Question

her local charity by encouraging people to lay quarters on the floor of a classroom in the shape of an equilateral triangle. She puts one quarter in the first row, two in the second, three in the third, and so on. Her target is to raise $400. How many rows does she need ?

Answer 1

The student needs 57 rows of quarters to reach her target of raising $400, as this number of rows results in the first triangle number greater than or equal to 1600 quarters, with each quarter worth $0.25.

Step-by-step explanation:

To determine how many rows are necessary to reach the target of $400 by placing quarters in the shape of an equilateral triangle, we can use the sequence of triangle numbers. Each row represents a triangle number, which can be found by the formula T(n) = n(n+1)/2, where n is the row number.

Since each quarter is worth $0.25, we need to reach a total of $400/$0.25 = 1600 quarters. We need to find the row number (n) such that the total number of quarters up to that row equals or exceeds 1600.

We are thus looking for the smallest integer n such that:
T(n) >= 1600
Let's solve for n using the triangle number formula.

Step 1: Set up the inequality
strong>n(n+1)/2 >= 1600

Step 2: Multiply both sides by 2 to get n(n+1) >= 3200

Step 3: Find the smallest integer n such that this condition is true.

By testing each successive integer, we find that n = 57 yields T(57) = 57(58)/2 = 1653, which is the first triangle number greater than or equal to 1600.

Therefore, the student needs to lay out 57 rows of quarters to raise at least $400.