A cylindrical can with both a top and bottom has the same surface area as a cube with edge 50mm. The height of the can if 6cm. Find the radius of the can to the nearest thousandth.
To solve this problem, we can start by finding the surface area of both the cylindrical can and the cube.
The surface area of the cylindrical can can be computed using the formula:
Surface Area of a Cylinder = 2πrh + 2πr^2,
where r is the radius of the circular ends of the cylinder and h is the height.
The surface area of the cube can be found using the formula:
Surface Area of a Cube = 6s^2,
where s is the length of the sides of the cube.
Given that the surface area of the cylindrical can is equal to that of the cube, we can set up the following equation:
2πrh + 2πr^2 = 6s^2.
Now, let's substitute the values we know into the equation and solve for the radius r.
Given:
h = 6 cm,
s = 50 mm = 5 cm (since 1 cm = 10 mm).
Substituting these values into the equation, we have:
2πr * 6 + 2πr^2 = 6 * (5^2)
12πr + 2πr^2 = 6 * 25
2πr^2 + 12πr - 6 * 25 = 0
To simplify the equation, divide all terms by 2:
πr^2 + 6πr - 75 = 0
Now, we have a quadratic equation. We can solve this using either factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:
r = (-b ± √(b^2 - 4ac)) / (2a).
For our equation, a = 1, b = 6π, and c = -75. Substituting these values:
r = (-(6π) ± √((6π)^2 - 4(1)(-75))) / (2(1))
r = (-6π ± √(36π^2 + 300)) / 2
r = (-6π ± √(36π^2 + 300)) / 2
r = -3π ± √(9π^2 + 75)
To find the radius, we need to evaluate -3π + √(9π^2 + 75). Since we are asked to round to the nearest thousandth, we can use a calculator to find a decimal approximation for the radius:
r ≈ -3π + √(9π^2 + 75) ≈ 0.787.
Hence, the radius of the cylindrical can, rounded to the nearest thousandth, is approximately 0.787.