Question 184037
Write a formula for the perimeter of a square as a function of its area:
:
we know the area of a square = side squared
Therefore, one side = square root of the area (a)
Perimeter (P) is the sum of all 4 sides:
so we have:
:
P(a) = {{{4*sqrt(a)}}}
:
:
and
2)
Area of a sign.
A sign is in the shape of a square with a semicircle of radius x adjoining one side and a semicircle of diameter x removed from the opposite side. If the sides of the square are length 2x, then write the area of the sign as a function of x.
:
Write an expression for the area of each part:
:
Attached semi circle: {{{(pi*x^2)/2}}}; (radius=x)
;
the square: {{{(2x)^2}}}; (side=2x)
:
removed semi circle: {{{(pi*(.5x)^2)/2}}}; (radius=.5x)
:
f(x) = area of the sign
Total area = large semi circle + square - small semicircle
:
f(x) = {{{(pi*x^2)/2}}} + {{{(2x)^2}}} - {{{(pi*(.5x)^2)/2}}}
f(x) = {{{(pi*x^2)/2}}} + {{{(4x^2)}}} - {{{(pi*(.25x^2))/2}}}
get each expression over 2
f(x) = {{{(pi*x^2)/2}}} + {{{(8x^2)/2)}}} - {{{(pi*(.25x^2))/2}}}
so we have
f(x) = {{{(pi*x^2 + 8x^2 - pi*.25x^2)/2}}}
Factor out x^2
f(x) = {{{(x^2(pi + 8 - .25pi))/2}}}
combine the pi's
f(x) = {{{(x^2(.75pi + 8))/2}}}