A 19,000 litre rainwater tank is to be built in the shape of a cylinder. It must

Question

A 19,000 litre rainwater tank is to be built in the shape of a cylinder. It must be at most 3 metres tall, and have radius at most 2 metres. Its base costs 20 dollars per square metre, and the sides and top cost 10 per square metre. What are the cheapest and most expensive shapes that satisfy these conditions? (a) Give the total cost as a function of the height H and radius R. Use units of metres for both H and R. Note that it is case sensitive - you must use exactly the variable names given in the question (b) Give an equation that connects the height H and radius R. Your answer must be an equation includingRemember to use consistent units.

Explanation / Answer

The shape of tank is a cylinder. Let say height of the cylinder is H and Radius is R.

Base Cost= $ 20 / m2 ;

Side and top cost = $ 10 / m2

Total Cost = Base cost + tope base cost + side cost =

TC = 20 * R2 + 10 * R2  + 10 * 2 RH = 10(3R2 + 2RH)

Here Constraints are

H 3 mt. and R 2 mt.

(b) Here the Volume of the tank is given, which is

V = 19000 litre rainwater = 19m3

R2 H = 19

(c) R2 H = 19

H = 19/ R2

Putting this value in cost function

TC = 10(3R2 + 2RH) = 10 (3R2 + 2 * R * 19/ R2)

TC = 10 (3R2 + 38/R)

(d) To find the local minima, we have to differentiate it with respect to R

d(TC)/ dR = 10 [6R - 38/R2]

it will be zero for minima

so 6R3 = 38

=> R = 1.2633 mt.= 126.33 cm

(e) For H = 3 metres

Rpermissible = (19/ H)0.5 = 1.4198 = 141.98 cm

so Rminima < Rpermissible

(f) Largest allowable radius R = 2 meter

(g) Cheapest cost when R = 1.4198 meter and H = 3 meter

TC = 10 [3R2 + 2RH] = 10 [ 3 * (1.4298)2 + 2 * 1.4298 * 3]

TC = $ 462.09

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