A pie diagram shows the percentage distribution of proteins, water and other dry elements in the human body. Given that proteins correspond to 16% and water corresponds to 70%. If both proteins and the other dry elements correspond to p%, then what is the central angle of the sector representing p on the pie diagram?

Correct Option

The problem provides the percentage distribution of components in the human body:

• Percentage of proteins = 16%
• Percentage of water = 70%

To determine the percentage of other dry elements, subtract the known percentages from the total (100%):

Percentage of other dry elements = $100\mathrm{\%}-(16\mathrm{\%}+70\mathrm{\%})$ = $100\mathrm{\%}-86\mathrm{\%}$ = $14\mathrm{\%}$

The variable 'p' represents the combined percentage of proteins and other dry elements:

p% = Percentage of proteins + Percentage of other dry elements = $16\mathrm{\%}+14\mathrm{\%}$ = $30\mathrm{\%}$

In a pie diagram, the total central angle is 360°. Therefore, 1% corresponds to $\frac{{360}^{\circ }}{100}$ = ${3.6}^{\circ }$.

The central angle for p% (30%) is calculated as:

Central angle = $30\times {3.6}^{\circ }$ = ${108}^{\circ }$

Thus, the central angle of the sector representing p on the pie diagram is 108°.

Incorrect Options

The calculated central angle for p% (30%) is 108°. The other options do not correspond to this value.

• Option 1 (54°): This angle would represent $\frac{{54}^{\circ }}{{3.6}^{\circ }/\mathrm{\%}}$ = 15%. This does not match the calculated p% (30%).
• Option 2 (96°): This angle would represent $\frac{{96}^{\circ }}{{3.6}^{\circ }/\mathrm{\%}}$ $\approx$ 26.67%. This does not match the calculated p% (30%).
• Option 4 (120°): This angle would represent $\frac{{120}^{\circ }}{{3.6}^{\circ }/\mathrm{\%}}$ $\approx$ 33.33%. This does not match the calculated p% (30%).

Therefore, options 1, 2, and 4 are incorrect as they do not represent the central angle corresponding to p%.