How To Roof My House

Find the minimum cost of construction (application of differentiation)?

A container has a square base, and 4 sides made of concrete, and an aluminium roof.
The container has a volume of 270m^3.
If the roof costs twice as much per unit area as the concrete, find the dimensions of the tank which would minimize construction cost.
Please explain/show working =S

Let side length of base = x

and let the height of the container = h

Area of concrete base = x²

Total area of the four concrete sides = 4xh

area of aluminum roof = x²

Now let construction cost = C

Let the cost of concrete = 1 unit

so cost of aluminum = 2 units

and C = 2x² + x² + 4xh

C = 3x² + 4xh … [eqn 1]

Now have to get C in terms of just 1 variable:

V = x²h

so x²h = 270

h = 270/x² [eqn 2]

subs [eqn 2] into [eqn 1] → C = 3x² + 4x * 270/x²

C = 3x² + 1080/x

C’ = 6x – 1080/x² and C” = 6 + 2160/x³

C’ = 0 when 6x – 1080/x² = 0

6x³ = 1080

x³ = 180

x = ³√180

when x = ³√180, C” > 0 … so local min when x = ³√180

subs x = ³√180 into [eqn 2] → h = 270/[180^(2/3)]

so the dimensions for minimum cost are:

base side length = ³√180 m … (approx 5.65 m to 2 dec pl)

and height = 270/[180^(2/3)] m … (approx 8.47 m to 2 dec pl)

Roof Dormer Costs – Building Design Money Saving Tips

Calculated Industries 4065 Construction Master Pro Advanced Construction Math Calculator $79.95 Advanced construction/math calculator is packed with solutions for building professionals. Calculates sheets of building material needed, solutions for stairs, roofing materials, compound miter cut angles, materials estimates and more. Also features cost per unit, multiple memory, exponential notation, backspace, user preferences. Stores last 20 entries. Standard calculations and auto shutoff. Per…

Lomanco BIB-12 MILL Whirlybird Turbine Ventilator $52.06 Featuring permanently lubricated upper and lower ball bearings, the Lomanco 12-inch whirlybird turbine ventilator ensures long life and no maintenance. It reduces winter ice build-up, as well as energy bills. Made with all-aluminum, rust-free construction, this internally braced turbine ventilator withstands winds up to 147 MPH. With a rigid, spider-type structure, it’s riveted at every connection…

Uni-Solar PVL-68 PowerBond PVL 68 Watt 12 Volt 112-Inch x 15.5-Inch Flexible Solar Panel $179.00 Grade A with 20 years warranty. UniSolar flexible solar panel. simple & easy peel and stick installation. Ideal for metal roof. RV /Boat, Home & Commercial applications. Battery charging or Grid tie Solar Systems. UniSolar is one of the largest manufacturer of Flexible Thin Film Solar Panels. Now with MC4 connectors. Made in USA….

Gloss BLACK VINYL ROOF overlay decal car wrap 4ftx6ft $68.99 Qty: 1 Decal Size: 72 inches in length x 48 inches in height Black Vinyl Roof Overlay with Rain Gutter Strip Overlay! The rain strips come precut to cover the rain gutter strips on the roof. The precut strips for the gutters allow a much cleaner and professional installation! Size: 4ft x 6ft roof overlay and 2 strips that are 1.5″ x 6ft (email me for other sizes!) Color: Gloss Black We only suppl…

Roofing Construction & Estimating $23.40 Editorial Reviews Product Description Installation, repair and estimating for nearly every type of roof covering available today in residential and commercial structures: asphalt shingles, roll roofing, wood shingles and shakes, clay tile, slate, metal, built-up, and elastomeric. Covers sheathing and underlayment techniques, as well as secrets for installing leakproof valleys. Many estimating tips…