4≤p≤9, -0.25≤q≤0.36, -0.49≤r≤-0.01 and r/s^2 = p^2/q, where p, q, r an

rq = (-0.25)(-0.49)
\(\sqrt{rq} = \sqrt{(-0.25)(-0.49)} = \sqrt{(-1)(.5)^2(-1)(.7)^2}= .5*.7= .35\)

Since we are looking for MAXs - MINs ——> first put the equation in terms of (s)^2

(s)^2 = (r * q) / (p)^2


Before simplifying the expression, we can look at the equation to deduce some qualities:


Since all the variables are non-zero, the output of (s)^2 will always be +positive

This means the right side of the equation must be positive.

Also, (p)^2 in the denominator will always have a +positive output. In order for the right hand side of the equation to be positive, the product of R and Q must result in a positive value in the Numerator.

Thus we either have to take both values of R and Q as positive or both values as negative.

Since R can take only negative values, this means Q will necessarily have to be negative ———> (neg) * (neg) = positive


With this knowledge, now we can take the Square Root of both sides of the equation to have the equation in terms of S

Rule: Sqrt ( (S)^2 ) = [S]

Thus:

[S] = Sqrt( R * Q ) / (P)

Opening the Modulus:

S = the (+) value or (-)value of the above expression on the right hand side of the equation


Remember, S is not limited by any given constraint. It can take a negative or positive value ——-> the possible values that S can take are only constrained by what we put in R, Q , and P (which themselves are constrained)


(1st) Maximize Value of S

In order to Maximize the Value of S, given that we know the Product of (r * q) must be positive:

We want to make the Magnitude of the DEN as small as possible

While making the Magnitude of the NUM as large as possible


The smallest we can make the DEN of P ———> 4

The Largest we can make the positive NUM of: Sqrt( R * Q) ———->

highest magnitude of the value of Q = -(25/100) ........ (switched to fractional form)

Highest magnitude of the value of R = -(49/100)

Q * R = (-) * (-) = (+)

Q * R = (25 * 49) / (100 * 100)

the NUM involves taking the Square Root of Q * R

Sqrt( Q * R) = (5 * 7) / (100) = 35/100


And finally to get the MAXs ——-> which has a DEN of 4:

(35/100) * (1/4) = .0875 = MAXs


(2nd) MINIMUM Value of S = MINs

Since we found that:

S = (+/-) (Sqrt( RQ) ) * (1/P)

and

The Higher the Magnitude of a Negative Number, the LOWER the Value of hat Negative Number

this means we can take the Highest Magnitude Positive Value and Negate it to make it the MIN Value of S

thus: MINs = (-).0875



Finally, answering the question:

(MAXs) - (MINs) = (.0875) - (-.0875) =

.0875 + .0875 =

(2) * (.0875) =

.1750

Answer C

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