core value. Cost $1,200 / m³

Steam turbine + generator: $180 / kW of gross electrical capacity

Enriched Li-6 feedstock (ongoing fuel, not capital): $3,500 / kg

Site, permitting, staffing (one-time): $28,000,000

Total construction budget: $650,000,000

Core Requirements

Feasibility curve. Solve the Lawson criterion for τ_E at n = 1×10²⁰ m⁻³. Repeat for at least 3 more density values spanning roughly 0.5–3×10²⁰ m⁻³, and graph τ_E vs. n. This is a rational function (a hyperbola) — describe its shape in your own words: is there a density "sweet spot," or does higher density always straightforwardly help? Research confinement times actually achieved in real tokamaks and comment on whether your required τ_E values are believable.
Size your output. Choose and justify a target net electrical output (must exceed 15 MW; discuss how close to the 22 MW peak you get, and whether the gap needs storage/backup). Work backward — through the 15% recirculating tax and 35% conversion efficiency — to your required gross electrical power, gross thermal power, and fusion reaction rate (reactions/second).
Blanket. Using the given inner radius (5.0 m) and thickness (0.8 m), calculate blanket volume and capital cost.
Fuel economics. From your reaction rate, calculate the mass of Li-6 consumed per year (mole/mass conversion; each reaction consumes one Li-6 atom via breeding). Calculate the annual fuel procurement cost at $3,500/kg, and compare it — explicitly, with both numbers stated — to your total capital cost. What does the size of that gap tell you about fusion economics versus a fossil-fuel plant, where fuel is typically the dominant ongoing cost?
Shielding. Determine the minimum thickness meeting the boundary requirement (this is an equation with thickness in an exponent — logarithms solve it directly), then its volume and cost.
Assemble the full budget (magnets + vessel + cooling + blanket + shielding + turbine sized to your gross capacity + fixed costs) and confirm it fits within $650,000,000. If it doesn't, revise your target output and explain the trade-off.
Reality check. Research real published costs for fusion demonstration/pilot projects (e.g., ITER, or Commonwealth Fusion Systems' SPARC and its planned successor ARC) and compare them to your numbers. This scenario explicitly assumes a mature, mass-produced module rather than a first-of-a-kind prototype — explain in your own words why that assumption is doing a lot of work to make this budget achievable, and what would have to become true in the real world for it to hold.
Compare to an alternative. Research approximate installed cost per kW for a solar-plus-battery system serving the same town, and discuss the trade-offs (cost, land use, continuous output vs. intermittency).
Discuss one safety/regulatory issue beyond shielding thickness (e.g., neutron activation, tritium handling, decommissioning).