Question

Shalini has to cut out circles of diameter $ 1\frac{1}{4} cm$ from an aluminum strip of dimensions $ 8\frac{3}{4} cm$ by $ 1\frac{1}{4} cm$. How many full circles can Shalini cut? Also calculate the wastage of the aluminum strip.

Answer 1

Step 1: Converting mix fraction into improper fraction

Diameter of the circle = Breadth of the aluminum strip

$1\frac{1}{4}=\frac{5}{4}cm$

Length of aluminum strip, $8\frac{3}{4}=\frac{35}{4}cm$

Step 2: Number of full circles can Shalini cut and also the wastage of the aluminum strip.

$$\begin{gathered} =\left(\frac{35}{4}\right)÷\left(\frac{5}{4}\right) \\ =\frac{35}{4}\times \frac{4}{5} \\ =7circles \end{gathered}$$

Step 3: Radius of circle

Radius of circle $=\frac{diameter}{2}$

$=\frac{5}{(4\times 2)}=\frac{5}{8}cm$

Step 4: Area to be cut by $7$ full circles

Since, Area of circle $=\pi r^2$

Area to be cut by one circle$=\left[\frac{22}{7}\right]\times {\left[\frac{5}{8}\right]}^2$ $\left[\because \pi =\frac{22}{7}\right]$

$=\left[\frac{22}{7}\right]\times \left[\frac{25}{64}\right]$

Area to be cut by $7$ full circles=$=7\times \frac{22}{7}\times \frac{25}{64}$

$$\begin{gathered} =\frac{(22\times 25)}{64} \\ =\frac{550}{64}c{m}^2 \end{gathered}$$

Step 5: Area of the aluminum strip

Since, $\text{Area of the aluminum strip = length × breadth}$

$$\begin{gathered} =\frac{35}{4}\times \frac{5}{4} \\ =\frac{175}{16}c{m}^2 \end{gathered}$$

Therefore, the wastage of aluminum strip$=\frac{175}{16}-\frac{550}{64}$

$$\begin{gathered} =\frac{(700-550)}{64} \\ =\frac{150}{64} \\ =\frac{75}{32}c{m}^2 \end{gathered}$$

Hence, number of circles cut by Shalini are $7$ & the wastage of the aluminum strip is $\frac{75}{32}c{m}^2$.