I have no idea how to solve this question.Help please!A silo is in the shape of a cylinder with a hemispherical cap. Determine the radius r of the silo if the height of the whole silo is 25 ft and the volume is 935 ft^3. Notice the height is not just the cylinder but also includes the hemisphere on topI just need the steps to find the radius
Well, I can point you in the right direction. Treat this problem as 2 parts...the volume of the cylinder and the volume of the hemisphere (1/2 x the volume of a full sphere). Then just plug in R for the radius (variable since it's unknown right now), 25 for the height and 935 for V, and solve for R like: Volume_Cylinder(R) + Volume_Hemisphere(R) = 935.
First a reminder of some formula for calculating the volumes of objects: Vcylinder = h * pi * r^2 Vsphere = 4/3 * pi * r^3 Now we know that we only have half a sphere, so we can change the 4/3 on the front to 2/3 right away: Vhemisphere = 2/3 * pi * r^3 For the cylinder, we can't simply plug in the height of the silo for h, since the height also includes the hemisphere on top. BUT we know that the hemisphere adds exactly r to the height of the cylinder, so the formula becomes: Vcylinder = (h-r) * pi * r^2 Vcylinder = (25-r) * pi * r^2 Vcylinder = 25pi * r^2 - pi * r^3 We also know that: Vtotal = Vcylinder + Vhemisphere 935 = 25pi * r^2 - pi * r^3 + 2/3 * pi * r^3 935 = 25pi * r^2 - 1/3 * pi * r^3 2805/pi = 75r^2 - r^3 r^3 - 75r^2 + 2805/pi = 0 Now there's a cubic to be solved, it is at this point I cheated and used an online cubic solver to get the answer r = 3.53463 feet to two significant figures. The other solutions to the cubic are 74.8 and -3.4, but these are clearly incorrect. If you plug 3.53463 feet in to check, you get: Vcylinder = (25-r) * pi * r^2 Vcylinder = (25-3.53463) * pi * 3.53463^2 Vcylinder = 21.46537 * pi * 12.4936092369 Vcylinder = 842.512 Vhemisphere = 2/3 * pi * r^3 Vhemisphere = 2/3 * pi * 3.53463^3 Vhemisphere = 2/3 * pi * 44.160286017023847 Vhemisphere = 92.489 Vtotal = Vcylinder + Vhemisphere Vtotal = 842.512 + 92.489 Vtotal = 935.001 Close enough I'd say.
