A 12 cm by 12 cm ice cube tray is in the shape of a cuboid with twelve hemispheres inset. The depth of each hemispheres is 1 cm.
a) If all the hemispheres were filled with water, calculate the total volume of water that the ice cube tray could hold.
It is now desired to paint the top of the ice cube tray a pleasant shade of blue.
b) calculate the rusface area of one hemisphere on the tray.
c) Calculate the total surface area of the top of the ice cube tray.
Thank you so much for huge help!!!!!
a) To calculate the total volume of water that the ice cube tray could hold, we need to find the volume of the cuboid and the volume of the hemispheres.
The volume of the cuboid is given by the formula V_cuboid = length × width × height. In this case, the dimensions are 12 cm × 12 cm × 1 cm, so the volume of the cuboid is V_cuboid = 12 cm × 12 cm × 1 cm = 144 cm³.
The volume of each hemisphere is given by the formula V_hemisphere = (4/3) × π × r³, where r is the radius of the hemisphere. Since the depth of each hemisphere is 1 cm, the radius of each hemisphere is half the width of the cuboid, which is 12 cm / 2 = 6 cm. Thus, the volume of each hemisphere is V_hemisphere = (4/3) × π × 6 cm × 6 cm × 6 cm = 288π cm³.
Since there are twelve hemispheres in the ice cube tray, the total volume of water that the tray can hold is 12 × 288π cm³ = 3456π cm³.
b) To calculate the surface area of one hemisphere, we need to find the area of the curved surface. The formula for the surface area of a sphere is given by A_sphere = 4πr², where r is the radius of the sphere. In this case, the radius of the hemisphere is 6 cm, so the surface area of one hemisphere is A_hemisphere = 4π × 6 cm × 6 cm = 144π cm².
c) Finally, to calculate the total surface area of the top of the ice cube tray, we add the surface area of the cuboid and the surface area of all the hemispheres. The surface area of the cuboid is given by A_cuboid = 2lw + 2lh + 2wh, where l, w, and h are the dimensions of the cuboid. In this case, the dimensions are 12 cm, 12 cm, and 1 cm, respectively, so the surface area of the cuboid is A_cuboid = 2 × 12 cm × 12 cm + 2 × 12 cm × 1 cm + 2 × 12 cm × 1 cm = 432 cm².
Since there are twelve hemispheres, the total surface area of the hemispheres is 12 × A_hemisphere = 12 × 144π cm² = 1728π cm².
Therefore, the total surface area of the top of the ice cube tray is A_cuboid + 1728π cm².