Basic radar equation scaling – Effect of peak transmitted power on maximum range If the peak transmitted power P_t in a monostatic radar is increased by a factor of 81 (all other parameters unchanged), by what factor does the maximum detection range R_max increase?

Introduction:
The basic radar range equation shows that received echo power falls with R^4 (two-way propagation and spreading). Consequently, the maximum range R_max scales with the fourth root of peak transmitted power P_t. This question checks your ability to apply the 4th-root dependence quickly.

Given Data / Assumptions:

• Monostatic radar; same antenna gains, target RCS, wavelength, receiver sensitivity, and SNR threshold.
• Only P_t is scaled by 81.
• Free-space propagation and the same processing gain.

Concept / Approach:

From the radar equation, keeping everything except P_t fixed, R_max ∝ (P_t)^(1/4). Therefore, the range factor equals the fourth root of the transmit-power factor.

Step-by-Step Solution:

Verification / Alternative check:

Doubling R requires 16× power (2^4). Here, 81× power gives 3× range, consistent with the 4th-power relationship.

Why Other Options Are Wrong:

9 or 27 imply square-root or cube-root scaling; 81 implies linear scaling; √81 = 9 is a misapplied square-root relation.

Common Pitfalls:

Forgetting the two-way path loss gives R^4; using square-root scaling appropriate to one-way links, not radar.

Final Answer:

3.