Checking units, constraints, and reasonableness — Topic practice

12,000 square centimeters

An area conversion squares the length conversion: 1.2 m²(100 cm/1 m)² = 1.2(10,000) = 12,000 cm². Multiplying by only 100 would convert one dimension but not the other. See Example 5.02.

4

The right side of $\sqrt{\text{x + 12}}$ = x must be nonnegative. Squaring gives x² − x − 12 = 0, so x = 4 or x = −3. In the original equation, 4 works because $\sqrt{\text{16}}$ = 4; −3 fails because $\sqrt{\text{9}}$ = 3 ≠ −3. Therefore the only real solution is 4. See Example 5.06.

7 buses

The quotient 221/36 lies between 6 and 7. Six buses provide only 6(36) = 216 seats, fewer than needed; seven provide 252. A minimum number of whole containers must be rounded up when any demand remains. Ordinary rounding to the nearest integer would be the wrong rule here. See Example 5.10.

26%

Combine the counts: (12 + 14)/(30 + 70) = 26/100 = 26%. The two group rates are 40% and 20%, but their groups are different sizes. Their unweighted average, 30%, therefore does not describe the combined group. See Example 5.12.

96 square units

The area is w(w + 4). At integer width w = 8, it is 8(12) = 96, which meets the bound. At w = 9, it is 117, too large. Because both positive dimensions increase as w increases, no larger integer width can work. Thus 96 is the greatest area. See Example 5.15.