A solid cylinder has a volume of 25π cm3. Determine its least possible surface area.
(a) 105.1 cm2
(b) 101.5 cm2
(c) 101.5 m2
(d) 105.1 m2
Answer (2)
Establish the relationship between the radius r and height h using the volume formula: V = π r 2 h = 25 π , leading to h = r 2 25 .
Express the surface area A in terms of r : A = 2 π r 2 + 2 π r h = 2 π r 2 + r 50 π .
Minimize A by finding its derivative with respect to r and setting it to zero: d r d A = 4 π r − r 2 50 π = 0 , which gives r = 3 2 25 .
Calculate the minimum surface area using the optimized radius: A = 6 π ( 2 25 ) 2/3 ≈ 101.5 cm 2 .
Explanation
Problem Introduction We are given a solid cylinder with a volume of 25 π cm 3 , and we want to find the least possible surface area.
Volume and Height Relation Let r be the radius and h be the height of the cylinder. The volume V is given by V = π r 2 h = 25 π . Therefore, we have the relation r 2 h = 25 , which implies h = r 2 25 .
Surface Area Formula The surface area A of the cylinder is given by A = 2 π r 2 + 2 π r h . Substituting h = r 2 25 into the surface area formula, we get
A = 2 π r 2 + 2 π r ( r 2 25 ) = 2 π r 2 + r 50 π .
Finding Critical Points To minimize the surface area, we need to find the critical points by taking the derivative of A with respect to r and setting it equal to zero:
d r d A = 4 π r − r 2 50 π .
Setting d r d A = 0 , we have
4 π r − r 2 50 π = 0 ⇒ 4 π r = r 2 50 π ⇒ 4 r 3 = 50 ⇒ r 3 = 4 50 = 2 25 .
Thus, r = 3 2 25 .
Calculating Height Now we find the corresponding height h :
h = r 2 25 = ( 2 25 ) 2/3 25 = 25 ⋅ ( 25 2 ) 2/3 = 2 2/3 ⋅ 2 5 1/3 = 2 ⋅ ( 2 25 ) 1/3 = 2 r .
Calculating Minimum Surface Area Substitute the value of r back into the surface area formula to find the minimum surface area:
A = 2 π r 2 + r 50 π = 2 π ( 2 25 ) 2/3 + ( 2 25 ) 1/3 50 π = 2 π ( 2 25 ) 2/3 + 50 π ( 25 2 ) 1/3 .
We can simplify this as:
A = 2 π ( 2 25 ) 2/3 + 50 π 2 5 1/3 2 1/3 = 2 π ( 2 25 ) 2/3 + 50 π 2 5 1/3 2 1/3 = 2 π ( 2 25 ) 2/3 + 25 π ⋅ 2 ⋅ ( 25 2 ) 1/3 .
A = 2 π ( 2 25 ) 2/3 + 2 π ⋅ 25 ⋅ ( 25 2 ) 1/3 = 2 π ( 2 25 ) 2/3 + 2 π ( 2 5 1/3 25 ⋅ 2 ) = 2 π ( 2 25 ) 2/3 + 2 π ( 2 ⋅ 2 5 2/3 ) .
A = 2 π ( 2 25 ) 2/3 + 4 π ( 2 25 ) 2/3 = 6 π ( 2 25 ) 2/3 .
Using a calculator, we find that A ≈ 101.525 cm 2 .
Final Answer The least possible surface area is approximately 101.5 cm 2 . Therefore, the answer is (b).
Examples
Cylinders are commonly used in various engineering applications, such as designing pressure vessels or storage tanks. Determining the minimum surface area for a given volume is crucial to minimize material usage and cost. For example, when designing a cylindrical propane tank with a fixed volume, engineers aim to minimize the surface area to reduce the amount of steel required, thereby lowering production costs and improving efficiency. This optimization problem ensures that the tank is both cost-effective and structurally sound.
The least possible surface area of a solid cylinder with a volume of 25π cm³ is approximately 101.5 cm². This is achieved by optimizing the dimensions of the cylinder using calculus. The chosen answer is (b) 101.5 cm².
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