Hey, that is true. In Math, we either over-simplify or over-complicate things. And if we get it just right enough, we were just lucky.

But then, who are to say what we did was just right enough?

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(*) A rectangle has an area of 38 square feet. Determine the perimeter as a function of the length l.

Rectangle.

width = w

length = L

Perimeter = w+L+w+L = 2w +2L = 2(w+L)

Area = w*L = wL

wL = 38

w = 38/L

Substitute that into the Perimeter,

Perimeter = 2(38/L +L) -------***

Over-simpliying that,

P = (76/L +2L)

P = (76 +2L^2)/L ----------****

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(*) The volume of a right circular cylinder is 540 cubic centimeters. Express the height of the cylinder as a function of its radius r.

Right circular cylinder.

Radius = r

Height = h

Volume = pi(r^2)h

pi(r^2)h = 540

h = 540 / pi(r^2) ----------***

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(*) The hypotenuse of a right triangle is twice as long as the height. Determine the height as a function of the base b.

Right triangle.

Base = b

Height = h

Hypotenuse = r

Given:

r = 2*h

By Pythagorean theorem,

r^2 = b^2 +h^2

Substitute 2h for r,

(2h)^2 = b^2 +h^2

4h^2 = b^2 +h^2

4h^2 -h^2 = b^2

3h^2 = b^2

h^2 = (b^2)/3

Get the square roots of both sides,

h = b / sqrt(3) ----------------------***

Over-simplifying that,

Rationalize the denominator,

Multiply both numerator and denominator by sqrt(3),

h = b*sqrt(3) / 3

h = (1/3)b*sqrt(3) ----------***