The fuel rods for a certain type of nuclear reactor are bundled into a cylindrical shell. Consider this cross-section showing 19 identical fuel rods:
a) If the diameter of the cylindrical shell is 12.35 cm, calculate the shaded area of the cross-section.
b) If the length of the cylindrical shell is 84.50 cm, what is the volume of the shaded space around the fuel rods?
c) What is the volume of a single fuel rod?
Got no diagram, so I have no idea how the rods are packed into the shell.
Anyway, assuming a rod of radius r, the cross-section of a rod is pi r^2
The cross-section of the shell is pi * 12.35^2 = 152.52 pi = 479.16 cm^2
The shaded space is thus 479.16 - 19* pi r^2 = 479.16 - 59.69r^2
Multiply that by 84.50 to get 40489.0 - 5043.8r^2 cm^3
Volume of a single rod (assuming the same length as the shell) is 84.5 * pi r^2 = 265.46 r^2 cm^3
4189.9
a math for a math
To solve these problems, we need to understand the geometry of the cylindrical shell and the fuel rods. Let's break down each question step by step:
a) To calculate the shaded area of the cross-section, we need to determine the area of the entire cross-section and subtract the area of the fuel rods.
1. Find the radius of the cylindrical shell:
The diameter is given as 12.35 cm, so the radius is half of that, which is 12.35 cm / 2 = 6.175 cm.
2. Calculate the area of the entire cross-section:
The area of a circle is given by the formula A = π * r^2, where A is the area and r is the radius.
Plug in the values: A = π * (6.175 cm)^2 ≈ 119.99 cm^2.
3. Determine the total area of the fuel rods:
The fuel rods are arranged in a circle, so each fuel rod will occupy a portion of the cross-sectional area.
Since there are 19 identical fuel rods, divide the total area of the cross-section by the number of fuel rods:
Shaded Area = Total Area - (Area of one fuel rod * number of fuel rods).
b) To determine the volume of the shaded space around the fuel rods, we need to calculate the volume of the cylindrical shell and subtract the volume of the fuel rods.
1. Find the radius of the cylindrical shell (already calculated in part a).
2. Calculate the volume of the cylindrical shell:
The volume of a cylinder is given by the formula V = π * r^2 * h, where V is the volume, r is the radius, and h is the height (or length) of the cylinder.
Plug in the values: V = π * (6.175 cm)^2 * 84.50 cm.
3. Determine the total volume of the fuel rods:
The volume of each fuel rod is given by the formula V = π * r^2 * h, where V is the volume, r is the radius, and h is the height of the fuel rod.
Since the fuel rods are identical, multiply the volume of one fuel rod by the number of fuel rods:
Shaded Volume = Total Volume - (Volume of one fuel rod * number of fuel rods).
c) To find the volume of a single fuel rod, we need to calculate the volume of a cylinder using the diameter of the fuel rod.
1. Find the radius of the fuel rod:
The diameter is given as the distance across the center of the fuel rod, which is twice the radius.
Divide the diameter by 2 to get the radius.
2. Calculate the volume of the fuel rod:
Use the formula V = π * r^2 * h, where V is the volume, r is the radius, and h is the height of the fuel rod.
Plug in the values: V = π * (radius)^2 * h.
By following these steps, you should be able to calculate the shaded area, shaded volume, and volume of a single fuel rod in the given cylindrical shell.