If x and y are positive real numbers such that x[x] = 36 and y[y] = 71,then x+y is equal to which number.[.]denotes the greatest integer function.
We know that x[x] = 36, which means x is between 6 and 7 (since 6[6] = 36 and 7[7] = 49). Let's set x = 6 + a, where a is the decimal part of x. Then we have:
x[x] = 6a = 36
a = 6
Therefore, x = 6.6
Similarly, y must be between 8 and 9 (since 8[8] = 64 and 9[9] = 81). Let's set y = 8 + b, where b is the decimal part of y. Then we have:
y[y] = 8b = 71
b = 8.875
Therefore, y = 8.875
So, x + y = 6.6 + 8.875 = 15.475. The greatest integer less than or equal to 15.475 is 15. Therefore, x + y = 15.
To solve this problem, let's break it down step-by-step:
Step 1: Find the value of x and y separately.
Since x[x] = 36, we know that the greatest integer function of x is equal to 36. Therefore, the value of x lies between 36 and 37.
Step 2: Determine the possible values of x and y.
We need to find the positive real numbers that will satisfy the equation x[x] = 36 and y[y] = 71.
x[x] = 36 implies that x is between 36 and 37. Similarly, y is between 71 and 72.
Step 3: Calculate x + y.
To determine the value of x+y, we need to find the sum of x and y.
Since x and y are positive real numbers, we can conclude that x+y will be greater than 107 (36+71) and less than 110 (37+72).
Therefore, x+y can be any number between 107 and 110, exclusive of these endpoints.
In conclusion, x+y can be any real number between 107 and 110, not including 107 or 110.