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The base of a right prism is an equilateral triangle. If its height is one-fourth and each side of the base is tripled, then the ratio of the volumes of the old to the new prism is
- a)4 : 3
- b)1 : 4
- c)1 : 2
- d)4 : 9
Correct answer is option 'D'. Can you explain this answer?
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Problem Analysis:
Let's consider the original prism as Prism 1 and the new prism as Prism 2. We are given that the base of Prism 1 is an equilateral triangle and its height is one-fourth. We are also given that each side of the base is tripled to form Prism 2. We need to find the ratio of the volumes of Prism 1 to Prism 2.
Key Points:
- Base of Prism 1: Equilateral triangle
- Height of Prism 1: One-fourth of the base
- Sides of Prism 2: Three times the sides of Prism 1
- We need to find the ratio of the volumes of Prism 1 to Prism 2
Solution:
To find the ratio of the volumes of Prism 1 to Prism 2, we need to calculate the volumes of both prisms and then find their ratio.
Volume of Prism 1:
The volume of a prism can be calculated using the formula: Volume = Base Area × Height.
Since the base of Prism 1 is an equilateral triangle, the area of the base can be calculated using the formula: Area = (√3/4) × (side^2), where side is the length of each side of the equilateral triangle.
Let's assume that the length of each side of the equilateral triangle is 'a'. So, the area of the base of Prism 1 is (√3/4) × (a^2).
The height of Prism 1 is given as one-fourth of the base. So, the height of Prism 1 is (1/4) × a.
Using the formula for volume, the volume of Prism 1 is:
Volume1 = (√3/4) × (a^2) × (1/4) × a
Volume1 = (√3/16) × (a^3)
Volume of Prism 2:
The base of Prism 2 is formed by tripling the sides of Prism 1. So, the length of each side of the base of Prism 2 is 3a.
The height of Prism 2 is given as one-fourth of the base. So, the height of Prism 2 is (1/4) × 3a.
Using the formula for volume, the volume of Prism 2 is:
Volume2 = (√3/4) × (3a)^2 × (1/4) × (1/4) × 3a
Volume2 = (√3/16) × 9a^3 × (1/16) × 3a
Volume2 = (√3/16) × (9/16) × (3/16) × (a^4)
Volume2 = (√3/16) × (27/256) × (a^4)
Volume2 = (√3/256) × (27/4) × (a^4)
Volume2 = (27√3/1024) × (a^4)
Ratio of Volumes:
To find the ratio of the volumes of Prism 1 to Prism 2, we divide the volume of Prism 1 by the volume of Prism 2.
Ratio = Volume1 / Volume2
Ratio = [ (√3/16
Let's consider the original prism as Prism 1 and the new prism as Prism 2. We are given that the base of Prism 1 is an equilateral triangle and its height is one-fourth. We are also given that each side of the base is tripled to form Prism 2. We need to find the ratio of the volumes of Prism 1 to Prism 2.
Key Points:
- Base of Prism 1: Equilateral triangle
- Height of Prism 1: One-fourth of the base
- Sides of Prism 2: Three times the sides of Prism 1
- We need to find the ratio of the volumes of Prism 1 to Prism 2
Solution:
To find the ratio of the volumes of Prism 1 to Prism 2, we need to calculate the volumes of both prisms and then find their ratio.
Volume of Prism 1:
The volume of a prism can be calculated using the formula: Volume = Base Area × Height.
Since the base of Prism 1 is an equilateral triangle, the area of the base can be calculated using the formula: Area = (√3/4) × (side^2), where side is the length of each side of the equilateral triangle.
Let's assume that the length of each side of the equilateral triangle is 'a'. So, the area of the base of Prism 1 is (√3/4) × (a^2).
The height of Prism 1 is given as one-fourth of the base. So, the height of Prism 1 is (1/4) × a.
Using the formula for volume, the volume of Prism 1 is:
Volume1 = (√3/4) × (a^2) × (1/4) × a
Volume1 = (√3/16) × (a^3)
Volume of Prism 2:
The base of Prism 2 is formed by tripling the sides of Prism 1. So, the length of each side of the base of Prism 2 is 3a.
The height of Prism 2 is given as one-fourth of the base. So, the height of Prism 2 is (1/4) × 3a.
Using the formula for volume, the volume of Prism 2 is:
Volume2 = (√3/4) × (3a)^2 × (1/4) × (1/4) × 3a
Volume2 = (√3/16) × 9a^3 × (1/16) × 3a
Volume2 = (√3/16) × (9/16) × (3/16) × (a^4)
Volume2 = (√3/16) × (27/256) × (a^4)
Volume2 = (√3/256) × (27/4) × (a^4)
Volume2 = (27√3/1024) × (a^4)
Ratio of Volumes:
To find the ratio of the volumes of Prism 1 to Prism 2, we divide the volume of Prism 1 by the volume of Prism 2.
Ratio = Volume1 / Volume2
Ratio = [ (√3/16
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The base of a right prism is an equilateral triangle. If its height is one-fourth and each side of the base is tripled, then the ratio of the volumes of the old to the new prism isa)4 : 3b)1 : 4c)1 : 2d)4 : 9Correct answer is option 'D'. Can you explain this answer?
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