Question

Consider a pyramid with a square base side of $6$inches and with a height of $12$inches, as shown below. If we cut off the top of the pyramid parallel to the base $3$inches from the tip, what is the volume of the remaining solid?

A.$141.75$
B.$140$
C.$135.48$
D.$144$
E.$130$

Answer 1

Hint: First, find the volume of the pyramid using formula,
Volume of pyramid=$\dfrac{1}{3} \times {\text{base area}} \times {\text{height}}$
Then again use this formula to find the volume of the cut-off pyramid. Then subtract the volume of the cut-off pyramid from the volume of the pyramid and you will get the volume of the remaining solid.

Complete step by step answer:

Given A pyramid has a square base side=$6$inches and height =$12$inches
Then base area of =${\left( {side} \right)^2}$
Then base area=${\left( 6 \right)^2}$
Then the volume of this pyramid is given by-
Volume of pyramid=$\dfrac{1}{3} \times {\text{base area}} \times {\text{height}}$
On putting the given values we get,
Volume of the pyramid=$\dfrac{1}{3} \times {\left( 6 \right)^2} \times 12$
On solving we get,
Volume of pyramid=$36 \times 4 = 144{\text{ inc}}{{\text{h}}^3}$
Now we cut off the top of the pyramid where the smaller pyramid has base =$1.5$ inches and height=$3$ inches.

Then base area=${\left( {side} \right)^2}$
On putting value we get,
Base area=${\left( {1.5} \right)^2} = 2.25$
Now using the formula of the volume of cut off pyramid,
Volume of cut off pyramid=$\dfrac{1}{3} \times {\text{base area}} \times {\text{height}}$
On putting the values we get,
Volume of cut off pyramid=\[\dfrac{1}{3} \times 2.25 \times 3\]
On solving we get,
Volume of cut off pyramid=$2.25{\text{ inche}}{{\text{s}}^3}$
Now we have to find the volume of remaining solid.
The volume of remaining solid=Volume of the pyramid- the volume of cut off the pyramid
On putting the values we get,
The volume of the remaining solid=$144 - 2.25$
On subtraction we get,
Volume of the remaining solid=$141.75{\text{ inc}}{{\text{h}}^3}$
Answer-Hence correct answer is A.

Note: Here, the student may get confused between base and base area. Both are different, the base is the base of the pyramid while the base area is the area of the base of the pyramid. Since the base of the pyramid is a square and we know the area of the square=${\left( {side} \right)^2}$, so we use this same formula to find the base area.