Answer:
All the sizes that satisfy [tex]kd^2 =144[/tex]
Step-by-step explanation:
To answer this question we first need to find the minimum wasted area of the dough.
Let us call the diameter of the cookie [tex]d[/tex], and [tex]a[/tex] the length of the dough sheet, then the [tex]n[/tex] number of cookies that fit into length [tex]a[/tex] will be
[tex]n = \dfrac{a}{d}[/tex]
and therefore, the number that will fit into the whole square sheet will be
[tex]n^2 = \dfrac{a^2}{d^2}[/tex]
Since the area of each cookie is
[tex]A = \pi \frac{d^2}{4}[/tex]
the area of n^2 cookies will be
[tex]A_n = n^2\pi \frac{d^2}{4}[/tex],
which is the area of all the cookies cut out from the dough sheet; therefore, after the cutting, the area left will be
(1). [tex]\text{area left}= a^2-n^2\pi \frac{d^2}{4}[/tex]
putting in the value of [tex]n^2[/tex] we get
[tex]a^2- \dfrac{a^2}{d^2}\pi \frac{d^2}{4}[/tex]
which simplifies to
area left = a^2( 1 - (π/4))
putting in a = 12 we get
area left = 30.902 in^2.
Going back to equation (1) we find that
a^2-n^2(πd^2/4) =30.902
12^2- n^2(πd^2/4) =30.902
and if we call k = n^2, we get
12^2- k(πd^2/4) =30.902
113.098 = k(πd^2/4)
simplifiying this gives
kd^2 = 144.
As a reminder, k here is the number of cookies cut from the dough sheet.
Hence, our cookie diameter must satisfy kd^2 = 144, meaning larger the diameter of the cookies less of the should you cut out to satisfy the above equality.
