Question

a cookie baker has an automatic mixer that turns out a sheet of dough in the shape of a square 12 inches wide. His cookie cutter cuts 3-inch diameter circular cookies as shown below. The supervisor complained that too much dough was being wasted and ordered the baker to find out what size cookie would have the least amount of waste.

Answer 1

Answer:

All the sizes that satisfy $kd^2 =144$

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 $d$, and $a$ the length of the dough sheet, then the $n$ number of cookies that fit into length $a$ will be

$n = \dfrac{a}{d}$

and therefore, the number that will fit into the whole square sheet will be

$n^2 = \dfrac{a^2}{d^2}$

Since the area of each cookie is

$A = \pi \frac{d^2}{4}$

the area of n^2 cookies will be

$A_n = n^2\pi \frac{d^2}{4}$,

which is the area of all the cookies cut out from the dough sheet; therefore, after the cutting, the area left will be

(1). $\text{area left}= a^2-n^2\pi \frac{d^2}{4}$

putting in the value of $n^2$ we get

$a^2- \dfrac{a^2}{d^2}\pi \frac{d^2}{4}$

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.