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The volume of a pyramid with a square base is 200 cm3. The height of the pyramid is 13 cm. What will be the length of the slant edges (i.e., the distance between the apex and any other vertex), rounded to the nearest integer?
- a)12 cm
- b)13 cm
- c)14 cm
- d)15 cm
- e)16 cm
Correct answer is option 'C'. Can you explain this answer?
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The volume of a pyramid with a square base is 200 cm3. The height of ...
Volume = 1/3 × Area of the base (A2) × Height of the pyramid
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(H = 13 cm) Thus, 200 = 1/3 × A2 × 13 ⇒ A2 = 600/13
Therefore, side of the square base = A =(√600 / 13)
Say, diagonal of the base = 2B
Then, B2 + B2 = A2 = 600/13
⇒ B2 = 300/13
(Slant edge of pyramid)2 = H2 + B2
Thus, length of the slant edge
Most Upvoted Answer
The volume of a pyramid with a square base is 200 cm3. The height of ...
Given information:
- Volume of the pyramid = 200 cm^3
- Height of the pyramid = 13 cm
To find:
- Length of the slant edges (distance between the apex and any other vertex)
Formula for the volume of a pyramid:
- Volume = (1/3) * base area * height
1. Calculation of base area:
- Since the base of the pyramid is square, we can find the area by squaring the length of one side.
- Let the length of one side of the square base be 'a'.
- Base area = a^2
2. Calculation of volume:
- Given volume = 200 cm^3
- Volume = (1/3) * base area * height
- Substituting the values, 200 = (1/3) * a^2 * 13
- Simplifying, 200 = (13/3) * a^2
- Multiplying both sides by 3/13, a^2 = (200 * 3) / 13
- a^2 = 600/13
3. Calculation of slant edge length:
- In a pyramid, the slant edge forms a right triangle with the height and one-half the diagonal of the square base.
- Let the slant edge length be 's' and the diagonal of the square base be 'd'.
- We can find the diagonal using the Pythagorean theorem: d^2 = a^2 + a^2
- Simplifying, d = √2 * a
4. Substituting the values:
- Since the diagonal is equal to √2 times the side length, we have d = √2 * a
- Substituting the value of a^2 from step 2, we have d = √2 * √(600/13)
- Simplifying, d = √(2 * 600/13) = √(1200/13) = (√1200) / (√13)
5. Rounding to the nearest integer:
- The value of (√1200) / (√13) is approximately 11.961.
- Rounding this to the nearest integer, we get 12.
Therefore, the length of the slant edges, rounded to the nearest integer, is 12 cm.
- Volume of the pyramid = 200 cm^3
- Height of the pyramid = 13 cm
To find:
- Length of the slant edges (distance between the apex and any other vertex)
Formula for the volume of a pyramid:
- Volume = (1/3) * base area * height
1. Calculation of base area:
- Since the base of the pyramid is square, we can find the area by squaring the length of one side.
- Let the length of one side of the square base be 'a'.
- Base area = a^2
2. Calculation of volume:
- Given volume = 200 cm^3
- Volume = (1/3) * base area * height
- Substituting the values, 200 = (1/3) * a^2 * 13
- Simplifying, 200 = (13/3) * a^2
- Multiplying both sides by 3/13, a^2 = (200 * 3) / 13
- a^2 = 600/13
3. Calculation of slant edge length:
- In a pyramid, the slant edge forms a right triangle with the height and one-half the diagonal of the square base.
- Let the slant edge length be 's' and the diagonal of the square base be 'd'.
- We can find the diagonal using the Pythagorean theorem: d^2 = a^2 + a^2
- Simplifying, d = √2 * a
4. Substituting the values:
- Since the diagonal is equal to √2 times the side length, we have d = √2 * a
- Substituting the value of a^2 from step 2, we have d = √2 * √(600/13)
- Simplifying, d = √(2 * 600/13) = √(1200/13) = (√1200) / (√13)
5. Rounding to the nearest integer:
- The value of (√1200) / (√13) is approximately 11.961.
- Rounding this to the nearest integer, we get 12.
Therefore, the length of the slant edges, rounded to the nearest integer, is 12 cm.
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