Question

The diameter of a sphere is decreased by \[25\% \] . By what percent does its curved surface decrease?

Answer 1

Hint: We are given the percentage decrease in diameter of the sphere and we are asked to find the percentage decrease in its curved surface area. Firstly we should know what percentage change is. Percentage change is a simple mathematical concept that represents the degree of change over time. Percentage decrease formula can be obtained by simply dividing the decreased value by the original value and multiplying that with \[100\]. We will use the formula for curved surface area of the sphere given by \[4\pi {r^2}\].

Complete step-by-step solution:
For a sphere of radius \[r\] units we have the following formulas:
Diameter of sphere \[ = 2r\]
Surface area of sphere \[ = 4\pi {r^2}\]
Volume of sphere \[ = \dfrac{4}{3}\pi {r^2}\]
Percentage change is a simple mathematical concept that represents the degree of change over time. Percentage decrease formula can be obtained by simply dividing the decreased value by the original value and multiplying that with \[100\].
Therefore percentage decreased=decreased value original value× \[100\]
Where decreased value=original value – new value
Let the radius of the sphere be \[\dfrac{r}{2}cm\].
Therefore its diameter \[ = 2\left( {\dfrac{r}{2}} \right)cm = rcm\]
Curved surface area of the original sphere \[ = 4\pi {\left( {\dfrac{r}{2}} \right)^2} = \pi {r^2}c{m^2}\]
New diameter (decreased) of the sphere \[ = r - r \times \dfrac{{25}}{{100}} = \dfrac{{3r}}{4}cm\]
Therefore radius of the new sphere \[ = \dfrac{1}{2}\left( {\dfrac{{3r}}{4}} \right)cm = \dfrac{{3r}}{8}cm\]
Therefore new curved surface area of the sphere \[ = 4\pi {\left( {\dfrac{{3r}}{8}} \right)^2} = \dfrac{{9\pi {r^2}}}{{16}}c{m^2}\]
Therefore decrease in original curved surface area \[ = \pi {r^2} - \dfrac{{9\pi {r^2}}}{{16}} = \dfrac{{7\pi {r^2}}}{{16}}\]
Percentage of decrease in the original curved surface area \[ = \dfrac{{\dfrac{{7\pi {r^2}}}{{16}}}}{{\pi {r^2}}} \times 100\% \]
\[ = 43.75\% \]
Hence the original curved surface area decreases by \[43.75\% \]

Note: To solve such types of questions one must have a strong grip over the concept of sphere and its related formulas. We should be aware of the concept of percentage change. Do the calculations very carefully and recheck them so as to get the required answer.