2011-2012 Mathcounts workout 8

#231: 0.4% = 0.004
100 x 1.004 = 10100.4
(100.4 + 100) x 1.004=201.20
(201.20 + 100 ) x 1.004 = 302.40
... The answer is $1231.66  (I don't like this question.)

 Examples of Applications of Exponential Functions

#232: Using distance formula you'll get the length of the unit to be √  (-4-4)² + (-5-2)²
= √ 113 , which is between 10 and 11. The sum is 21.

#233: 8/81 = 0.098765432 
2012/9 = 223 R5, The answer is 6.

#234: 19.5 x (1.2 + 0.06)= 24.57
100%(the bill) + 20% tip = 1205 = 1.2
6% tax = 0.06

#235: There are total 3 x 10 (from 1 to 10) + 2 x 7 (from 11 to 17)+ 1 x 8
(from 18 to 25) = 52 cards.
Among all, 30 cards are 10 0r less than 10.
(30/52)(29/51)(28/50)(27/49)(26/48) about 0.055

#236: 5 √ x - 30 = 2√ x + 54, 3√ x = 84  Divided both sides by 3 and you have √ x   = 28
 Square both sides and you have x = 28 x 28 = 784

#237: The dimension is 1: 2 : 3 ratio and the shortest leg is 2 ft, which means the other two legs are
4 and 6.  The longest distance between and two vertices are the space diagonal. 
√ 2² + 4² + 6² = √56 = 7.48 ft

# 238:The answer is 30.25 cm²

#239:SAT has similar question.
(x² + ax + 6) /(x +1) = x + b (given)
x² + ax + 6 = (x + b) (x + 1) = x² + x + bx + b = x² + x (b+1) + b
b = 6 and since b +1 = a, a = 7. 6a - 7b = 0

#240: 

Using 45-45-90 degree angle ratio, you get two legs of the right triangle DBC each as 3 √ 2.
 CE + CB = 3√ 2  and their ratio is 2 to 1.
CE = 2√ 2  and BE =√ 2.

Using Pythagorean theorem, you have
6² = √ 2² + (3 √ 2 + x)²
34 = (3 √ 2 + x)²
√ 34 = 3 √ 2 + x
x = 1.58831... about 1.59