Rational and irrational numbers Transitional Reasoning Download PDF

Questions

1. 1.

   An acoustic engineer is designing a sensor array that requires the frequency ratio between two resonant chambers to be a rational number so that interference patterns remain strictly periodic over time. The engineer calculates four candidate ratios from the chamber dimensions. Which frequency ratio confirms that the resulting interference pattern will be strictly periodic?

   • A) $\sqrt{1.\overline{7}}$
   • B) $\sqrt{7.2}$
   • C) $\sqrt{0.45}$
   • D) $\sqrt{18} - \sqrt{2}$
2. 2.

   A telecommunications engineer is testing four test signal intervals to identify one that produces an irrational duration in microseconds, ensuring that the signal will not synchronize with periodic clock cycles. The engineer evaluates four candidate configurations based on their calculated durations. Which configuration results in an irrational duration?

   • A) Configuration A: $\sqrt{\frac{49}{81}} + 0.\overline{4}$ microseconds
   • B) Configuration B: $\sqrt{18} - \sqrt{2}$ microseconds
   • C) Configuration C: $\sqrt{0.04} \times \sqrt{225}$ microseconds
   • D) Configuration D: $\frac{\sqrt{75}}{\sqrt{3}} - 1.25$ microseconds
3. 3.

   A chemist analyzes four experimental solutions to find which one can be prepared in exact integer volume proportions. A solution qualifies only if its concentration factor evaluates to a rational number. Based on the calculated concentration factors, which solution meets the chemist's criterion?

   • A) Solution 1 with factor $\sqrt{0.4}$
   • B) Solution 2 with factor $\sqrt{50} - \sqrt{18}$
   • C) Solution 3 with factor $\frac{\sqrt{24}}{\sqrt{2}}$
   • D) Solution 4 with factor $\sqrt{1.\overline{7}}$
4. 4.

   An engineering team is calibrating the gear ratio $k$ of an automated vehicle, requiring $k$ to be a rational number so that the control system can track position without accumulating irrational rounding errors. The team evaluates four proposed expressions for the gear ratio based on different component configurations. Which of the following expressions results in a rational number?

   • A) $k = \sqrt{0.16} + \sqrt{\frac{25}{9}}$
   • B) $k = \sqrt{0.4} + \sqrt{\frac{16}{9}}$
   • C) $k = \sqrt{18} - \sqrt{2}$
   • D) $k = \pi \cdot \frac{7}{22}$
5. 5.

   A landscape architect plans to construct a stone border along one of four decorative garden features, choosing the feature whose boundary length in meters is a rational number so that it can be built from an exact number of identical modular pavers. The architect evaluates four different proposals based on their planned dimensions. Which proposal results in a border length that is a rational number?

   • A) One side of a square patio with an area of 20 square meters
   • B) The perimeter of a square flower bed with an area of 5.76 square meters
   • C) The diagonal of a rectangular courtyard with side lengths of $\sqrt{5}$ meters and $\sqrt{7}$ meters
   • D) The circumference of a circular fountain with a diameter of 3.5 meters