Ava Richardson, the proprietor of the Ava's Elegance store, predicts that when a specific type of hand lotion is priced at p dollars per tube, she will sell a B(p) number of tubes per monthB(p)=p+3500? p?5

each at a total cost ofC(p)=0.2p2+3p+200

a. Represent Ava's profit P(p) as a function of the price p per tube?

b. What is the rate of change in profit in relation to p when the price is set at $12 per tube? Is the profit trend ascending or descending at this price point?

-> ap ( b ) -Soo 10.41 + 2/+ dip 500 10.2/ 4 2 +200 - _- 500 / 0.4 p + 2 ) = =500 0. 21 2 4 2 p+ 200 p+ 3 0 . 4 / 2 + 31 + 1. 2p +6 = 0. 2 12 + 21+200 7 0. 202 + 1. 26 - 198 = 0 2p2 + 12p - 1940 = 0 12 +6p - 970 = 0 p = - 62536- 4x(- 970) 2 .= 28.28 02 -34.2 8X Ip = 28.28 500 ( 0 . 4 / + 2) - 500 dep 2 ( p + 3 ) 2 p = 28. 28 +( 2 ) X 500 fo. 21 + 20 + 200 (+ 3 1 2 p = 28.28 = -6.39 p = 28. 28 is a Maxima P (P = 12 ) ! Ascending285 . price = p dollars/ tube . number cof tubes she will sell = BEP ) ( per month ) B ( P ) = 500 pt3 ;p35 ". each tube at a total cost of (CP) = 0.212 + 3p + 200 " per month ) PCP= price x number of fubed Total cost for all tubes = P X B ( P ) - CCP ) X B ( P ) = BCP ) ( P - CCP) ) = 500 P - (0 . 2 p 2 + 3p + 200 ) pts 1. P(p) = - 500 0 . 2 1 2 + 2 p + 200 (b ) - 500 ( 0, 4 p + 2 + 500 10.2/2 + 2/+ 200 p +3 (3 + 3 ) 2 P = 12 p = 12 - 500 / 6.8 + 500 ( 28.8+ 24+200) 15 ( 15 ) 2 "MUM= 3 35.1)