Math questions ?

Question

1) A cylindrical container has a volume of 759 cm³.

a) Determine the dimensions of the container with minimum surface area.

b)Determine the minimum surface area.

How do I determine the dimensions and determine the minimum surface area??

Anything helps, and is greatly appreciated!!

Answer 1

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Starting with the equation of the volume of a cylinder:

V = πr²h

And we are told this is fixed at 759 cm³:

759 = πr²h

Given that, we are then asked to find the dimensions of the container with the minimum surface area.  So the next thing I'll do is put the general form for the surface area below:

A = 2πr² + 2πrh

Two bases (πr²) and a rectangle with the height (h) and circumference as the width (2πr)

If we solve the first equation for h in terms of r we can substitute it into the area equation:

759 = πr²h

759 / (πr²) = h

A = 2πr² + 2πrh

A = 2πr² + 2πr * 759 / (πr²)

Some of the factors will cancel out of the second term:

A = 2πr² + 2 * 759 / r

A = 2πr² + 1518 / r

We can now find the "r" that makes "A" a minimum by solving for the zero of the first derivative:

A' = 4πr - 1518 / r²

0 = 4πr - 1518 / r²

Multiply both sides by r²:

0 = 4πr³ - 1518

1518 = 4πr³

759 / (2π) = r³

cube root of both sides:

r = ∛[759 / (2π)]

If we rationalize the denominator by multiplying both halves by ∛(4π²):

r = ∛[3036π² / (8π³)]

r = ∛(3036π²) / (2π)

Now that we have a value for "r" we can find the value for "h":

h = 759 / (πr²)

Using the value found before we rationalized the denominator:

h = 759 / {π ∛[759 / (2π)]²}

Rationalizing this denominator by multiplying both halves by ∛[759 / (2π)]:

h = 759 ∛[759 / (2π)] / {π ∛[759 / (2π)]³}

h = 759 ∛[759 / (2π)] / [π * 759 / (2π)]

h = 759 ∛[759 / (2π)] / (759 / 2)

h = 759 ∛[759 / (2π)] * (2 / 759)

h = 2 ∛[759 / (2π)]

This is twice the value found for the radius so we can use the same steps to rationalize the denominator one more time to get:

h = 2 * ∛(3036π²) / (2π)

h = ∛(3036π²) / π

The exact values of the height and radius are:

h = ∛(3036π²) / π cm and r = ∛(3036π²) / (2π)

Which are approximately equal to:

h = 9.887 cm and r = 4.943 cm (rounded to 3DP)

The value of this surface area is:

A = 2πr² + 1518 / r

A = 2π∛[759 / (2π)]² + 1518 / ∛[759 / (2π)]

simplifying and rationalizing denominators:

A = 2π∛[759² / (4π²)] + 1518 ∛[759 / (2π)]² / ∛[759 / (2π)]³

A = 2π∛[759² * 2π / (8π³)] + 1518 ∛[759 / (2π)]² / [759 / (2π)]

A = 2π∛(576081 * 2π) / (2π) + 1518 ∛[759 / (2π)]² * 2π / 759

A = ∛(1152162π) + 2 ∛[759 / (2π)]² * 2π

A = ∛(1152162π) + 4π ∛[759 / (2π)]²

A = ∛(1152162π) + 4π ∛[759² / (4π²)]

A = ∛(1152162π) + 4π ∛[759² * 2π / (8π³)]

A = ∛(1152162π) + 4π ∛(576081 * 2π) / (2π)

A = ∛(1152162π) + 2∛(1152162π)

A = 3∛(1152162π) cm²

Which is approximately:

A = 460.620 cm² (rounded to 3DP)

Testing this with the decimal approximation for r:

A = 2πr² + 1518 / r

A = 2π(4.943)² + 1518 / 4.943

A = 48.866498π + 307.10095

A = 153.51863 + 307.10095

A = 460.620 cm³ (rounded to 3DP)

So this is the correct answer.

Again, the exact and approximate answers to this question are:

h = ∛(3036π²) / π cm and r = ∛(3036π²) / (2π)

h = 9.887 cm and r = 4.943 cm (rounded to 3DP)

A = 3∛(1152162π) cm²

A = 460.620 cm³ (rounded to 3DP)

Answer 2

r = radius, h = height of cylinder, C = circumference of circle, d = diameter of circle

Volume of a cylinder (VoC) is (pi * r^2 * h) = 759

Surface area (SA) of a cylinder is (2 * pi * r^2) + (h * C)

C = pi * d and d = 2r

So SA = (2 * pi * r^2) + (h * pi * r * 2)

Divide both sides of the VoC equation by pi you get hr^2 = 759/pi