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 length of the top. The prism's height is # 9 #, the cap's height is #8 #, and the cap's radius is #7 #. 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 length of the top.  The prism's height is # 9 #, the cap's height is #8 #, and the cap's radius is #7 #.  What is the object's volume?

Wyatt Anderson · Accepted Answer

#V ~~ 3200.76#

The sphere and the prism are the two components we wish to work on.

We know the diameter is equal to the length of the top, and we know that #d=2r#

By doing this, we are able to calculate that the diameter is 14 and the radius is 7.

Let us now begin by discussing the volume of the prism. A prism's volume is:

#V = lwh#

Given that the top is square and that both length and width equal 14, which is the diameter of the cap, we plug everything in to obtain:

#V = 14^2*9# #V = 1764#

Now for the sphere portion. A sphere's volume is equal to:

#V = 4/3pir^3# But since we are dealing with half a sphere, we will use this: #V = 2/3pir^3#

If we plug everything in, we obtain:

#V = 2/3pi(7)^3# #V = 1372pi/3#

To find your answer, add the two volumes together:

#1372pi/3+1764 = V# #V ~~ 3200.76#

Mason Adams · Answer

undefined To find the volume of the object, we need to calculate the volumes of both the prism and the spherical cap, then sum them.

The volume $ V_p $ of the prism is given by:

$$ V_p = \text{Base area} \times \text{Height} $$

Since the top of the prism is square-shaped, with a side length equal to the diameter of the cap's base, the base area of the prism is the square of this side length.

Given that the side length (and diameter) is equal to 7 (the radius of the cap), the base area is $ 7^2 = 49 $ square units.

The height of the prism is given as 9 units.

$$ V_p = 49 \times 9 = 441 $$ cubic units.

Next, let's find the volume $ V_c $ of the spherical cap.

The formula for the volume of a spherical cap is:

$$ V_c = \frac{1}{3} \pi h^2 (3R - h) $$

Where:
- $ h $ is the height of the cap (given as 8 units).
- $ R $ is the radius of the cap (given as 7 units).

$$ V_c = \frac{1}{3} \pi \times 8^2 \times (3 \times 7 - 8) $$
$$ V_c = \frac{64}{3} \pi \times (21 - 8) $$
$$ V_c = \frac{64}{3} \pi \times 13 $$
$$ V_c = \frac{832}{3} \pi $$

Now, the total volume of the object is the sum of the volumes of the prism and the spherical cap:

$$ V_{\text{total}} = V_p + V_c $$
$$ V_{\text{total}} = 441 + \frac{832}{3} \pi $$

Therefore, the volume of the object is $ 441 + \frac{832}{3} \pi $ cubic units.