Question

A circular metal plate expands under heating so that its radius increases by 2%. Find the approximate increase in the area of the plate if the radius of the plate before heating is 10cm.

Answer 1

Hint: Given the radius of the plate before heating is 10cm and find its area say x. The radius of the plate after heating will be $ 10 + \dfrac{{10 \times 2}}{{100}} $ cm. Find the area of the plate with the new radius say y. Subtract the value of x from y to get the increase in the area of the plate after heating.

Complete step-by-step answer:
Area of the circle is $ \pi {r^2} $ , where r is the radius of the circle and the value of $ \pi $ is 22/7 or 3.14.

We are given that a circular metal plate expands under heating so that its radius increases by 2% and the radius of the plate before heating is 10cm.
We have to find the increase in the area of the plate after heating.
Radius of the circular plate before heating = 10cm.
Area of the circular plate is $ \pi {r^2} $
 $
\Rightarrow Area = \pi {r^2} \\
  r = 10cm \\
   \to Area = \pi \times {10^2} \\
  \therefore Area = 100\pi c{m^2} \\
  $
Radius of the plate is increased by 2% after heating.
The new radius R is $ 10 + \left( {10 \times \dfrac{2}{{100}}} \right) = 10 + 0.2 = 10.2cm $

The area of the plate after heating is $ \pi {R^2} $
 $
\Rightarrow Area = \pi {R^2} \\
  R = 10.2cm \\
\Rightarrow Area = \pi \times {\left( {10.2} \right)^2} \\
  \therefore Area = 104.04\pi c{m^2} \\
  $
Increase in the area is $ 104.4\pi - 100\pi = 4.4\pi c{m^2} \simeq 4\pi c{m^2} $
The approximate increase in the area of the circular plate after heating is $ 4\pi c{m^2} $

Note: Approximate increase must be in integers not in decimals and if the digit after the decimal point is greater than 5 then sealing of the number must be taken else the floor of the number. Circle is a two-dimensional figure whereas a Sphere is a three-dimensional object. A circle has all points at the same distance from its centre along a plane, whereas in a sphere all the points are equidistant from the centre at any of the axes. Sphere and circle both have the same shape but different areas. So do not confuse a circle with a sphere.