Test questions on applied polynomial functions pre-calculus

📚 Quick Study Guide

• 📈 Polynomial functions are functions that can be written in the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$, where $n$ is a non-negative integer and the coefficients $a_i$ are constants.
• 👷‍♀️ Applied polynomial functions often model real-world scenarios such as projectile motion, optimization problems, and growth/decay models.
• 📐 Key concepts include finding zeros (roots), determining end behavior, identifying local maxima and minima, and understanding the relationship between the equation and the graph.
• 💡 When solving applied problems, carefully define your variables, set up the polynomial equation that models the situation, and use algebraic techniques or graphing calculators to find the solution.
• 💸 Optimization problems often involve finding the maximum or minimum value of a polynomial function within a given domain. This can be solved using calculus (derivatives) or by analyzing the graph of the function.

🧪 Practice Quiz

1. A projectile is launched vertically upward from a height of 5 feet with an initial velocity of 100 ft/s. The height $h(t)$ of the projectile after $t$ seconds is given by $h(t) = -16t^2 + 100t + 5$. What is the maximum height reached by the projectile?
   1. 156.25 feet
   2. 161.25 feet
   3. 100 feet
   4. 5 feet
2. A rectangular garden is to be fenced off using 200 feet of fencing. If one side of the garden is $x$ feet, express the area $A$ of the garden as a function of $x$.
   1. $A(x) = x(200 - x)$
   2. $A(x) = x(100 - x)$
   3. $A(x) = 2x(100 - x)$
   4. $A(x) = x^2 - 100x$
3. The profit $P(x)$ (in dollars) from selling $x$ units of a product is given by $P(x) = -0.1x^2 + 50x - 1000$. How many units must be sold to maximize profit?
   1. 100
   2. 250
   3. 500
   4. 1000
4. A box with an open top is to be constructed from a square piece of cardboard, 12 inches on a side, by cutting equal squares from the corners and turning up the sides. If the side length of the cut-out square is $x$ inches, what is the volume $V(x)$ of the box?
   1. $V(x) = x(12 - 2x)^2$
   2. $V(x) = x^2(12 - x)$
   3. $V(x) = x(12 - x)^2$
   4. $V(x) = (12 - 2x)^2$
5. The population of a town is modeled by the function $P(t) = 0.001t^3 - 0.5t^2 + 100t + 5000$, where $t$ is the number of years since 2000. What was the population in 2000?
   1. 0
   2. 5000
   3. 5099
   4. 5000.001
6. The cost $C(x)$ of producing $x$ items is given by $C(x) = 0.01x^3 - 0.6x^2 + 13x$. What is the cost of producing 10 items?
   1. $71
   2. $70
   3. $64
   4. $60
7. A farmer wants to enclose a rectangular field with a fence. He has 500 feet of fencing. One side of the field is along a river and does not need fencing. What is the maximum area that can be enclosed?
   1. 62500 sq ft
   2. 31250 sq ft
   3. 15625 sq ft
   4. 50000 sq ft