A cylinder has a radius and height that are whole numbers (measured in cm), and a volume of 36 x π cm². What is the smallest surface area it can have?
Let's start by using the formula for the volume of a cylinder:
V = πr²h
We know that the volume is 36π, so we can write:
36π = πr²h
Simplifying this equation, we can divide both sides by π:
36 = r²h
We want to find the smallest surface area, which means we want to minimize the sum of the lateral area (the curved part of the cylinder) and the area of the two bases (top and bottom). We can write this as:
A = 2πrh + 2πr²
We can use the equation we found for the volume to eliminate one of the variables (either r or h) from this expression. Let's solve for h:
h = 36/r²
Now we can substitute this expression for h into the formula for the surface area, and we get:
A = 2πr(36/r²) + 2πr²
Simplifying this expression:
A = 72/r + 2πr²
To find the minimum value of this expression, we can take the derivative with respect to r and set it equal to zero:
dA/dr = -72/r² + 4πr
0 = -72/r² + 4πr
72/r² = 4πr
r³ = 18/π
r ≈ 1.7109
Now we can use this value of r to find the corresponding value of h:
h = 36/r² ≈ 12.305
Finally, we can plug these values of r and h into the formula for the surface area:
A = 2πrh + 2πr²
A ≈ 68.001 cm²
Therefore, the smallest surface area the cylinder can have is approximately 68.001 cm² when the radius is about 1.7109 cm and the height is about 12.305 cm.
During medieval times Europe's major cities would most likely be located near which of the following?
A.
mountain ranges
B.
deserts
C.
forests
D.
rivers
D. Rivers.
Which Germanic tribe defeated the last Roman army in Western Europe?
A.
the Jutes
B.
the Saxons
C.
the Huns
D.
the Franks
B. The Saxons.
What was one step that Charlemagne took to help unite his kingdom?
A.
He chose a single strong heir to inherit his throne.
B.
He drove Muslim forces out of Europe.
C.
He created and supported schools.
D.
He declared himself head of the Catholic Church.
C. He created and supported schools.
How did Pope Leo III's crowning of Charlemagne as emperor of the Romans symbolize an increase in the Church's power?
A.
It suggested that emperors have the right to choose the pope.
B.
It suggested that the pope had the power to choose an emperor.
C.
It gave the pope the authority to rule the empire.
D.
It made the pope part of the emperor's court.