while shopping for wine for his wedding, Kepler noticed that the price of a barrel of wine (here assumed to be a cylinder) was determined solely by the length d of a dipstick that was inserted diagonally through a hole in the top of the barrel to the edge of the base of the barrel. Kepler realized that this measurement does not determine the volume of the barrel and that for a fixed value of d, the volume varies with the radius r and height h of the barrel. for a fixed value d, what is the ratio r/h that maximizes the volume of the barrel?
To find the ratio r/h that maximizes the volume of the barrel, we need to consider the relationship between the variables involved. Let's break it down step by step:
1. Define the variables:
- d: The length of the dipstick (fixed value).
- r: The radius of the barrel.
- h: The height of the barrel.
- V: The volume of the barrel.
2. Establish the equation linking the variables:
We know that the dipstick is inserted diagonally from the top of the barrel to the edge of the base. Using the Pythagorean theorem, we can express the relationship between r, h, and d:
d² = r² + h²
3. Express the volume V in terms of r and h:
The volume of a cylinder is given by V = π * r² * h.
However, in this case, we need to express the volume in terms of the fixed value d instead of r and h. To do this, we can solve the equation obtained in step 2 for r:
r² = d² - h²
Substituting this value of r² into the volume equation, we have:
V = π * (d² - h²) * h
4. Identify the objective:
We want to find the ratio r/h that maximizes the volume of the barrel. In other words, we want to find the maximum value of V in terms of h, while considering the constraint that d is a fixed value.
5. Optimize the volume:
To find the maximum value of V, we can take the derivative of V with respect to h, set it equal to zero, and solve for h. This will give us the critical point(s) where the maximum occurs.
Differentiating V with respect to h:
dV/dh = π * (2d²h - 3h³)
Setting dV/dh equal to zero:
2d²h - 3h³ = 0
Factoring out an h from the equation:
h(2d² - 3h²) = 0
Setting each factor equal to zero:
h = 0 (Not a practical solution for the height of the barrel)
2d² - 3h² = 0
Solving the second equation for h:
2d² = 3h²
h² = (2d²) / 3
h = √((2d²) / 3)
6. Find the ratio r/h:
Now that we have the value of h, we can substitute it back into the equation obtained in step 2 to find r:
r² = d² - h²
Substituting the value of h, we have:
r² = d² - ((2d²) / 3)
r² = (3d² - 2d²) / 3
r² = d² / 3
r = √(d² / 3)
Finally, the ratio r/h is given by:
(r/h) = (√(d² / 3)) / (√((2d²) / 3))
(r/h) = √(d² / (2d²))
(r/h) = √(1 / 2) = 1 / √2 = √2 / 2
Therefore, for a fixed value of d, the ratio r/h that maximizes the volume of the barrel is √2 / 2.
To determine the ratio r/h that maximizes the volume of the barrel for a fixed value of d, we need to consider the relationship between the variables and find the maximum value.
Let's denote the radius of the barrel as "r" and the height of the barrel as "h".
The volume of a cylinder is given by the formula:
V = πr^2h
Since we are given that the measurement of the dipstick does not determine the volume, we can assume that the dipstick length, d, is constant.
From the information given, we can deduce that the dipstick's diagonal length is equal to the square root of (r^2 + h^2), based on the Pythagorean theorem.
Therefore, we have the equation:
d = √(r^2 + h^2)
To maximize the volume of the barrel, we need to express the volume formula as a single variable. So, solve the equation for h:
h^2 = d^2 - r^2
Substituting this into the volume equation, we have:
V = πr^2(d^2 - r^2)
Now, let's find the maximum value of V. To do this, we can take the derivative of V with respect to r and set it equal to zero:
dV/dr = 2πr(d^2 - 2r^2) = 0
Simplifying the equation, we have:
d^2 - 2r^2 = 0
Rearranging further, we get:
2r^2 = d^2
Dividing by 2, we find:
r^2 = d^2/2
Taking the square root of both sides, we get:
r = √(d^2/2)
Now, we can substitute this value of r back into the equation for h:
h^2 = d^2 - r^2
Substituting the above value of r, we have:
h^2 = d^2 - (d^2/2)
Simplifying further, we obtain:
h^2 = d^2/2
Finally, solving for h, we find:
h = √(d^2/2)
To determine the ratio r/h, we divide r by h:
r/h = (√(d^2/2)) / (√(d^2/2))
Simplifying, we get:
r/h = 1
Therefore, for a fixed value of d, the ratio r/h that maximizes the volume of the barrel is 1.