Question

Consider the volume of a rectangular pyramid, one whose base is not square. Does changing one of the three variables (length, width, or height) by a given fraction have a greater effect on the volume of the pyramid than changing either of the other two variables? That is, which variable, if any, has the largest influence on the volume? Assume that the other two variables remain constant. Explain your answer.

Answer 1

PLATO SAMPLE ANSWER:

No. Changing one of the three variables (l, w, or h) by a given fraction has the same effect on the volume of a rectangular pyramid as changing either of the other two variables because each variable appears as a factor in the volume formula.

Explanation:

Answer 2

Let

L------> the length of the base of the pyramid

W-----> the width of the base of the pyramid

H-----> the height of the pyramid

V------> Volume of a rectangular pyramid

we know that

V=(1/3)*[L*W]*H

so

the three variables are going to have the same influence on the volume

example

case a) let's change the variable L (we reduce the variable by half)

L=(1/2)*L

the new volume is equal to

V new=(1/3)*[(L/2)*W]*H-----> V new=(1/2)*V

the new volume is half the original volume

case b) let's change the variable W (we reduce the variable by half)

W=(1/2)*W

the new volume is equal to

V new=(1/3)*[(W/2)*L]*H-----> V new=(1/2)*V

the new volume is half the original volume

case c) let's change the variable H (we reduce the variable by half)

H=(1/2)*H

the new volume is equal to

V new=(1/3)*[L*W]*H/2-----> V new=(1/2)*V

the new volume is half the original volume

therefore

the answer is

the three variables have the same influence on the volume