Question

1. An irrigation channel made of concrete is to have a cross section in the form of an isosceles trapezoid, three of whose sides are 4 feet long. How should the trapezoid be shaped if it is to have the maximum possible area?

2. A truck is to be driven 130 km at a constant speed of X km/hr . Speed laws require that 50 <= x <= 100. Assume that gasoline costs $2.90 per gallon and is consumed at the rate of 2 + [(x^2) / 360] gallons per hour. If the driver paid $15 dollars per hour, find the most economical speed and the most economical total cost for the trip.

Answer 1

are 4 x-4 1) Three sides the feet long each side be a feet 2-4 2 Let fourth bi Height ha v n?pet (2-4): 42 h. V 16 (2-972 of Trapezium - 놀 xhx [uta f(x) = x/16 - (2-4) 2 x [atu] Area 2 For Maximum Area f(n)=0 f'(a) = [(at) (V16-( dn ₃ / 10- (2-4)2 + (atu) 2x2 / 16 - (2-4) 2 *[-2 (ay) Ž V 16 (2-472 (2+4) (x-4) 816-(3-4)2 f'(x)=0 x²-16 = £ / 16- (2-4) ² 8/16 (2-4)2 4 (16-(Y) 2) = x2-16 " (16 (22 - 2x +16)] =x²-16 2 [-x² +82 +16]=x²-16 3x² - 160 -48=0 ) a= 7.474 ft. 7.474 ft 4. y fu YA

* Let the a km/h constant speed of truck= taken for the journey t = hr Time 130 2 Total x time consumed Gasoline Gasoline hour (2 + x 2 ) t gallons x Total Gasoline cost Total Gasoline Cost of Gasoline Gallon 12 + 22 t.(2.90) 360 Amount to be paid to driver = Time x 15 15t Driver's salary Total cost for Trip = Gasoline cost & ist (5.8+ 2912 3600) + + But t= 130 5.8 + 2an? Total Cost = 3600 3600 +15) (139) - 2704 t 377 a 360 -) fl(a)=0 2 cost For Minimum fald [2704 + 377 a 360 - ] 377 + - 2704 360 712 =) 7= f(n)=0 =) - 2704 +377 0 50.81 km/hr 360 cost = f(50.817= 2704 & 377x5o. 81 = $ 106.13 50.81 360 50.81 km/hr Economic Total cost $106.43 Minimum Economic Speed -