Sections of cylindrical tubing are to be made from thin rectangular sheets that have an area of S = 600 in2. Find the radius r and height h needed to construct a tube that has a volume of 600 in3. (Round your answers to two decimal places.)
A) r= ? in
B) h= ? in
Thank you!
To find the radius (r) and height (h) needed to construct a tube with a volume of 600 in³, we can use the following formula:
Volume of a cylinder = π * r² * h
Step 1: Substitute the given volume (600 in³) into the formula:
600 = π * r² * h
Step 2: We also know that the area of the rectangular sheet is 600 in². The rectangular sheet can be rolled into the shape of the cylinder, so its area is equal to the lateral surface area of the cylinder:
Area of a cylinder = 2πrh
Since we have an expression for the volume in terms of r and h, we can substitute it into the area formula:
600 = 2πrh
Step 3: Now we have a system of two equations:
600 = π * r² * h (Equation 1)
600 = 2πrh (Equation 2)
From Equation 2, we can isolate h:
h = 600 / (2πr)
Step 4: Substitute this expression for h into Equation 1:
600 = π * r² * (600 / (2πr))
Simplify:
600 = π * r * 300
Divide both sides by π:
600 / π = r * 300
Simplify:
2r = (600 / π)
Divide both sides by 2:
r = (600 / (2π))
Simplify further:
r ≈ 47.75 in
Step 5: Substitute the value of r back into Equation 2 to find h:
600 = 2π * (47.75) * h
Simplify:
600 = 95πh
Divide both sides by 95π:
h = (600 / (95π))
Simplify further:
h ≈ 2.02 in
Therefore, the radius (r) is approximately 47.75 in and the height (h) is approximately 2.02 in.