Modeling with functions Knowledge-Based Practice

Questions

1. 1.

   A wire of total length 60 cm is cut into two pieces, where one piece of length $x$ cm is bent to form a square and the remaining piece is bent to form a circle. Which function models the total combined area $A(x)$, in square centimeters, enclosed by the two geometric figures?

   • A) $A(x) = \frac{x^2}{4} + \frac{(60 - x)^2}{2\pi}$
   • B) $A(x) = \frac{x^2}{16} + \frac{(60 - x)^2}{4\pi}$
   • C) $A(x) = \frac{x^2}{16} + \frac{(60 - x)^2}{2\pi}$
   • D) $A(x) = \frac{x^2}{8} + \frac{(60 - x)^2}{4\pi^2}$
2. 2.

   A right circular cylinder of radius $r$ and height $h$ is inscribed in a right circular cone of height 12 cm and base radius 4 cm. Which function expresses the volume $V(r)$ of the cylinder, in cubic centimeters, in terms of $r$?

   • A) $V(r) = 4\pi r^2 - 12\pi r^3$
   • B) $V(r) = 12\pi r - 3\pi r^2$
   • C) $V(r) = 3\pi r^2 - \frac{3}{4}\pi r^3$
   • D) $V(r) = 12\pi r^2 - 3\pi r^3$
3. 3.

   An open-top rectangular container has a square base of side length $x$ meters and a fixed volume of 50 cubic meters. If base material costs 6 dollars per square meter and side material costs 4 dollars per square meter, which function models the total material cost $C(x)$ in dollars?

   • A) $C(x) = 6x^2 + \frac{50}{x}$
   • B) $C(x) = 6x^2 + \frac{200}{x}$
   • C) $C(x) = 24x^2 + \frac{800}{x}$
   • D) $C(x) = 6x^2 + \frac{800}{x}$
4. 4.

   An open-top box is made from a square sheet of cardboard with a side length of 30 cm by cutting identical squares of side length $x$ cm from each of the four corners and folding up the flaps. Which function models the volume $V(x)$ of the box in cubic centimeters?

   • A) $V(x) = x(30 - x)^2$
   • B) $V(x) = (30 - 2x)^2$
   • C) $V(x) = x^2(30 - 2x)$
   • D) $V(x) = x(30 - 2x)^2$
5. 5.

   A rectangular garden is enclosed along a straight brick wall, requiring fencing only along the other three sides. If 120 meters of fencing are used and $x$ represents the length of each of the two sides perpendicular to the wall, which function models the area $A(x)$ of the garden in square meters?

   • A) $A(x) = 120x - 2x^2$
   • B) $A(x) = 60x - x^2$
   • C) $A(x) = 120x - x^2$
   • D) $A(x) = 60x - 2x^2$

Answers and step-by-step solutions appear after you finish the test.