Question

click an item in the list or group of pictures at the bottom of the prob correct position in the answer box. release your mouse button whe the item to the trashcan. click the trashcan to clear all your answers.

the figure shows a composite solid with a pyramid on top, a rectangular prism in the middle, and a pyramid cut out from the bottom part of the prism. the slant edges of the top pyramid are 3\, the side of the square base of the top pyramid (and also the side of the square face of the prism) is 3\, the height of the prism is 6\, and there is a small square (with side length implied by the diagram) at the bottom right inside the prism related to the cut - out pyramid.

v =

(there is a trashcan icon, a square root icon, and a number pad with digits 0 - 9 and a k key at the bottom.)

Explanation:

Step1: Identify the 3D shape

This is a square pyramid on top of a square prism (cube-like base). The base edge length for both is 3 inches, pyramid height can be found via Pythagoras, prism height is 6 inches.

Step2: Calculate pyramid base area

Base is square: $A = 3^2 = 9$ square inches

Step3: Find pyramid height

Slant height = 3 inches, half base = $\frac{3}{2}=1.5$ inches. Use Pythagorean theorem:
$$h_p = \sqrt{3^2 - 1.5^2} = \sqrt{9 - 2.25} = \sqrt{6.75} = \frac{3\sqrt{3}}{2}$$

Step4: Calculate pyramid volume

Formula: $V_p = \frac{1}{3}Ah_p$
$$V_p = \frac{1}{3} \times 9 \times \frac{3\sqrt{3}}{2} = \frac{9\sqrt{3}}{2}$$

Step5: Calculate prism volume

Formula: $V_r = Ah_r$
$$V_r = 9 \times 6 = 54$$

Step6: Total volume

Add pyramid and prism volumes:
$$V = 54 + \frac{9\sqrt{3}}{2} = \frac{108 + 9\sqrt{3}}{2}$$

Answer:

$\frac{108 + 9\sqrt{3}}{2}$ cubic inches (or $54 + \frac{9\sqrt{3}}{2}$ cubic inches)