The problem asks us to find the total volume of a composite figure consisting of a square pyramid on top of a rectangular prism. The square pyramid has a base side length of $6.5$ inches and a height of $6.5$ inches. The rectangular prism has a square base with side length $6.5$ inches and a height of $12$ inches. We need to find the total volume, rounded to the nearest tenth.

1. Problem Description

The problem asks us to find the total volume of a composite figure consisting of a square pyramid on top of a rectangular prism. The square pyramid has a base side length of $6.5$ inches and a height of $6.5$ inches. The rectangular prism has a square base with side length $6.5$ inches and a height of $12$ inches. We need to find the total volume, rounded to the nearest tenth.

2. Solution Steps

First, we find the volume of the square pyramid. The formula for the volume of a pyramid is:

$V_{pyramid} = \frac{1}{3} \times base \ area \times height$

The base is a square, so the area of the base is $(side)^2 = (6.5)^2 = 42.25$ square inches.

The height of the pyramid is $6.5$ inches.

$V_{pyramid} = \frac{1}{3} \times 42.25 \times 6.5 = \frac{274.625}{3} \approx 91.5416667$ cubic inches.

Next, we find the volume of the rectangular prism. The formula for the volume of a rectangular prism is:

$V_{prism} = length \times width \times height$

Since the base is a square, the length and width are both $6.5$ inches. The height of the prism is $12$ inches.

$V_{prism} = 6.5 \times 6.5 \times 12 = 42.25 \times 12 = 507$ cubic inches.

Finally, we add the volumes of the pyramid and the prism to find the total volume:

$V_{total} = V_{pyramid} + V_{prism} = 91.5416667 + 507 = 598.5416667$ cubic inches.

Rounding to the nearest tenth, we get $598.5$ cubic inches.

3. Final Answer

598.5