Word Problems
Apply function concepts to real-world scenarios involving physics, economics, biology, and everyday situations.
📖 Learn
Word problems require translating real-world situations into mathematical functions. This lesson helps you develop a systematic approach to solving applied problems.
Problem-Solving Strategy
- Read the problem carefully (at least twice)
- Identify the unknown quantities and assign variables
- Determine the type of function that models the situation
- Write the equation using given information
- Solve and check your answer in context
Common Function Types in Applications
| Function Type | Real-World Application |
|---|---|
| Linear | Constant rate of change (distance, cost, temperature conversion) |
| Quadratic | Projectile motion, area optimization, revenue/profit |
| Exponential | Population growth, compound interest, radioactive decay |
| Square Root | Distance formulas, period of pendulum |
💡 Worked Examples
Example 1: Projectile Motion
Problem: A ball is thrown upward from ground level with an initial velocity of 64 ft/s. The height h(t) in feet after t seconds is given by h(t) = -16t^2 + 64t. Find the maximum height and when the ball hits the ground.
Solution:
- This is a quadratic function (parabola opening downward)
- Maximum occurs at vertex: t = -b/(2a) = -64/(2(-16)) = 2 seconds
- Maximum height: h(2) = -16(4) + 64(2) = -64 + 128 = 64 feet
- Ball hits ground when h(t) = 0: -16t^2 + 64t = 0, so t(−16t + 64) = 0
- t = 0 (initial) or t = 4 seconds (landing)
Answer: Maximum height is 64 feet at t = 2 seconds; ball lands at t = 4 seconds.
Example 2: Exponential Growth
Problem: A population of bacteria doubles every 3 hours. If the initial population is 500, write a function for the population P(t) after t hours, and find the population after 9 hours.
Solution:
- Exponential growth model: P(t) = P_0 * 2^(t/d) where d is doubling time
- P_0 = 500, d = 3
- P(t) = 500 * 2^(t/3)
- P(9) = 500 * 2^(9/3) = 500 * 2^3 = 500 * 8 = 4000
Answer: P(t) = 500 * 2^(t/3); after 9 hours, population is 4,000 bacteria.
✏️ Practice Problems
Solve each word problem by identifying the function type and writing an equation.
Problem 1: Falling Object
An object is dropped from a height of 400 feet. Its height h(t) after t seconds is h(t) = 400 - 16t^2. How long until it hits the ground?
Show Hint
Set h(t) = 0 and solve for t. Only consider positive values.
Problem 2: Revenue Function
A company sells x units of a product at price p = 100 - 0.5x dollars per unit. Write the revenue function R(x) and find the number of units that maximizes revenue.
Show Hint
Revenue = price times quantity. R(x) = x * p(x). Find the vertex of the resulting parabola.
Problem 3: Compound Interest
$5,000 is invested at 6% annual interest compounded monthly. Write a function A(t) for the amount after t years, and find the balance after 5 years.
Show Hint
Use A(t) = P(1 + r/n)^(nt) where n = 12 for monthly compounding.
Problem 4: Temperature Conversion
The function F(C) = (9/5)C + 32 converts Celsius to Fahrenheit. Find the inverse function and use it to convert 98.6 F to Celsius.
Show Hint
Swap F and C, then solve for C to find the inverse.
Problem 5: Fencing Problem
A farmer has 200 feet of fencing to enclose a rectangular area along a river (no fence needed on the river side). Express the area A as a function of width w, and find dimensions that maximize the area.
Show Hint
If w is width, express length in terms of w using the constraint 2w + L = 200.
Problem 6: Radioactive Decay
A radioactive substance decays according to A(t) = 100e^(-0.05t), where t is in years. How long until only 25 grams remain?
Show Hint
Set A(t) = 25 and solve for t using natural logarithms.
Problem 7: Pendulum Period
The period T of a pendulum (in seconds) is given by T(L) = 2*pi*sqrt(L/32), where L is the length in feet. Find the period for a 2-foot pendulum and the length needed for a 1-second period.
Show Hint
For the second part, set T = 1 and solve for L.
Problem 8: Break-Even Analysis
A company's cost function is C(x) = 500 + 15x and revenue function is R(x) = 35x, where x is units produced. Find the break-even point.
Show Hint
Break-even occurs when R(x) = C(x). Solve for x.
Problem 9: Population Model
A city's population is modeled by P(t) = 50000(1.03)^t, where t is years since 2020. What will the population be in 2030? When will it reach 75,000?
Show Hint
For 2030, t = 10. For the second part, use logarithms to solve for t.
Problem 10: Profit Optimization
A theater charges $12 per ticket and sells 400 tickets per show. For each $0.50 increase in price, 10 fewer tickets are sold. What ticket price maximizes revenue?
Show Hint
Let x = number of $0.50 increases. Price = 12 + 0.5x, Quantity = 400 - 10x. Write and maximize R(x).
✅ Check Your Understanding
Can you confidently answer these questions?
- How do you identify whether a problem requires a linear, quadratic, or exponential model?
- What does the vertex of a parabola represent in optimization problems?
- How do you set up an equation from a word problem?
- When do you use the compound interest formula vs. continuous growth?
- How do you verify that your answer makes sense in context?
🚀 Next Steps
- Practice translating word problems into equations
- Review any function types you found challenging
- Move on to Common Mistakes to avoid typical errors
- Try creating your own word problems to deepen understanding