The Quest for 700: Weekly GMAT Challenge (Answer)

First, set up an equation for the area of the rectangle. If x is the width, then we have

x(x + 4) = 221

Note that if we put this into standard quadratic form and then try to factor, we wind up back where we started, in some sense: we are looking for two numbers that multiply to 221 and that differ by 4.

x2 + 4x – 221 = 0

Beyond pure trial and error, we can look for nearby squares. 221 is nearly 225, which equals 152. So we might try numbers near 15. As it turns out, 221 = 13 × 17. We might even get there by noticing a difference of squares: 221 = 225 – 4 = 152 – 22 = (15 – 2)(15 + 2) = 13 × 17. As a last resort, we could always use the quadratic formula, which gets us the roots of the equation as well.

Now, the diagonal of the rectangle will be given by the Pythagorean Theorem:

d2 = 132 + 172

= 169 + 289

= 458.

The square root of 458 is definitely larger than 20, since 202 = 400. Going up, we can compute 212 = 441 < 458, whereas 222 = 484 > 458. So the length of the diagonal must be between 21 and 22.