Question

Find the volume of the pyramid with height h and rectangular base with dimensions b and 2b

How do i do this using Calculus?

Answer 1

\[A = 1/3* h *3b\]
Since Calculus is the study of change, answering in general terms like this is "using Calculus" however, if you want to look all fancy, you can define it a bit better, and use limits or integrals if you feel so needed. for instance, this equation works IFF h>0 and b>0

Answer 2

You would have to have some other data, where the height is dependent on the base.

Then you'll end up building small rectangular prisms from a double integral.
\[\int\limits_{0}^{b}\int\limits_{0}^{2b}f(x,y)dxdy\]

Where f(x,y) is the height.

Answer 3

okay, thanks everyone

Answer 4

\[\int\limits_{0}^{h}2\left(b-\frac{bx}{h}\right)^2dx=\frac{2b^2h}{3}\]

Answer 5

thanks Mr.Zarkon

Answer 6

but is it possible can you explain just alittle bit , why you multiplied by 2 and why you got b-bx/h

Answer 7

the dimensions of the base are b and 2b. for any height x (0<x<h)

the area of the cross section is \[\left(b-\frac{bx}{h}\right)\cdot\left(2b-\frac{2bx}{h}\right)=\left(b-\frac{bx}{h}\right)\cdot2\left(b-\frac{bx}{h}\right)=2\left(b-\frac{bx}{h}\right)^2\]

Answer 8

Why is it divided by 3?