A cylinder has a height that is 3 times as large as it's radius. The lateral area of the cylinder is 54π square units. What is the length of the cylinder?
Show work.
(This is 10th grade, please don't make it advanced.)
Let's start by using the formula for the lateral surface area of a cylinder:
Lateral Surface Area = 2πrh
where r is the radius and h is the height.
We know that the lateral surface area is 54π, so:
54π = 2πr(3r) (since h is 3 times r)
54π = 6πr^2
Divide both sides by 6π:
9 = r^2
Take the square root of both sides:
r = 3
Now that we know the radius is 3 units, we can use the height equation to find the height:
h = 3r = 3(3) = 9
Therefore, the height of the cylinder is 9 units. To find the length (which is the same as the height in this case), we simply double the height:
Length = 2h = 2(9) = 18
So the length of the cylinder is 18 units.
To solve this problem, we'll use the formula for the lateral area of a cylinder:
Lateral Area = 2πrh
Given that the lateral area of the cylinder is 54π square units, we can set up the equation:
54π = 2πrh
To find the height of the cylinder, we should first determine the radius. Let's assume the radius is "r".
According to the given information, the height of the cylinder is 3 times the radius, which means the height is 3r.
Substituting the height and radius values into the equation, we have:
54π = 2πr(3r)
To simplify further, we can cancel out π from both sides:
54 = 2(3r)(r)
54 = 6r^2
Dividing both sides of the equation by 6:
r^2 = 9
Taking the square root of both sides:
r = 3
Now that we have the value of the radius, we can find the height by multiplying it by 3:
Height = 3r
Height = 3(3)
Height = 9 units
Therefore, the length of the cylinder is 9 units.