Application of Derivatives

Question

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m^3. If building of tank costs Rs. 70 per sq. metres for the base and Rs. 45 per square metre for sides. What is the cost of least expensive tank ?

Solution

Let x metres be length and y metres be breadth of base.
Now length of tank = 2 metres
Let V be volume of tank

therefore space space space space space straight V space equals 2 xy

But

straight V space equals space 8 space straight m cubed

(given)

therefore space space 2 xy space equals 8 space space space space rightwards double arrow space space space xy space equals 4 space space space space rightwards double arrow space space space straight y space equals space 4 over straight x

...(1)
Area of base = xy = 4m²

open square brackets because space of space left parenthesis 1 right parenthesis close square brackets

Area of four sides =

left parenthesis 2 straight x plus 2 straight y plus 2 straight x plus 2 straight y right parenthesis straight m squared space equals space left parenthesis 4 straight x plus 4 straight y right parenthesis straight m squared space equals space 4 left parenthesis straight x plus straight y right parenthesis straight m squared space space space space space space space space space space space space space space space space space

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open square brackets because space space of space left parenthesis 1 right parenthesis close square brackets

Let C be the total cost of building tank.

therefore space space space space straight C space equals space Rs. space open square brackets 4 cross times 70 plus 4 open parentheses straight x plus 4 over straight x close parentheses cross times 45 close square brackets space equals space Rs. space open square brackets 280 plus 180 space open parentheses straight x plus 4 over straight x close parentheses close square brackets.

therefore space space dC over dx space equals space 0 plus 180 space open parentheses 1 minus 4 over straight x squared close parentheses space equals space 180 space open parentheses 1 minus 4 over straight x squared close parentheses

space dC over dx equals 0 space space space space space space space space space space space space space rightwards double arrow space space space 180 space open parentheses 1 minus 4 over straight x squared close parentheses space equals space 0 space space space space space space space rightwards double arrow space space space 1 minus 4 over straight x squared space equals 0

therefore space space space straight x squared minus 4 space equals space 0 space space space space rightwards double arrow space space straight x squared space equals space 4 space space space space space rightwards double arrow space space space straight x space equals space minus 2 comma space space 2

Rejecting x = -2 as x cannot be negative, we get x =2

fraction numerator straight d squared straight C over denominator dx squared end fraction space equals space 180 space open parentheses 0 plus 8 over straight x cubed close parentheses space equals space 1440 over straight x cubed

When

straight x space equals 2 comma space space fraction numerator straight d squared straight C over denominator dx squared end fraction space equals 1440 over 8 greater than 0

therefore space space space straight C space is space least space when space straight x space equals space 2 therefore space space space space space space space least space cost space equals space 280 plus 180 open parentheses 2 plus 4 over 2 close parentheses space equals space 280 plus 180 cross times 4 space equals space 280 plus 720 space equals space Rs. space 1000

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Application Of Derivatives

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m^3. If building of tank costs Rs. 70 per sq. metres for the base and Rs. 45 per square metre for sides. What is the cost of least expensive tank ?

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

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For Daily Free Study Material Join wiredfaculty

Application Of Derivatives

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m3. If building of tank costs Rs. 70 per sq. metres for the base and Rs. 45 per square metre for sides. What is the cost of least expensive tank ?

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m^3. If building of tank costs Rs. 70 per sq. metres for the base and Rs. 45 per square metre for sides. What is the cost of least expensive tank ?