Some diffuicult Maxima and Minima Questions. (3U) (1 Viewer)

[currysauce]

1. An open rectangular gutter is being constructed. The cross sectional area is constant at A sq. m. The reduce friction, the area of the walls in contact must be least. Dimensions in picture are (L length, y Width and x Altitude)

(i) If the length of the gutter is L meters, show that the surface area S in contact is given by

S = 2AL/y + yL

(ii) Hence, show that for least friction the width of the gutter its twice its altitude.

2. A bioler is to be made from sheet metal. It is to be in the form of a cylinder bounded by a hemisphere at each end. The radius of the cylinder - and hence the hemispheres - is r meters and the height is h meters.

i) The surface area (S) is given by S=2(pi)rh + 4(pi)r^2. Show that the volume V of the boiler is given by V = rS /2 + 2(pi)r^3/3.

ii) Find the dimensions of the boiler (rad and height.) for th volume to be greatest.


3. In the following question, the pronumerals a and b are positive.

i) Find the turning point on the curve y=ax+b/x x>0

ii) Sketch this curve

iii) The line, y=c, lies wholly below the curve y =ax+b/x. Using ur graph or otherwise, prove that ab> c^2/4

 

[shafqat]

3s the one i can solve in the fewest no of lines
y'= a-bx^2
turning point when x = sqrt (b/a)
it is a min (u can check)
when x = sqrt (b/a), y = 2sqrt(ab) by substitution
as this is the min, and is always above y = c (given),
2sqrt(ab) > c
ab > c^2/4

 

[currysauce]

any ideas?

 

[maths > english]

my solution to the first ones attached