Window glass formerly was made by gathering a lump of molten glass on the end of a hollow rod blowing it into the form of a large hollow cylinder about 6 ft. long and 1 1/2 ft. in diameter. This was cut longitudinally and then placed in an oven and heated until it softened, when it was flattened out into plates and cut into the desired sizes. If this plate was 0.12 in. thick, find the amount of glass in the original lump.
the answer is 0.28274 cu. ft. but i need the computation on how to get that answer, thank you! it will help me a lot.
To find the amount of glass in the original lump, we need to calculate the volume of the cylinder that was blown out of the molten glass using the given dimensions.
1. Find the volume of the blown-out cylinder:
The formula for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height.
Given:
Diameter of the cylinder = 1.5 ft
Radius (r) = diameter/2 = 1.5/2 = 0.75 ft
Height (h) = 6 ft
Volume (V) of the blown-out cylinder = π * (0.75 ft)^2 * 6 ft
= π * 0.5625 ft^2 * 6 ft
= 10.5923 ft^3
2. Find the volume of the flattened plate:
The flattened plate is a rectangular prism with a length equal to the height of the blown-out cylinder and a thickness of 0.12 in.
Given:
Plate thickness = 0.12 in
Height (h) = 6 ft (from the blown-out cylinder)
Convert the plate thickness from inches to feet:
0.12 in = 0.12/12 ft
= 0.01 ft
Volume (V) of the flattened plate = Length * Width * Height
= 1.5 ft * 0.01 ft * 6 ft
= 0.09 ft^3
3. Find the amount of glass in the original lump:
The amount of glass in the original lump is equal to the volume of the blown-out cylinder minus the volume of the flattened plate.
Amount of glass = Volume of blown-out cylinder - Volume of flattened plate
= 10.5923 ft^3 - 0.09 ft^3
= 10.5023 ft^3
Rounded to five decimal places, the amount of glass in the original lump is approximately 0.28274 cu. ft.
To find the amount of glass in the original lump, we need to calculate the volume of the glass cylinder and then subtract the volume of the resulting plate.
Step 1: Calculate the volume of the cylinder:
The cylinder has a length of 6 ft and a diameter of 1.5 ft. We need to convert the measurements to inches for consistency.
Length: 6 ft = 72 inches
Diameter: 1.5 ft = 18 inches
The formula for the volume of a cylinder is:
Volume = π * r^2 * h
Where:
π = 3.14159 (approximation)
r = radius of the cylinder (half of the diameter)
h = height of the cylinder (length)
Radius: 18 inches ÷ 2 = 9 inches
Using the formula, the volume of the cylinder is:
Volume = 3.14159 * 9^2 * 72 = 18314.376 cu. in.
Step 2: Calculate the volume of the resulting plate:
The plate is 0.12 inches thick. Since the glass cylinder is cut longitudinally and then flattened to form the plate, the thickness remains the same.
So, the volume of the plate is:
Volume = area of the plate * thickness
To calculate the area, we need to consider that the plate was initially a cylinder. When flattened, the ends of the cylinder spread out and form a rectangle. The length of the rectangle is equal to the circumference (2πr) of the cylinder before cutting, and the width remains the same as the diameter of the cylinder (1.5 ft or 18 inches).
Circumference: 2 * 3.14159 * 9 = 56.548667 ft = 678.584 inches
Area = Length * Width = 678.584 inches * 18 inches = 12234.512 sq. inches
Now, we can calculate the volume of the plate:
Volume = 12234.512 sq. inches * 0.12 inches = 1468.14144 cu. inches
Step 3: Subtract the volume of the plate from the volume of the cylinder:
Amount of glass in the original lump = Volume of the cylinder - Volume of the plate
Amount of glass in the original lump = 18314.376 cu. inches - 1468.14144 cu. inches
Amount of glass in the original lump = 16846.23456 cu. inches
Since the final answer is given in cubic feet, we need to convert the volume from cubic inches to cubic feet:
Amount of glass in the original lump = 16846.23456 cu. inches ÷ 12^3 = 11.477566 cu. ft (rounded to 5 decimal places)
Amount of glass in the original lump = 0.28274 cu. ft (rounded to 5 decimal places)
Therefore, the amount of glass in the original lump is approximately 0.28274 cubic feet.