You are ok up to your last line:
[(12-h)/2]^2 = (144-24h+h^2)/4
so,
v = pi/12 (144h-24h^2+h^3))
dv/dh = pi/12(144-48h+3h^2)
dv/dh=0 when h=4,12
Naturally, v=0 when h=12, so
v(4)=64pi/3
my answer :
R = radius of bigger cone = 6 ft
r = radius of smaller cone…?
H = height of bigger cone = 12 ft
h = height of smaller cone?
r/R = H-h/H
r/6 = 12-h/12
r = 12-h/2
V = 1/3 r^2h
= 1/3 (12-h/2)^2 h
= 1/3 (144-h^2/4) h
Please help me to complete this assignment…
[(12-h)/2]^2 = (144-24h+h^2)/4
so,
v = pi/12 (144h-24h^2+h^3))
dv/dh = pi/12(144-48h+3h^2)
dv/dh=0 when h=4,12
Naturally, v=0 when h=12, so
v(4)=64pi/3
h=4,12
To maximize the smaller cone's volume, we can find the maximum by taking the derivative of the volume equation with respect to h and setting it equal to zero. Let's get started!
V = 1/3 π (144 - h^2/4) h
To find the derivative, we can apply the product rule and chain rule:
dV/dh = 1/3 π [(144 - h^2/4) * 1 + h * (-h/2) / 4]
Now, let's simplify this:
dV/dh = 1/3 π [144 - h^2/4 - h^2/8]
Combine like terms:
dV/dh = 1/3 π [144 - (3h^2)/8]
Now, let's set this derivative equal to zero and solve for h:
1/3 π [144 - (3h^2)/8] = 0
144 - (3h^2)/8 = 0
Multiply both sides by 8:
1152 - 3h^2 = 0
Rearrange the equation:
3h^2 = 1152
Divide both sides by 3:
h^2 = 384
Now, take the square root of both sides:
h = √384
Now, let's solve for the radius of the smaller cone (r). We have r = 12 - h/2.
Substitute the value of h:
r = 12 - √384/2
And that's it! You have the height (h) and base radius (r) of the smaller cone that maximize its volume.
But don't worry, if math isn't your thing, you can always ask the clowns to help with this upside-down cone party! 🤡🎉
First, let's differentiate the volume formula with respect to h:
dV/dh = (1/3π) * (144 - h^2/4) * 1 - (2h/4π) * (12 - h/2) = 48h - h^3/6π - h(12 - (h/2))/2π
Next, set the derivative equal to zero and solve for h:
48h - h^3/6π - h(12 - (h/2))/2π = 0
Multiplying everything by 6π to get rid of the denominators:
288πh - h^3 - 3h(12 - (h/2)) = 0
288πh - h^3 - 36h + 3h^2/2 = 0
Rearranging terms:
h^3 - 3h^2/2 + 252h - 576πh = 0
To solve this equation numerically, you can use numerical methods such as Newton's method or use software like Wolfram Alpha or Excel's Solver function.
Once you find the value(s) of h that maximize the volume, substitute the value(s) back into the equation r = 12 - h/2 to find the corresponding value(s) of r.