Find the maximum or minimum value for the following problems. 1.) A man wishes t

Question

Find the maximum or minimum value for the following problems.

1.) A man wishes to fence in a rectangular plot lying next to a river. No fencing is required along the river bank. If he has 800m of fence and he wishes to maximize are to be fenced, find the dimensions of the desired enclosed plot.

Ans: 200m x 400m

2.) A rectangular box, open at the top, with a square base is to have a volume of 4000 cm3. Find the dimensions if the box is to contain the least amount of material.

Ans: 20cm x 20cm x 10cm

3.) Find the maximum slope of a tangent line to the curve y = 3x2 - 2x3.

Ans: m = 3/2 at x = 1/2

Explanation / Answer

1) 2l + b = 800

b = (800 - 2l)

Area = l*b = l*(800 - 2l)

A = 800l - 2l^2

For Area to be max,

A' = 0

800 - 4l = 0

l = 800/4 = 200

b = (800 - 2*200) = 400

Dimensions of the rectangle are 200m x 400m

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2) V = s^2 * h = 4000

h = 4000 / s^2

Surface Area A = 4hs + s^2

= 4s(4000/s^2) + s^2

= 16000/s + s^2

For A to be minimum,

A' = 0

-16000/s^2 + 2s = 0

s^3 = 8000

s = 20 cm

h = 4000 / 20^2 = 4000/400 = 10 xm

Dimensions of the box are 20cm x 20cm x 10 cm

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3) y = 3x^2 - 2x^3

Slope m = y' = 6x - 6x^2

For slope to be maximum,

m' = 0

6 - 12x = 0

x = 1/2

Slope m = 6(1/2) - (6 (1/2)^2 ) = 3 - 3/2 = 3/2

m = 3/2 at x = 1/2

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