#### To determine

**To find:**

The volume of a pyramid with height h and rectangular base with dimensions b and 2b.

#### Answer

V= 23b2h.

#### Explanation

**1) Concept:**

Definition of volume:

Let S be a solid that lies between x=a and x=b. If the cross sectional area of S in the plane Px, through x and perpendicular to the x-axis, is A(x), where A is a continuous function, then the volume of S is

V=limn→∞∑i=1nAxi*∆x=∫abAxdx

**2) Given:**

Height of pyramid is h, base is b, and 2b.

**3) Calculations:**

Consider the pyramid having the apex point at the origin and its axis coinciding with the x-axis.

From figure,

Consider the similar triangles, ∆ABC & ∆ADE,

For a cross-section at height h-x and base b,

Using the properties of similar triangles,

BCDE=ABBD

That is,

bx/2b/2=xh

Therefore, solving for bx,

bx=b·xh

Similarly,

For a cross-section at height h-x and base 2b,

lx=2bxh

Therefore, we have a rectangular cross section of width bx=bxh and length lx=2bxh and having thickness dx

Then, the volume of elementary cuboid is

dV=area of rectangular cross section·thickness

dV= bx·lx dx=bxh·2bxh·dx=2b2x2h2dx

Hence, the total volume of the pyramid is

V=∫dV=∫2b2h2x2dx

Taking limits from 0 to h,

V=∫0h2b2h2x2dx

After integrating,

V= 2b2h2x33h0

By substituting the limits,

V=2b2h2h33-0

That is V=23b2h

**Conclusion:**

The volume of a pyramid with height h and rectangular base with dimensions b and 2b is 23b2h