6. Volumes of Pyramids - 11. Circumference, Area, and Volume

Let's analyze the given composite solid. We assume that the base is a square, not another rhombus. Otherwise, we have insufficient data to solve the exercise.

Notice that the solid is a cube from which a pyramid has been cut out. c Volume of Composite Solid = c Volume of Cube - c Volume of Pyramid

The side length of the cube is 12 inches. Therefore, its volume is V_1=12^3=1728 cubic inches. Now, let's find the volume of the pyramid. Let's start by finding the volume of its base. The base of the pyramid is the smaller square below.

To find its area, B, we will subtract the areas of four right triangles, A_(Δ), from the area of big square, A_(□). The big square has a side length of 12 inches. This tells us that A_(□)=12^2=144 square inches. Since both of the legs of the right triangle are 6 inches, A_(Δ)= 6* 62=18 square inches.

B=A_(□)-4A_(Δ)

▼

Substitute values and evaluate

SubstituteII

A_(□)= 144, A_(Δ)= 18

B= 144-4( 18)

Multiply

B=144-72

SubTerms

Subtract terms

B=72

Now, using the formula for the volume of a pyramid, we will find the volume of the pyramid, V_2.

V_2=1/3Bh

▼

Substitute values and evaluate

SubstituteII

B= 72, h= 12

V_2=1/3( 72)( 12)

Multiply

V_2=1/3(864)

MoveRightFacToNumOne

1/b* a = a/b

V_2=864/3

CalcQuot

Calculate quotient

V_2=288

This tells us that the volume of the pyramid is V_2=288 cubic inches. Finally, we will find the volume of the given composite solid. c Volume of Composite Solid = c Volume of Cube - c Volume of Pyramid ⇓ V= 1728- 288= 1440 Therefore, the volume of the composite solid is 1440 cubic inches.

Exercise 20 Page 640

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