Modeling with functions Knowledge-Based Download PDF

Questions

1. 1.

   A rectangle is inscribed in a semicircle of radius 6 such that its base lies along the diameter. If $x$ denotes half the length of the base along the diameter, which function $A(x)$ models the area of the rectangle?

   • A) $A(x) = x \sqrt{36 - x^2}$
   • B) $A(x) = 2x \sqrt{36 - x^2}$
   • C) $A(x) = 2x(36 - x^2)$
   • D) $A(x) = x^2 \sqrt{36 - x}$
2. 2.

   A closed right circular cylindrical tank has a volume of $500\pi$ cubic meters. If the radius of the circular base is $r$ meters, which function $S(r)$ models the total surface area of the tank in square meters?

   • A) $S(r) = \pi r^2 + \frac{500\pi}{r}$
   • B) $S(r) = 2\pi r^2 + \frac{500\pi}{r}$
   • C) $S(r) = 2\pi r^2 + \frac{1000\pi}{r}$
   • D) $S(r) = 2\pi r^2 + \frac{1000\pi}{r^2}$
3. 3.

   A manufacturer determines that the cost to produce $x$ items is $C(x) = 50x + 800$ dollars, and the selling price per item is $p(x) = 250 - 2x$ dollars. Which function $P(x)$ represents the manufacturer's total profit from producing and selling $x$ items?

   • A) $P(x) = -2x^2 + 300x - 800$
   • B) $P(x) = -2x + 200$
   • C) $P(x) = -2x^2 + 200x + 800$
   • D) $P(x) = -2x^2 + 200x - 800$
4. 4.

   An open-top box is constructed from a rectangular sheet of metal measuring 16 inches by 24 inches by cutting congruent squares of side length $x$ inches from each corner and folding up the sides. Which function $V(x)$ models the volume of the box in cubic inches?

   • A) $V(x) = 4x^3 - 80x^2 + 192x$
   • B) $V(x) = 4x^3 - 80x^2 + 384x$
   • C) $V(x) = 2x^3 - 40x^2 + 384x$
   • D) $V(x) = x^3 - 40x^2 + 384x$
5. 5.

   A rectangular plot of land is enclosed on three sides using 60 meters of fencing, while the fourth side is bounded by a straight brick wall. If each of the two fenced sides perpendicular to the wall has length $x$ meters, which function $A(x)$ models the area of the plot in square meters?

   • A) $A(x) = 60x - x^2$
   • B) $A(x) = 30x - x^2$
   • C) $A(x) = 60x - 4x^2$
   • D) $A(x) = 60x - 2x^2$