# 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 # 12 #, the cap's height is #8 #, and the cap's radius is #4 #. What is the object's volume?

It's a sphere over square prism with side of square base and the diameter of sphere same. i.e.

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

- Volume of the prism: [ V_{\text{prism}} = \text{Base area} \times \text{Height} ] The base area of the prism is the square-shaped top, so if its side length is equal to the diameter of the spherical cap, which is 8, then the area is ( 8 \times 8 = 64 ) square units.

[ V_{\text{prism}} = 64 \times 12 = 768 ] cubic units.

- Volume of the spherical cap: [ V_{\text{cap}} = \frac{1}{3} \pi h^2 (3r - h) ] Given that the height of the cap, ( h = 8 ), and the radius, ( r = 4 ), we can plug in these values:

[ V_{\text{cap}} = \frac{1}{3} \pi (8^2) (3 \times 4 - 8) ] [ V_{\text{cap}} = \frac{1}{3} \pi (64) (12 - 8) ] [ V_{\text{cap}} = \frac{1}{3} \pi (64) (4) ] [ V_{\text{cap}} = \frac{1}{3} \pi (256) ] [ V_{\text{cap}} = \frac{256}{3} \pi ]

Now, sum up the volumes of the prism and the spherical cap to find the total volume of the object:

[ V_{\text{total}} = V_{\text{prism}} + V_{\text{cap}} ] [ V_{\text{total}} = 768 + \frac{256}{3} \pi ]

Thus, the volume of the object is ( 768 + \frac{256}{3} \pi ) cubic units.

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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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