Answer:
To determine the dimensions of a can that will maximize volume, we can use the following steps:
Step 1: Identify the variables:
Let's assume the dimensions of the can are the height (h) and the radius of the base (r). These are the variables that we need to find to maximize the volume.
Step 2: Set up the equation for the volume of the can:
The volume of a cylindrical can is given by the formula V = πr²h, where V is the volume, r is the radius, and h is the height.
Step 3: Set up the constraint equation:
The total surface area of the can is given as 600 cm². The surface area consists of the curved surface area and the area of the circular base. The curved surface area is given by A = 2πrh, and the area of the circular base is given by B = πr². The total surface area can be expressed as: A + B = 600 cm².
Step 4: Solve the constraint equation for h:
Since we want to express h in terms of r, we rearrange the constraint equation: h = (600 - πr²) / (2πr).
Step 5: Substitute the value of h into the equation for the volume:
Substituting the value of h into the volume equation, we get: V = πr²[(600 - πr²) / (2πr)].
Step 6: Simplify the equation for the volume:
Simplifying the equation further, we get: V = (600r - πr³) / 2.
Step 7: Find the derivative of V with respect to r:
To find the maximum volume, we take the derivative of V with respect to r and set it equal to zero, then solve for r.
dV/dr = (600 - 3πr²) / 2 = 0.
Step 8: Solve for r:
Solving the equation, we have:
600 - 3πr² = 0
3πr² = 600
r² = 200
r = √200 ≈ 14.14 cm.
Step 9: Find the corresponding value of h:
Using the constraint equation, we substitute the value of r into the equation for h:
h = (600 - πr²) / (2πr)
h = (600 - π(14.14)²) / (2π(14.14))
h ≈ 10 cm.
Therefore, the dimensions that will maximize the volume of the can are approximately:
- Radius (r) ≈ 14.14 cm
- Height (h) ≈ 10 cm.