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 # 3 #, the cap's height is #7 #, and the cap's radius is #8 #. What is the object's volume?

Answer 1

# 1693.05 cu#

I employ this formula:

Volume of the sphere's cone-shaped portion with radius a

#= 4 / 3 a^3 ( alpha ) sin alpha#,
where #alpha (rad)# is the semi-vertical angle of the bounding

cone, extending from the sphere's center to the cap's edge.

Using the opposite spherical cap's dimensions,

the semi-angle at the center of which this opposing cap subtends

within its sphere,

Here, #alpha# rad #
#= arccos ( ( 8 - 7 ) / 8) = arccos ( 1 / 8 )= arcsin ( sqrt ( 63 )/8 ) #
#= 82.82^o = #
# = 1.4455 rad#.,

The prism's square-top's side length measures

#2 (sqrt( 8^2 - 1^2) ) = sqrt63#.

The whole book

V is the opposite spherical's volume.

cap plus the rectangular cylinder below's volume

The spherical cap's volume

= the condenser volume

section of the sphere that resembles ice and has this cap as its top; the volume of

the cone portion. Currently,

#V = 4/3 ( 8^3 )(1.4455 )( sqrt63/8 )#
#- 1 / 3 pi ( (sqrt63)^2 )(1)) + (3)(2sqrt63)^2#
#= 979.05 - 42 + 756#
#= 1693.05 cu#
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Answer 2

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

Volume of the prism: The volume of a prism is given by the formula (V_{\text{prism}} = \text{Base Area} \times \text{Height}). Since the base of the prism is square-shaped and its side length is equal to the diameter of the spherical cap, which is (8), the base area is (8 \times 8). Given the height of the prism is (3), the volume of the prism is (V_{\text{prism}} = 8 \times 8 \times 3).

Volume of the spherical cap: The volume of a spherical cap is given by the formula (V_{\text{cap}} = \frac{1}{3} \pi h^2 (3r - h)), where (h) is the height of the cap and (r) is the radius. Given the height of the cap is (7) and the radius is (8), we can calculate the volume of the cap using these values.

Finally, the total volume of the object is the sum of the volume of the prism and the volume of the spherical cap.

[V_{\text{total}} = V_{\text{prism}} + V_{\text{cap}}]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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