# hw15 - Â 6.5 15 p x = 01 x 3 A = 200 So in the notation of...

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Unformatted text preview: Â§ 6.5 15. p ( x ) = . 01 x + 3; A = 200. So in the notation of Figure 8 in the book, the selling price is B = p ( A ) = p (200). The producersâ€™ surplus is integraldisplay 200 B âˆ’ p ( x ) dx = integraldisplay 200 5 âˆ’ . 01 x âˆ’ 3 dx = integraldisplay 200 2 âˆ’ . 01 x dx = 2 x âˆ’ . 01 2 x 2 vextendsingle vextendsingle vextendsingle 200 = 400 âˆ’ . 01 2 (200 2 ) = 400 âˆ’ 200 = 200 dollars. 16. p ( x ) = x 2 / 9 + 1; A = 3. Producersâ€™ surplus is integraldisplay 3 p (3) âˆ’ p ( x ) dx = integraldisplay 3 2 âˆ’ x 2 9 âˆ’ 1 dx = integraldisplay 3 1 âˆ’ x 2 9 dx = x âˆ’ x 3 27 vextendsingle vextendsingle vextendsingle 3 = 3 âˆ’ 3 3 27 = 2 dollars. 17. p ( x ) = x/ 2 + 7; A = 10. integraldisplay 10 p (10) âˆ’ p ( x ) dx = integraldisplay 10 12 âˆ’ x/ 2 âˆ’ 7 dx = integraldisplay 10 5 âˆ’ x/ 2 dx = 5 x âˆ’ x 2 / 4 vextendsingle vextendsingle vextendsingle 10 1 = 50 âˆ’ 10 2 / 4 = 25 . 18. p ( x ) = 1 + 1 2 âˆš x ; A = 36. integraldisplay 36 p (36) âˆ’ p ( x ) dx = integraldisplay 36 4 âˆ’ 1 âˆ’ 1 2 âˆš x dx = integraldisplay 36 3 âˆ’ 1 2 âˆš x dx = 3 x âˆ’ ( 1 2 )( 2 3 ) x 3 / 2 vextendsingle vextendsingle vextendsingle 36 = 3(36) âˆ’ 1 3 (36 3 / 2 ) = 108 âˆ’ 1 3 6 3 = 108 âˆ’ 72 = 36 . 19. Demand: p D ( x ) = 12 âˆ’ x/ 50; Supply: p S ( x ) = x/ 20 + 5. Optimal production is when p D ( x ) = p S ( x ) 12 âˆ’ x/ 50 = x/ 20 + 5 7 = x/ 50 + x/ 20 7 = 7 x/ 100 x = 100 . Moreover, p D (100) = 12 âˆ’ 2 = 10 (and this agrees with p S (100) = 10). So the point of intersection of the curves is ( A, B ) = (100 , 10). Consumersâ€™ surplus: integraldisplay 100 p D ( x ) âˆ’ p D (100) dx = integraldisplay 100 12 âˆ’ x/ 50 âˆ’ 10 dx = integraldisplay 100 2 âˆ’ x/ 50 dx = 2 x âˆ’ x 2 / 100 vextendsingle vextendsingle vextendsingle 100 = 2(100) âˆ’ 100 2 / 100 = 100 dollars. Producersâ€™ surplus: integraldisplay 100 p S (100) âˆ’ p S ( x ) dx = integraldisplay 100 10 âˆ’ x/ 20 âˆ’ 5 dx = integraldisplay 100 5 âˆ’ x/ 20 dx 2 = 5 x âˆ’ x 2 / 40 vextendsingle vextendsingle vextendsingle 100 = 5(100) âˆ’ 100 2 / 40 = 500 âˆ’ 100( 100 40 ) = 500 âˆ’ 250 = 250 dollars. 20. Demand: p D ( x ) = âˆš 25 âˆ’ . 1 x . Supply: p S ( x ) = âˆš . 1 x + 9 âˆ’ 2. Solving for the in- tersection point takes some work here, and one must be careful with the logic, as we discuss below. Curves intersect where p D ( x ) = p S ( x ) âˆš 25 âˆ’ . 1 x = âˆš . 1 x + 9 âˆ’ 2 We can square both sides and the equality still holds: 25 âˆ’ . 1 x = ( . 1 x + 9) + 4 + 2( âˆ’ 2 âˆš . 1 x + 9) Now weâ€™ll isolate the square root term: 12 âˆ’ . 2 x = âˆ’ 4 âˆš . 1 x + 9 Now square both sides again: 144 + . 04 x 2 âˆ’ 24( . 2 x ) = 16( . 1 x + 9) Now simplify into standard quadratic form: 144 âˆ’...
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## This note was uploaded on 04/03/2008 for the course MATH 16A taught by Professor Stankova during the Spring '07 term at Berkeley.

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hw15 - Â 6.5 15 p x = 01 x 3 A = 200 So in the notation of...

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