a cylindrical cup is made so that the sum of its diameter and its height is 25 cm. find the dimensions of the cup so that the surface area of the cup is a maximum. The answers are. D= 50/3. H=25/3. But how do u get that
To find the dimensions of the cup that will maximize its surface area, we need to use calculus.
Let's assume that the diameter of the cup is D and the height of the cup is H. The sum of the diameter and height is given as 25 cm, so we can write an equation: D + H = 25.
The surface area of a cylinder is given by the formula: A = 2πr(r+h), where r is the radius and h is the height. Since the diameter is twice the radius, we can express the radius as D/2.
Now we need to express the surface area in terms of a single variable, so we substitute r = D/2 into the surface area formula:
A = 2π(D/2)((D/2) + H)
= πD(D + 2H)/2
= (π/2)D^2 + πDH
Now, we need to find the maximum value of A. To do that, we differentiate A with respect to D and set it equal to zero:
dA/dD = (π/2)2D + πH = 0
Simplifying the equation, we get:
πD + 2πH = 0
πD = -2πH
D = -2H
Since D must be positive, we discard the negative solution.
D = 2H
We also know that D + H = 25, substituting D = 2H into this equation:
2H + H = 25
3H = 25
H = 25/3
Substituting H = 25/3 back into D = 2H:
D = 2(25/3)
D = 50/3
Therefore, the dimensions that maximize the surface area of the cup are D = 50/3 cm and H = 25/3 cm.
To find the dimensions of the cylindrical cup that maximize its surface area, we can use calculus. The surface area of a cylinder is given by the formula:
S = 2πr^2 + 2πrh,
Where:
S is the surface area,
r is the radius of the base, and
h is the height of the cylinder.
Given that the sum of the diameter and height of the cylinder is 25 cm, we can write:
d + h = 25,
Since the diameter (d) is twice the radius (r), we have:
2r + h = 25,
Solving this equation for h, we get:
h = 25 - 2r.
Now, substitute this expression for h into the surface area formula:
S = 2πr^2 + 2πr(25 - 2r).
Simplifying, we have:
S = 2πr^2 + 50πr - 4πr^2.
To find the value of r that maximizes the surface area, we need to differentiate the surface area formula with respect to r and set it equal to zero:
dS/dr = 4πr + 50π - 8πr = 0.
Simplifying, we get:
-4πr + 50π = 0,
4πr = 50π,
r = 50/4,
r = 25/2.
Substituting this value of r back into the equation d + h = 25, we can solve for h:
2(25/2) + h = 25,
25 + h = 25,
h = 25/3.
Thus, the dimensions of the cup that maximize its surface area are:
Diameter (D) = 2r = 2(25/2) = 25 cm,
Height (H) = h = 25/3 cm.
Therefore, the answers are D = 50/3 cm and H = 25/3 cm.
2r+h = 25
a = πr(r+2h) = πr(r+25-2r) = π(25r-3r^2)
da/dr = 0 when r = 25/3
d = 2r = 50/3
h = 25 - 50/3 = 25/3