Please answer all A hot air balloon takes off from the edge of a plateau. Impose

Question

Please answer all

A hot air balloon takes off from the edge of a plateau. Impose a coordinate system as pictured below and assume that the path the balloon follows is the graph of the quadratic function y = f(x) = -4/2500 x^2 + 4/5 x. The land drops at a constant incline from the plateau at the rate of 1 vertical foot for each 5 horizontal feet. Answer the following questions. (a) What is the maximum height of the balloon above plateau level? (b) What is the maximum height of the balloon above ground level? (c) Where does the balloon land on the ground? (d) Where is the balloon 20 feet above the ground? (Round your answers to two decimal places.)

Explanation / Answer

y = -(4/2500)x^2 + (4/5)x

a) max. height of ballon above plateau

x = -(4/5 /- 8/2500) = (4*2500/40) = 250; x = 250 ; y = 100

b) slope of land , tantheta = 1/5 = y/250 ; y = 50 ft from horizontal line of plateau

So, max, height from plateau = 100+50 = 150 ft

c) ballon land on horizontal line joing plateau or take off ; y = 0 ;

0 = -(4/2500)x^2 + (4/5)x

x = 0, 500 ft

So, at 500 ft from take off point

landing point of ground = length of hypotenuse of rtight angled triangle

y/x = 1/5 ;   y/500 = 1/5 ; y = 100 ft

So, hypotenuse = sqrt(100^2 +500^2) = 509.90 ft

So, ballon lands on ground at 509.90 ft from take off point

d) 20 ft above ground means ; x/20 = 1/5 ; x = 4 ft

So find y at x= 5 ft ; y = -(4/2500)x^2 + (4/5)x

y = 3.96 ft

x = 4ft ; y = 3.96 ft

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