A closed cylindrical metal container has a total surface area of 54\pi\text{ cm}^2. What r… — JAMB UTME

A closed cylindrical metal container has a total surface area of $54\pi\text{ cm}^2$ . What radius, in centimeters, of the circular base will yield the maximum volume for the container?

Cevap: 3 cm

Cevap

The radius of the circular base that maximizes the volume is 3 cm.

To find the radius that yields maximum volume, we first express height

h

in terms of radius

r

using the total surface area formula

2\pi r^2 + 2\pi rh = 54\pi

, giving

h = \frac{27 - r^2}{r}

. Substituting this into the volume equation

V = \pi r^2 h

gives

V(r) = 27\pi r - \pi r^3

. Setting the first derivative

\frac{dV}{dr} = 27\pi - 3\pi r^2

to zero yields

3\pi r^2 = 27\pi

, so

r^2 = 9

and

r = 3\text{ cm}

. The second derivative

\frac{d^2V}{dr^2} = -6\pi r

evaluated at

r = 3

-18\pi

, which is strictly negative, confirming that

r = 3\text{ cm}

maximizes volume.

Adım Adım Çözüm

Set up the surface area equation for a closed cylinder with the given value.

2\pi r^2 + 2\pi r h = 54\pi

A closed cylinder consists of two circular bases (

2\pi r^2

) and a curved lateral surface (

2\pi r h

Express

h

in terms of

r

h = \frac{27 - r^2}{r}

Dividing the surface area equation by

2\pi

yields

r^2 + rh = 27

, allowing

h

to be isolated.

Substitute

h

into the volume formula

V = \pi r^2 h

to write volume as a function of

r

only.

V(r) = 27\pi r - \pi r^3

To maximize volume using calculus, the volume equation must be expressed in terms of a single variable.

Differentiate

V(r)

with respect to

r

and set the derivative equal to zero to find stationary points.

\frac{dV}{dr} = 27\pi - 3\pi r^2 = 0 \implies r = 3

Maximum volume occurs at a stationary point where the first derivative is zero.

Verify that

r = 3

produces a maximum using the second derivative test.

\frac{d^2V}{dr^2} = -6\pi r

; at

r = 3

\frac{d^2V}{dr^2} = -18\pi < 0

A negative second derivative indicates a local maximum.

Anahtar Kavram

Optimization and Stationary Points in Mensuration

Tahmini Süre:2m 0s

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