Pyramid $A$ is a square pyramid with a base side length of 14 inches and a height of 6 inches. Pyramid $B$ has a volume of 3,136 cubic inches. How many times bigger is the volume of pyramid B than pyramid A? Give your answer as a percentage. Provide an explanation and proof for your answer to receive full credit.
Answer (2)
Calculate the base area of pyramid A: ba se a re a = 14 in c h es × 14 in c h es = 196 in c h e s 2 .
Calculate the volume of pyramid A: V o l u m e A = 3 1 × 196 in c h e s 2 × 6 in c h es = 392 in c h e s 3 .
Determine the ratio of the volume of pyramid B to pyramid A: r a t i o = 392 in c h e s 3 3136 in c h e s 3 = 8 .
Calculate the percentage increase: p erce n t a g e in cre a se = ( 8 − 1 ) × 100% = 700% . Therefore, pyramid B is 700% bigger than pyramid A.
Explanation
Problem Analysis First, let's analyze the information we have. We know that pyramid A is a square pyramid with a base side length of 14 inches and a height of 6 inches. Pyramid B has a volume of 3,136 cubic inches. Our goal is to find out how many times bigger the volume of pyramid B is than pyramid A, expressed as a percentage.
Calculate Base Area of Pyramid A To solve this, we need to find the volume of pyramid A first. The formula for the volume of a pyramid is: V o l u m e = 3 1 × ba se a re a × h e i g h t Since pyramid A has a square base, the base area is the side length squared. So, the base area of pyramid A is: ba se a re a = s i d e × s i d e = 14 in c h es × 14 in c h es = 196 in c h e s 2
Calculate Volume of Pyramid A Now we can calculate the volume of pyramid A: V o l u m e A = 3 1 × 196 in c h e s 2 × 6 in c h es = 3 1 × 1176 in c h e s 3 = 392 in c h e s 3
Calculate the Ratio of Volumes Next, we need to find the ratio of the volume of pyramid B to the volume of pyramid A: r a t i o = V o l u m e A V o l u m e B = 392 in c h e s 3 3136 in c h e s 3 = 8
Convert Ratio to Percentage To express this ratio as a percentage increase, we subtract 1 from the ratio and multiply by 100%: p erce n t a g e in cre a se = ( r a t i o − 1 ) × 100% = ( 8 − 1 ) × 100% = 7 × 100% = 700% So, the volume of pyramid B is 700% bigger than the volume of pyramid A.
Final Answer Therefore, pyramid B is 700% bigger than pyramid A.
Examples
Imagine you're designing two different-sized tents in the shape of pyramids for a camping trip. If one tent (Pyramid A) has a base side length of 14 inches and a height of 6 inches, and you want to create another tent (Pyramid B) that is significantly larger with a volume of 3,136 cubic inches, calculating the percentage difference in their volumes helps you understand how much more material you'll need for the larger tent. In this case, Pyramid B is 700% bigger than Pyramid A, indicating a substantial increase in size and material requirements.
Pyramid B is 700% bigger than pyramid A, with volume calculations showing that pyramid A has a volume of 392 cubic inches compared to pyramid B's 3,136 cubic inches. The comparison involves calculating the ratio of their volumes and converting it into a percentage. This demonstrates a substantial increase in size between the two pyramids.
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