An artisan makes hand-crafted painted benches to sell at a craft mall. Her weekly revenue and costs (not including labor) are given in the table. Weekly Revenue and Costs for Hand-crafted Benches Weekly revenue, R (dollars) 301 873 1381 1742 1978 Number of benches sold each week, n 1 3 5 7 9 148 11 181 13 1701 207 (a) Find a quadratic model for revenue. (Round all numerical values to three decimal places.) R(n) = -10.857n² + 91.714n dollars X 1949 Find a cubic model for cost. (Round all numerical values to three decimal places.) C(n) = 18.143n³-65.857n²+61.143n+76 (b) Write the derivative formula for profit. P'(n) = per bench Weekly cost, C (dollars) 64 73 103 110 X dollars Using the previous models for revenue and cost, find a cubic model for profit, P. (Round all numerical values to three decimal places.) P(n)= dollars (c) Find and interpret the rates of change of profit when the artisan sells 5, 9, and 10 benches, respectively. (Round your answers to two decimal places.) P'(5) = $206.43 X per bench When the artisan makes 5 benches, the profit is ---Select--- by $ P'(9) = $ per bench When the artisan makes 9 benches, the profit is ---Select--- by $ P'(10) = $ per bench When the artisan makes 10 benches, the profit is ---Select--- by $ per bench. per bench. per bench. (d) Use the information in part (c) to determine the number of benches the artisan should produce each week for maximum profit. benches
An artisan makes hand-crafted painted benches to sell at a craft mall. Her weekly revenue and costs (not including labor) are given in the table. Weekly Revenue and Costs for Hand-crafted Benches Weekly revenue, R (dollars) 301 873 1381 1742 1978 Number of benches sold each week, n 1 3 5 7 9 148 11 181 13 1701 207 (a) Find a quadratic model for revenue. (Round all numerical values to three decimal places.) R(n) = -10.857n² + 91.714n dollars X 1949 Find a cubic model for cost. (Round all numerical values to three decimal places.) C(n) = 18.143n³-65.857n²+61.143n+76 (b) Write the derivative formula for profit. P'(n) = per bench Weekly cost, C (dollars) 64 73 103 110 X dollars Using the previous models for revenue and cost, find a cubic model for profit, P. (Round all numerical values to three decimal places.) P(n)= dollars (c) Find and interpret the rates of change of profit when the artisan sells 5, 9, and 10 benches, respectively. (Round your answers to two decimal places.) P'(5) = $206.43 X per bench When the artisan makes 5 benches, the profit is ---Select--- by $ P'(9) = $ per bench When the artisan makes 9 benches, the profit is ---Select--- by $ P'(10) = $ per bench When the artisan makes 10 benches, the profit is ---Select--- by $ per bench. per bench. per bench. (d) Use the information in part (c) to determine the number of benches the artisan should produce each week for maximum profit. benches
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:An artisan makes hand-crafted painted benches to sell at a craft mall. Her weekly revenue and costs (not including labor) are given in the table.
Weekly Revenue and Costs for Hand-crafted Benches
Weekly revenue, R
(dollars)
301
Number of benches
sold each week, n
R(n)
=
1
3
5
P'(9) = $
7
P'(10)
9
=
11
(b) Write the derivative formula for profit.
P'(n) =
873
1381
1742
1978
per bench
1949
13
1701
207
(a) Find a quadratic model for revenue. (Round all numerical values to three decimal places.)
-10.857n² + 91.714n
dollars
Weekly cost, C
(dollars)
64
73
Find a cubic model for cost. (Round all numerical values to three decimal places.)
C(n) = 18.143n³ – 65.857n² +61.143n +76
dollars
103
110
Using the previous models for revenue and cost, find a cubic model for profit, P. (Round all numerical values to three decimal places.)
P(n)=
dollars
148
181
(c) Find and interpret the rates of change of profit when the artisan sells 5, 9, and 10 benches, respectively. (Round your answers to two decimal places.)
P'(5) = $206.43
X per bench
When the artisan makes 5 benches, the profit is ---Select--- by $
per bench
When the artisan makes 9 benches, the profit is ---Select--- by $
per bench
When the artisan makes 10 benches, the profit is ---Select--- by $
per bench.
per bench.
per bench.
(d) Use the information in part (c) to determine the number of benches the artisan should produce each week for maximum profit.
benches
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