Question

A sheet is 11cm long and 2cm wide. Circular pieces 0.5cm in diameter are cut from it to prepare discs. Calculate the number of discs that can be prepared.

Answer 1

Hint: Find the area of the sheet using the formula of area of rectangle given as: - Area = length \[\times \] breadth. Now, assume that the number of discs that can be prepared from the given sheet is ‘n’. Find the area of one disc using the formula of area of circle given as: - Area = \[\pi {{r}^{2}}\], where ‘r’ is the radius given by: - radius = $\dfrac{\text{Diameter}}{2}$. Finally, find the value of ‘n’ by dividing the area of the sheet by the area of the disc.

Complete step-by-step solution
Here, we have provided a sheet of dimensions 11cm long and 2cm wide. We have to find the total number of discs that can be formed out of this sheet with the diameter of discs being 0.5cm.
Now, since the given sheet is in the form of a rectangle whose length is 11cm and breadth is 2cm. Therefore, its area can be given as: -
\[\Rightarrow \] Area of sheet = length \[\times \] breadth
\[\Rightarrow \] Area of sheet = 11 \[\times \] 2
\[\Rightarrow \] Area of sheet = 22 \[c{{m}^{2}}\]
Now, the discs are in the form of a circle whose diameter is given to us as 0.5cm. Therefore, applying the formula of the relationship between radius and diameter given as - radius = $\dfrac{\text{Diameter}}{2}$, we get,
\[\Rightarrow \] radius = \[\dfrac{0.5}{2}\]
\[\Rightarrow \] r = \[\dfrac{0.5}{2}\], where ‘r’ denotes radius.
Therefore, applying the formula of area of a circle to find the area of given disc, we get,
\[\Rightarrow \] Area of disc = \[\pi {{r}^{2}}\]
\[\Rightarrow \] Area of disc = \[\pi \times {{\left( \dfrac{0.5}{2} \right)}^{2}}\]
Let us assume that the number of discs that can be formed using the given sheet is ‘n’. So, the area of the sheet must be equal to the sum of areas of these ‘n’ discs. So, we have,
\[\Rightarrow \] Area of sheet = n \[\times \] Area of one disc
So, equating the respective area of sheet and disc in the above relation, we get,
\[\Rightarrow 22=n\times \pi \times {{\left( \dfrac{0.5}{2} \right)}^{2}}\]
\[\Rightarrow n=\dfrac{22\times {{2}^{2}}}{\pi \times {{\left( 0.5 \right)}^{2}}}\]
Substituting \[\pi =\dfrac{22}{7}\] in the above relation, we get,
\[\begin{align}
  & \Rightarrow n=\dfrac{22\times {{2}^{2}}}{\dfrac{22}{7}\times {{\left( 0.5 \right)}^{2}}} \\
 & \Rightarrow n=\dfrac{22\times 4\times 7}{22\times 0.25} \\
 & \Rightarrow n=112 \\
\end{align}\]
Hence, a total of 112 discs can be formed out of the given sheet.

Note: One may note that here in the above question, we do not have any other method but to equate the areas of the two shapes. So, you must memorize the formulas of the area of the rectangle and circle. The relation between radius and diameter must be known because we needed this conversion. Now, you may see that we have substituted \[\pi =\dfrac{22}{7}\] to calculate the answer. This was done to cancel the common factors and get the value of ‘n’ in the whole number.