An object is made of a prism with a spherical cap on its square shaped top. The cap's base has a diameter equal to the lengths of the top. The prism's height is # 7 #, the cap's height is #4 #, and the cap's radius is #8 #. What is the object's volume?

Question

An object is made of a prism with a spherical cap on its square shaped top. The cap's base has a diameter equal to the lengths of the top.  The prism's height is # 7 #, the cap's height is #4 #, and the cap's radius is #8 #.  What is the object's volume?

Noah Atkinson · Accepted Answer

It's a trickily-written question: we need to deduce that it's a sphere on a square prism. It is also not clear whether the top is spherical or hemispherical. The volume of the object is #716# or #582# cubic units (we are not told what units are used).

Volume of the sphere: #V=4/3pir^3=4/3xx3.14159xx8^2=268# cubic units

If the cap is a hemisphere, its volume will be half this, #134# cubic units

The square has side length #8# units (we also need to deduce this from the question).

Volume of square prism: #V=lxxbxxh=8xx8xx7=448# cubic units.

If the cap is a full sphere, the total volume is #268+448=716# cubic units.

If the cap is a hemisphere, the total volume is #134+448=582# cubic units.

Sadie Adams · Answer

undefined To find the volume of the object, you need to calculate the volumes of the prism and the spherical cap separately and then add them together.

1. Volume of the prism:
   Volume = base area × height
   Since the base is square, its area is the side length squared:
   Base area = (side length)^2 = (8)^2 = 64 square units
   Height of the prism = 7 units
   Volume of the prism = 64 square units × 7 units = 448 cubic units

2. Volume of the spherical cap:
   Given the cap's radius (r) is 8 units and height (h) is 4 units, you can use the formula for the volume of a spherical cap:
   Volume = (1/3)πh^2(3r - h)
   Volume of the cap = (1/3)π(4^2)(3(8) - 4) = (1/3)π(16)(24 - 4) = (1/3)π(16)(20) = (320/3)π cubic units

3. Total volume of the object:
   Total volume = Volume of prism + Volume of cap
   Total volume = 448 cubic units + (320/3)π cubic units

So, the volume of the object is 448 cubic units + (320/3)π cubic units.