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Marina86 [1]
3 years ago
14

2 Construct a rational function that will help solve the problem. Then, use a calculator to answer the question.

Mathematics
1 answer:
vaieri [72.5K]3 years ago
7 0

Answer:

<u><em>Dimension of box:-</em></u>

Side of square base = 10 in

Height of box = 5 in

Minimum Surface area, S = 300 in²

Step-by-step explanation:

An open box with a square base is to have a volume of 500 cubic inches.

Let side of the base be x and height of the box is y

Volume of box = area of base × height

                 500=x^2y

Therefore, y=\dfrac{500}{x^2}

It is open box. The surface area of box, S .

S=x^2+4xy

Put  y=\dfrac{500}{x^2}

S(x)=x^2+\dfrac{2000}{x}

This would be rational function of surface area.

For maximum/minimum to differentiate S(x)

S'(x)=2x-\dfrac{2000}{x^2}

For critical point, S'(x)=0

2x-\dfrac{2000}{x^2}=0

x^3=1000

x=10

Put x = 10 into y=\dfrac{500}{x^2}

y = 5

Double derivative of S(x)

S''(x)=2+\dfrac{4000}{x^3} at x = 10

S''(10) > 0

Therefore, Surface is minimum at x = 10 inches

Minimum Surface area, S = 300 in²

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