If the following two statements are true: 3 < a < 4 4 < b < 5 Find all the possible values of the expression: (b)/(3a) - (a)/(3b)

Question

If the following two statements are true:

3 < a < 4
4 < b < 5

Find all the possible values of the expression:
`(b)/(3a) - (a)/(3b)`

Answer 1

Answer: 0 < b/(3a) - a/(3b) < 4/9

Explanation

Let's simplify the given expression a bit

b/(3a) - a/(3b)

(b^2)/(3ab) - (a^2)/(3ab)

(b^2 - a^2)/(3ab)

We end up with a single giant fraction

• b^2-a^2 up top
• 3ab down below

The lower bound of this expression is when these two cases happen

• b^2-a^2 reaches its lower bound
• 3ab reaches its upper bound

We want the numerator to be as small as possible. At the same time we want the denominator to be as large as possible.

b^2-a^2 has the lower bound when making 'b' as small as possible and 'a' as large as possible. We go for b = 4 and a = 4

The lower bound of b^2-a^2 would be 4^2-4^2 = 0.

b^2-a^2 has the upper bound when making 'b' as large as possible and 'a' as small as possible. We go for b = 5 and a = 3

The upper bound of b^2-a^2 would be 5^2-3^2 = 16.

We can state that 0 < b^2-a^2 < 16.

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Now to find the lower and upper bound of 3ab.

The lower bound occurs when both 'a' and 'b' are as small as possible.

The lower bound is 3ab = 3*3*4 = 36

On the other end of the spectrum, the upper bound is 3ab = 3*4*5 = 60

Therefore: 36 < 3ab < 60

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We have found the following so far

• 0 < b^2-a^2 < 16
• 36 < 3ab < 60

This establishes the lower and upper bounds of numerator and denominator.

The fraction (b^2 - a^2)/(3ab) reaches its lower bound when we make the numerator reach its lower bound and the denominator its upper bound.

The lower bound of (b^2 - a^2)/(3ab) is 0/60 = 0.

The upper bound of (b^2 - a^2)/(3ab) is 16/36 = 4/9

Therefore 0 < (b^2 - a^2)/(3ab) < 4/9

This leads us back to: 0 < b/(3a) - a/(3b) < 4/9

Answer 2

(0, 16/45)

Explanation:

You want the range of possible values of the expression ...

`(b)/(3a)-(a)/(3b)`

subject to the restrictions ...

• 3 < a < 4
• 4 < b < 5

Rewrite

If we define x = b/a, the expression can be written as ...

f(x) = 1/3(x - 1/x)

For positive x, this is a monotonically increasing function of x. To find its extreme values, we can look for the extreme values of x = b/a.

Minimum

The value of x will be a minimum for minimum b and maximum a:

xmin = 4/4 = 1

f(1) = 1/3(1 -1/1) = 0

Maximum

The value of x will be a maximum for maximum b and minimum a:

xmax = 5/3

f(5/3) = 1/3(5/3 -3/5) = 1/3(25 -9)/15 = 16/45

Values of the expression

The values of the expression will lie in the open interval ...

0 to 16/45 . . . . . . . . . . endpoint values not included

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