the ratio between the radius of the base and height of a cylinder is 2:3.if its volume is 1617.find the total surface area of cylinder
r/h = 2/3 so h = 3r/2
v = πr^2h = 3π/2 r^3 = 1617
r = 7
so, h = 21/2
Now just find the area
a = 2πr(r+h)
To find the total surface area of a cylinder, we need to know both the radius and height of the cylinder. In this case, we are given the ratio between the radius and height, and the volume of the cylinder.
Let's assume that the radius of the base is 2x and the height is 3x, based on the given ratio.
The volume of a cylinder is given by the formula:
V = π * r^2 * h
where V is the volume, r is the radius, and h is the height. We are given that the volume is 1617, so we can substitute these values into the formula:
1617 = π * (2x)^2 * (3x)
Simplifying this equation:
1617 = 4πx^2 * 3x
1617 = 12πx^3
Now, we can solve for x by dividing both sides by 12π:
1617 / (12π) = x^3
x^3 = 13.61
Taking the cube root of both sides to solve for x:
x = ∛(13.61)
x ≈ 2.39
Now that we have the value of x, we can find the radius and height:
Radius (r) = 2x ≈ 2 * 2.39 ≈ 4.78
Height (h) = 3x ≈ 3 * 2.39 ≈ 7.18
Now that we have the radius and height of the cylinder, we can find the total surface area using the formula:
Total Surface Area = 2πr(r + h)
Substituting the values:
Total Surface Area = 2π * 4.78(4.78 + 7.18)
Calculating this:
Total Surface Area ≈ 2π * 4.78 * 11.96
Total Surface Area ≈ 360.92π
So, the total surface area of the cylinder is approximately 360.92π square units.