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某兴趣小组准备将一棱长为

a

的正方体木块打磨成圆锥，则圆锥的最大体积为

Mar 8, 2024

某兴趣小组准备将一棱长为

a

的正方体木块打磨成圆锥，则圆锥的最大体积为

Solution by Steps

step 1

Consider a cone inscribed in a cube with side length

a

. The cone's height

h

and base radius

r

will be related to the cube's side length

step 2

The volume

V

of a cone is given by the formula

V = \frac{1}{3}\pi r^2 h

step 3

To maximize the volume, we need to express

V

in terms of a single variable. Since the cone is inscribed in the cube, the height

h

plus twice the radius

2r

equals the side length

a

of the cube, so

h = a - 2r

step 4

Substitute

h

into the volume formula to get

V = \frac{1}{3}\pi r^2 (a - 2r)

step 5

To find the maximum volume, take the derivative of

V

with respect to

r

and set it equal to zero:

\frac{dV}{dr} = \frac{1}{3}\pi (2ra - 6r^2)

step 6

Solve

\frac{dV}{dr} = 0

for

r

to find the critical points

step 7

The critical point that maximizes the volume occurs when

r = \frac{a}{3}

step 8

Substitute

r = \frac{a}{3}

back into the expression for

h

to find

h = a - 2(\frac{a}{3}) = \frac{a}{3}

step 9

Now substitute

r = \frac{a}{3}

and

h = \frac{a}{3}

into the volume formula to find the maximum volume:

V_{max} = \frac{1}{3}\pi (\frac{a}{3})^2 (\frac{a}{3})

step 10

Simplify the expression for

V_{max}

to get

V_{max} = \frac{1}{3}\pi \frac{a^3}{27} = \frac{\pi a^3}{81}

Answer

V_{max} = \frac{\pi a^3}{81}

Key Concept

Maximizing the volume of a cone inscribed in a cube

Explanation

The maximum volume of a cone inscribed in a cube is found by expressing the volume in terms of one variable, taking the derivative, finding the critical points, and then substituting back to find the maximum volume.