Alice is constructing rectangular prisms to represent buildings in a diorama for her social studies class.
• The length of each rectangular prism exceeds the width by 6 cm.
• The height of each rectangular prism is 5 cm less than the width.
• The volume of each rectangular prism must be less than or equal to 50 cm3.
What are possible width values in centimeters for each rectangular prism?

L = w + 6

w = l - 6

h = w - 5

h = l - 11

V = lwh

V = l(l - 6)(l - 11)

But V <= 50

therefore,

l^3 - 17l^2 + 66l - 50 <= 0

since all the coefficients add to 0, 1 is a root. Therefore, it can be factored into:

(l-1)(l-8+sqrt(14))(l-8-sqrt(14))<= 0

but at the extremes,

h >= 0

l -11 >= 0

l >= 11

therefore, considering this,

we can see that

11 <= l <= 8 + sqrt(14)

11 - 6 <= l - 6 <= 8 + sqrt(14) - 6

5 <= w <= 2 + sqrt(14)

(In cm)

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