a time capsule is in the shape of a cylinder with a hemisphere at each end. if the capsule is surrounded with a metal outer layer 4 cm thick as shown( there's a picture) then determine the volume of the metal outer layer. round your answer to the nearest cubic centimeter. the capsule is 50 centimeter long and 10 cm wide
To determine the volume of the metal outer layer of the time capsule, we need to break it down into individual shapes and calculate their volumes.
1. Start by calculating the volume of the main cylindrical section:
- The length of the capsule is given as 50 cm, and the width (diameter) is given as 10 cm.
- The formula to calculate the volume of a cylinder is V = πr^2h, where r is the radius and h is the height/length.
- The radius of the cylinder is half the width, so r = 10 cm / 2 = 5 cm.
- Therefore, the volume of the cylindrical section is V_cylinder = π(5 cm)^2 * 50 cm.
2. Next, calculate the volume of the hemispherical ends:
- Since the capsule has a hemisphere at each end, we need to calculate the volume of two hemispheres.
- The formula for the volume of a hemisphere is V_hemisphere = (2/3)πr^3, where r is the radius.
- The radius of the hemispheres is equal to the radius of the cylindrical section, so r = 5 cm.
- Therefore, the volume of both hemispheres is V_hemisphere = (2/3)π(5 cm)^3.
3. Finally, subtract the volume of the inner space (time capsule) from the volume of the metal outer layer to get the desired volume:
- The inner volume of the time capsule is V_inner = V_cylinder + 2V_hemisphere.
- The total volume of the metal outer layer is V_outer = V_cylinder_outer + 2V_hemisphere_outer, where V_cylinder_outer is the volume of the outer cylinder and V_hemisphere_outer is the volume of the outer hemisphere.
- V_outer = V_cylinder_outer + 2V_hemisphere_outer - V_inner.
After calculating the values for V_cylinder_outer, V_hemisphere_outer, and V_inner using the given formulas, you can subtract V_inner from V_outer to determine the volume of the metal outer layer of the time capsule. Remember to round the answer to the nearest cubic centimeter.