Mathematics 30 June 1999 Grade 12 Diploma Exam

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Use the following information to answer the next question. A sports store is planning a sales promotion of mountain bikes. Each day that a bike remains unsold, the price will be discounted by 2% of the previous day’s price. The table below gives the successive daily prices of a mountain bike with an original price of $1 224.49. Day Bike Price 1 $1 200.00 2 $1 176.00 3 $1 152.48 4 $1 <strong>12</strong>9.43 5 $1 106.84 Written Response—10% 1. Successive daily prices of the mountain bike can be approximated by using the recursive formula t 1 = $1 200.00 and t n + 1 = 0.98t n . • Determine the price of the mountain bike on day 8 of this sales promotion. A POSSIBLE SOLUTION t 5 + 1 = 0.98(1 106.84) = 1 084.70 t 6 + 1 = 0.98(1 084.70) = 1 063.00 t 7 + 1 = 0.98(1 063.00) = 1 041.75 On the 8 th of the sales promotion the cost of the bike is $1 041.75. • Write an expression for t n as a function of n, n ∈ N. A POSSIBLE SOLUTION t 1 = 1 200.00 t 2 = 0.98(1 200.00) hence r = 0.98 t n = 1 200(0.98) n – 1 , n ∈ N 4
• The store will lose money on the mountain bike if its price falls below $600. On which day of the promotion will this occur? 600 = 1 200(0.98) n – 1 1 2 = 0.98n – 1 log 0.5=(n – 1)log 0.98 log 05 . log 098 . = n n ≈ 35.3 A POSSIBLE SOLUTION On the 36 th day, the cost of the bike is $591.69 which is the first price below $600. 5
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