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!
2pi r h = 600, so
pi rh = 300
pi r^2 h = pi rh * r = 300r = 600, so
r = 2
h = 150/pi
To find the radius and height needed to construct a tube with a volume of 600 in^3, we can use the formulas for the volume and surface area of a cylinder.
The volume of a cylinder is given by the formula V = πr^2h, where V is the volume, r is the radius, and h is the height.
Given that the volume of the cylinder is 600 in^3, we can set up the equation as follows:
600 = πr^2h (equation 1)
Next, we need to find the surface area of the thin rectangular sheet that will be used to create the tube. The surface area of the cylindrical tube is given by the formula S = 2πrh + 2πr^2, where S is the surface area.
Given that the area of the rectangular sheet is 600 in^2, we can set up the equation as follows:
600 = 2πrh + 2πr^2 (equation 2)
Now we have a system of equations (equations 1 and 2) that we can solve to find the values of r and h.
To solve the system of equations, we can use substitution or elimination.
Let's use the substitution method:
From equation 1: h = 600 / (πr^2)
Substituting h in equation 2, we get:
600 = 2πr * (600 / πr^2) + 2πr^2
600 = 1200 / r + 2πr^2 (simplifying)
600r = 1200 + 2πr^3 (multiplying both sides by r)
2πr^3 - 600r + 1200 = 0 (rearranging the equation)
This is a cubic equation, so we can solve it using numerical methods or approximations.
Using a numerical solver or calculator, we find that r ≈ 2.77 inches.
Substituting this value of r back into equation 1, we can solve for h:
h ≈ 600 / (π * (2.77)^2)
h ≈ 61.15 inches
So, the values are approximately:
A) r ≈ 2.77 inches
B) h ≈ 61.15 inches