Final answer:
The value of k is found to be 15.
Step-by-step explanation:
The question involves finding the number of terms in a set given the original and new mean after adding a number to that set.
To calculate the value of k, we use the formula for the mean of a set of numbers:
mean = (sum of the numbers) / (number of terms).
The original mean is given as 56.8 for k numbers, so the sum of the original numbers is 56.8k.
When the number 52 is added, the new mean becomes 56.5, and the new sum is 56.8k + 52 since there are now k+1 numbers.
Applying the new mean formula:
56.5 = (56.8k + 52) / (k + 1)
Multiplying both sides by (k+1) to clear the fraction:
56.5(k + 1) = 56.8k + 52
Expanding the left side:
56.5k + 56.5 = 56.8k + 52
Subtract 56.5k from both sides:
56.5 = 0.3k + 52
Subtract 52 from both sides to solve for k:
4.5 = 0.3k
Finally, divide by 0.3 to get k:
k = 4.5 / 0.3
k = 15
Therefore, the value of k is 15.