What is the central angle of the sector corresponding to the expenditure incurred on Royalty? Which two expenditures together have a central angle of 108°? If the difference betwee... What is the central angle of the sector corresponding to the expenditure incurred on Royalty? Which two expenditures together have a central angle of 108°? If the difference between two expenditures is represented by 18° in the pie-chart, what are those expenditures? If for an edition of the book the cost of paper is Rs. 56250, find the promotion cost for this edition.
Understand the Problem
The question set is based on a pie chart that illustrates the percentage distribution of expenditures involved in publishing a book. It includes questions about calculating angles corresponding to expenditures, identifying combinations of expenditures with specific angles, and determining costs based on given data. The questions require an understanding of basic percentages and geometry related to pie charts.
Answer
- Central angle for Royalty: $54^\circ$; Promotion cost: Rs. $5625$.
Answer for screen readers
- The central angle for Royalty is $54^\circ$.
- The angles for expenditure pairs to reach $108^\circ$ require further confirmation based upon total constraints, specifically $63^\circ$ and $45^\circ$. The promotion cost calculated is $5625$ Rupees.
Steps to Solve
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Calculate the central angle for Royalty To find the central angle corresponding to the Royalty expenditure, use the formula: $$ \text{Central angle} = \left(\frac{\text{Percentage}}{100}\right) \times 360^\circ $$ For Royalty, the percentage is 15. Thus: $$ \text{Central angle} = \left(\frac{15}{100}\right) \times 360 = 54^\circ $$
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Identify two expenditures with a central angle of 108° From the pie chart, we have:
- Binding Cost: 30%
- Printing Cost: 20%
- Royalty: 15%
- Transportation Cost: 10%
- Promotion Cost: 10%
- Paper Cost: 25%
We check combinations:
- Binding Cost (30%) + Transportation Cost (10%) = 40°
- Binding Cost (30%) + Printing Cost (20%) = 50°
- Royalty (15%) + Binding Cost (30%) = 45°
- Paper Cost (25%) + Printing Cost (20%) = 45°
- Royalty (15%) + Promotion Cost (10%) = 25°
- Paper Cost (25%) + Royalty (15%) + Promotion Cost (10%) = 50°
- Printing Cost (20%) + Paper Cost (25%) = 45°
None combine to exactly 108°, which suggests examining alternate combinations.
- Finding the expenditure pair for the angle of 108° If we observe the existing angles, we notice that two expenditure combinations might include:
- Binding Cost (30%) + Promotion Cost (10%) + Paper Cost (25%) = 65° To analyze multiple variables or overlaps, we can iterate through pairs and double-check combinations to yield 108° or explore combined sets.
- Difference represented by 18° For two angles contributing to 108°, with a difference of 18°, denote them ( x ) and ( y ). According to our assumption, $$ x + y = 108^\circ $$ $$ x - y = 18^\circ $$
Solving these:
- Adding: $$ 2x = 126 \implies x = 63^\circ $$
- Substituting back: $$ 63 - y = 18 \implies y = 45^\circ $$
The two angles ( x=63^\circ ) and ( y=45^\circ ) represent expenditures potentially as Binding Cost and another expenditure forming:
- Binding Cost & Paper Cost (30% + 25%) or vary potential values.
- Calculate promotion cost from paper cost Given:
- Cost of paper = Rs. 56250 Calculating Promotion Cost:
- Given:
- The percentage for Promotion cost is 10%. $$ \text{Promotion Cost} = \left(\frac{10}{100}\right) \times 56250 = Rs. 5625 $$
- The central angle for Royalty is $54^\circ$.
- The angles for expenditure pairs to reach $108^\circ$ require further confirmation based upon total constraints, specifically $63^\circ$ and $45^\circ$. The promotion cost calculated is $5625$ Rupees.
More Information
- Central angles correspond directly to percentages of 360 degrees to visualize spending portions for proper expenditure management in book publishing.
- The calculated promotion cost is indicative of budget estimates alongside graphic analyses, summing up total expenditures in publishing.
Tips
- Miscalculating the total angle resulting from sub-additions of its constituent components, or overlooking needed pairs within available data often leads to confusion.
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