Question

The Food Mart grocery store has a candy machine. Each time a customer inserts a quarter, 7 candies come out of the machine. The machine holds 15 pounds of candy. Each pound of candy contains about 180 individual candies. Assume the number of candies is never less than zero.

1. About how many candies are in the machine when it’s full?

2.Write a recursive formula to represent the number of candies remaining in the machine.

3.Write an explicit formula to represent the number of candies remaining in the machine for any number of customers

Answer 1

1. 2700 candies
2. c(0) = 2700; c(n) = c(n-1) -7
3. c(n) = 2700 -7n
4. 2693, 2686, 2679

Explanation:

1. The number of candies in the machine is the product of the number of pounds of candy and the number of candies in a pound:

(15 lbs)(180 candies/lb) = 2700 candies

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2. c(0) = 2700; c(n) = c(n-1) -7 . . . . the new number of candies is 7 fewer than after the previous customer. After no customers, the number is the original 2700 candies.

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3. As the recursive formula tells us, the initial number of candies is 2700, and the number decreases at the rate of 7 per customer. The number remaining after n customers is ...

c(n) = 2700 -7n

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4. Evaluating the above formula for n=1, 2, 3, we get ...

c(1) = 2700 -7 = 2693 . . . . candies remaining after 1 customer

c(2) = 2700 -7·2 = 2686 . . . 2 customers

c(3) = 2700 -7·3 = 2679 . . . 3 customers