Find all possible values of b, where (a,b,c) satisfies \[⌊a⌋⋅b⋅c=3a⋅⌊b⌋⋅c=4a⋅b⋅⌊c⌋=5\] and a, b, and c are positive. Enter all values, separated by commas.
We are tasked with finding all possible values of b that satisfy the following system of equations involving the floor functions:
⌊a⌋⋅b⋅c=3
a⋅⌊b⌋⋅c=4
a⋅b⋅⌊c⌋=5
where a, b, and c are positive real numbers. We will solve this step by step by exploring possible values for a, b, and c.
### Solution By Steps
#### Step 1: Analyze the first equation
From Equation (1):
⌊a⌋⋅b⋅c=3
The floor function ⌊a⌋ represents the greatest integer less than or equal to a. Since a is positive, the possible values of ⌊a⌋ are integers.
We first assume ⌊a⌋=1, which is the smallest possible value because a>0. Substituting this into Equation (1):
1⋅b⋅c=3
Thus, we have:
b⋅c=3
#### Step 2: Analyze the second equation
Now, consider Equation (2):
a⋅⌊b⌋⋅c=4
We already know b⋅c=3 from Equation (4). To satisfy this equation, we explore possible values of ⌊b⌋.
Let’s assume ⌊b⌋=1 first. Substituting into Equation (2):
a⋅1⋅c=4
So,
a⋅c=4
#### Step 3: Analyze the third equation
Now consider Equation (3):
a⋅b⋅⌊c⌋=5
We already have b⋅c=3 and a⋅c=4. To satisfy this equation, we explore possible values of ⌊c⌋.
Let’s assume ⌊c⌋=1. Substituting into Equation (3):
a⋅b⋅1=5
So:
a⋅b=5
#### Step 4: Solve the system of equations
Now, we have three equations:
1. b⋅c=3
2. a⋅c=4
3. a⋅b=5
We can solve this system step by step. First, solve for c from Equation (5):
c=4a
Substitute this into Equation (4):
b⋅4a=3
Simplifying:
b=3a4
Now substitute this into Equation (6):
a⋅3a4=5
Simplifying:
3a24=5
Multiply both sides by 4:
3a2=20
Solve for a2:
a2=203
So:
a=√203=2√153
#### Step 5: Find the value of b
Now that we have a=2√153, substitute this back into the expression for b:
b=3a4=3×2√1534=2√154=√152
Thus, the value of b is:
b=√152