Final answer:
By using the formula for the nth triangular number, we can determine that the largest equilateral triangle that can be made with 52 marbles will consist of 45 marbles, leaving 7 marbles left over.
Step-by-step explanation:
To solve the problem of finding the biggest equilateral triangle that can be made with 52 marbles and determine how many marbles will be left over, one should consider the pattern of numbers required for each successive layer of the triangle.
An equilateral triangle with n layers has 1 + 2 + 3 + ... + n marbles, with the total number of marbles being equal to the n-th triangular number, represented by the formula (n(n + 1))/2. We can therefore create an inequality to find the maximum n such that (n(n + 1))/2 ≤ 52.
Here is how we can approach this problem step by step:
- Write out the formula for the nth triangular number: (n(n + 1))/2.
- Set up an inequality where this number is less than or equal to 52: (n(n + 1))/2 ≤ 52.
- Solve this inequality for n to find the largest integer value that satisfies it.
- Calculate the number of marbles used in the largest equilateral triangle by plugging the integer value of n back into the formula.
- Subtract this number from the total of 52 marbles to find the number left over.
After solving the inequality, we find that the greatest value of n that satisfies it is 9, since (9(10))/2 = 45.
Therefore, the largest equilateral triangle we can make will have 45 marbles, and we will have 52 - 45 = 7 marbles left over.