express cost in terms of width

Question

the company manufactures another tent that is triangular prism but has a fixed volume of 1.8 cubic meters. the dimensions of this tent are such that the occupants can sleep either parallel and perpendicular to the triangular end. the sleeping orientation depends on the dimensions of the tent floor.

the floor of this tent is made of a material that is 1..5 times more expensive per square meter that the material used for the remainder of the tent. owing to construction costs the entrance of the tent is 1.2 times is more expensive per square meter that the remainder of the top section of the tent.

express C cost in terms of W width. sketch the graph of C verses W.

Answer 1

The volume and cost must be found in terms width. The volume = 1.8 = floor area x height x width/2. The floor area is length x width. Thus 1.8 = length x height x width^2/2 . length = 1.8 /(height x width^2/2) or

length = 0.9 / (height x width^2)

The top is the width x upper length. The upper length = 2 hypotenuse of the right triangle of width and height.

Cost = 1.5 floor area + 1.2 (front area) + top + back area

Cost = 1.5 ( length x width) + 1.2 ( width x height/2) + width x 2 sq rt ( length^2 + height^2)+ width x height /2

Cost = width( 1.5 length + 1.2/2 height x 2 sq rt (length^2 + height^2) + height /2)

Cost = width { 1.5 (0.9 /height x width ^ 2) +1.2 /2 height x 2 sq rt ( 0.9 / [(height x width ^2) + height^2]+ height /2}.

If you let height be 3 then the equation becomes ( people have to have room to move in and the less height the more floor space.)

Cost = width { 1.5( 0.9/3 width ^ 2) + 1.2 (3) /2 x 2 sq rt ( 0.9/[(3x width^2) + 9]+ 3/2}

Cost = width {( 0.45 width ^2) + (3.6 sq rt [( 0.3/ width ^2) + 9] ) + 1.5}