Question

An energy drink container in the shape of a right circular cylinder must have a volume of 19 fluid ounces (1 fluid ounce is approximately 1.80469 cubic inches). The cost per square inch of constructing the top and bottom is twice the cost of constructing the lateral side. Find the dimensions that will minimize the cost. (Round your answers to two decimal places.)

Answer 1

Answer:

the dimension that will minimize cost are radius of 2.79 inch and height of cylinder of 1.40 inch.

Step-by-step explanation:

volume of a circular cylinder = πr²h

volume of 19 fluid ounces (1 fluid ounce is approximately 1.80469 cubic inches) = 19* 1.80469 =34.28911 in³

The cost per square inch of constructing the top and bottom is twice the cost of constructing the lateral side, means

surface area of top + bottom = surface area of side

area of circle(top + bottom) = πr²+πr² = 2πr²

area of the side = 2πrh

the cost of top and bottom = twice the cost of side

2πr² = 2( 2πrh) =  4πrh

divide both side by  2πr

r=  4πrh/ 2πr =2h

r = 2h

for volume = πr²h = 34.28911 in³

and r = 2h

π(2h)²h =34.28911 in³

4πh³ = 34.28911 in³

h³ = 34.28911 /(4*3.142) = 2.7283 in³

h = cube root of 2.7283 in³ = 1.397 inch

r = 2*  1.76 = 2.794 inch

the dimension that will minimize cost are radius of 2.79 inch and height of cylinder of 1.40 inch.