A metal solid cylinder of radius 6 cm and length 12 cm is ,melted and recast into another solid cylinder of length 3 cm.
A. Find the base radius of the new cylinder.
B. Find the change in total surface area in terms of 3.14.
volume of old = π r^2 h
= π(36)(12)= 432π
this must stay the same
we know the new h, but not the r
new volume = 3π r^2
so 2π r^2 = 432π
divide by 2π
r^2 = 216
r =√216 = appr 14.7
SA = 2πr^2 + 2πrh
old SA = 2π(36) + 2π(6)(12) = 216π
new SA = 2π(√216)^2 + 2π(√216)(3)
= 432π + 6√216π
change = 432π + 36√6π - 216π
= 216π + 36√6 π
Thanks
To find the answers to the given questions, we'll use the concept of conservation of volume.
Volume of a cylinder = π * radius^2 * height
Let's solve each question step by step:
A. Find the base radius of the new cylinder:
1. The volume of the original cylinder is given by V1 = π * (6 cm)^2 * 12 cm = 432π cm^3.
2. Since the metal solid cylinder is melted and recast into another solid cylinder, the volume remains unchanged.
3. The volume of the new cylinder is given by V2 = π * radius^2 * 3 cm.
4. Equating V1 and V2, we have 432π cm^3 = π * radius^2 * 3 cm.
5. Canceling out π and rearranging the equation, we have radius^2 = (432 cm^3) / (3 cm) = 144 cm^2.
6. Taking the square root of both sides, we get the radius of the new cylinder as sqrt(144 cm^2) = 12 cm.
Therefore, the base radius of the new cylinder is 12 cm.
B. Find the change in total surface area in terms of π:
1. The total surface area of a cylinder is given by A = 2πrh + 2πr^2, where r is the base radius, and h is the height.
2. For the original cylinder, the total surface area is A1 = 2π(6 cm)(12 cm) + 2π(6 cm)^2 = 288π cm^2.
3. For the new cylinder, the total surface area is A2 = 2π(12 cm)(3 cm) + 2π(12 cm)^2 = 216π cm^2.
4. The change in total surface area is ΔA = A2 - A1 = 216π cm^2 - 288π cm^2 = -72π cm^2.
Therefore, the change in total surface area of the new cylinder in terms of π is -72π cm^2.