Master Pre-Calculus
with Hard Problems

A factory's daily profit (in dollars) depends on the number of units produced: $P(u) = 3u^2 - 12u + 5$. The number of units produced depends on the number of workers on shift: $u(w) = 2w + 1$. A manager wants to know the profit when 4 workers are on shift.

A chemist converts Celsius to Fahrenheit using $F(C) = \tfrac{9}{5}C + 32$. A student accidentally records the temperature as a Fahrenheit value of $F = 68$ and needs to find the original Celsius reading.

A drone's altitude (in meters) over time $t$ (seconds) is modeled by $h(t) = \sqrt{t}$. An engineer redesigns the drone so that it starts 4 seconds later and begins 5 meters higher than the original model.

A packaging company models its monthly waste (kg) as $W(x) = 2x^3 - 5x^2 + kx - 3$, where $x$ is weeks into the month. An auditor finds that at the end of week $x = 3$, the waste remainder when divided by $(x-3)$ is exactly 9 kg.

A pharmacist models the concentration of a drug in the bloodstream (mg/L) as: $$C(t) = \frac{6t^2 + 2t}{3t^2 - 12}$$ where $t > 0$ is time in hours. A patient wants to know the long-run concentration as $t \to \infty$, and also asks at what time(s) the formula is undefined.

Sarah invests $\$5{,}000$ in an account at 6% annual interest, compounded quarterly. Her brother invests $\$4{,}500$ at 6.2% continuous interest. After 10 years, who has more money, and by approximately how much?

Use $e \approx 2.71828$.

A bacteria colony triples every hour. Scientists measure the initial population at 200 cells. They need to find the exact time $t$ (in hours) when the colony reaches 50,000 cells.

A sound engineer measures the decibel level using $L = 10\log\!\left(\dfrac{I}{I_0}\right)$, where $I_0 = 10^{-12}$ W/m². Two speakers produce intensities $I_1$ and $I_2 = 8I_1$. The engineer adds the logs of the two ratios: $$\log\!\left(\frac{I_1}{I_0}\right) + \log\!\left(\frac{I_2}{I_0}\right) = \log(I_0^{-2} \cdot I_1 \cdot I_2)$$ Find the combined value in terms of $\log(I_1/I_0)$.

A Ferris wheel of radius 10 m has its center at height 12 m. A passenger starts at the rightmost point. The angle of rotation from the starting position is $\theta = \dfrac{7\pi}{6}$ radians (measured counterclockwise). Find the passenger's exact height above the ground.

A physics student is simplifying a formula for wave interference. They need to prove that the following expression equals $\sin\theta$: $$\frac{\sin^2\!\theta + \sin\theta\cos^2\!\theta}{\sin^2\!\theta + \cos^2\!\theta} - \frac{\sin\theta\cos^2\!\theta}{\sin^2\!\theta + \cos^2\!\theta} + \cos^2\!\theta \cdot \tan\theta$$

Two park rangers at stations $A$ and $B$ are 8 km apart. Ranger A spots a wildfire at an angle of 58° from the line $AB$. Ranger B spots the same fire at an angle of 73° from the line $AB$ (same side). Find the distance from station B to the fire, to the nearest tenth.

Ocean tides at a harbor follow a sinusoidal pattern. The water depth ranges from a minimum of 2 m to a maximum of 10 m. One full tidal cycle takes 12 hours. At $t=0$ hours, the depth is at its maximum. Write the equation of depth $d(t)$ and determine the depth at $t = 5$ hours. Round to the nearest tenth.

Three numbers form an arithmetic sequence. If 2 is added to the second term and 14 is added to the third term, the three numbers form a geometric sequence. The first term of the arithmetic sequence is 1. Find the common difference of the arithmetic sequence.

A bouncy ball is dropped from a height of 20 m. Each bounce reaches 60% of the previous height. Calculate the total vertical distance the ball travels (both up and down) before coming to rest.

A statistician expands $(2x - 3)^7$ to find a specific coefficient for a probability model. They need the 5th term of the binomial expansion. What is the 5th term?

A satellite orbits Earth in an elliptical path with Earth's center at one focus. The closest point to Earth (perigee) is 800 km from Earth's center, and the farthest point (apogee) is 4200 km from Earth's center. Find the distance from the center of the ellipse to each focus.

Two tugboats pull a barge. Tugboat A pulls with force vector $\vec{F_A} = \langle 300, 400 \rangle$ N and Tugboat B pulls with $\vec{F_B} = \langle 100, -200 \rangle$ N. Find the angle between the two force vectors, rounded to the nearest degree.

An architect designs a curved roofline (parabola) and a straight structural beam (line). The roofline follows $y = -x^2 + 6x - 5$ and the beam follows $y = x + 1$. At how many points do they intersect, and what are the coordinates?

A baseball is hit with parametric equations describing its position (in feet): $$x(t) = 80t, \quad y(t) = -16t^2 + 64t + 4$$ where $t$ is in seconds. A fielder is standing at a horizontal distance of 240 feet from home plate. At what height does the ball pass over the fielder? Is the ball still rising or falling at that point?

A sonar system uses polar coordinates. A submarine is detected at polar coordinates $\left(10,\, \dfrac{2\pi}{3}\right)$. A torpedo is launched from the origin toward rectangular coordinates $(-5,\, 5\sqrt{3})$. Are the submarine and the torpedo's target at the same location?