The slan heights of two similar pyramids are in the ratio 1:5. The volume of the smaller pyramid is 60m^3. What is the volume in cubic meters of the larger pyramid?
To find the volume of the larger pyramid, we can use the fact that the volumes of similar solids are proportional to the cubes of their corresponding dimensions.
Let's denote the volume of the smaller pyramid as V₁ and the volume of the larger pyramid as V₂. Also, let the slant heights of the pyramids be h₁ and h₂, respectively.
Given that the slant heights are in the ratio 1:5:
h₁/h₂ = 1/5
Since the slant heights are in inverse proportion, we can write:
h₂/h₁ = 5/1
Now, let's use the formula for the volume of a pyramid:
V = (1/3) * base area * height
Since the bases of the two pyramids are similar, their base areas are also in the ratio 1:5.
Let the base area of the smaller pyramid be A₁ and the base area of the larger pyramid be A₂.
Since the volumes of the pyramids are proportional to the cubes of their corresponding dimensions, the volumes can also be written as:
V₁/V₂ = (A₁ * h₁) / (A₂ * h₂)
Now, we have two ratios:
h₁/h₂ = 1/5 and V₁/V₂ = 1/5
We are given that V₁ = 60 m². Plugging in the values, we can solve for V₂:
60/V₂ = 1/5
Cross-multiplying:
5 * 60 = V₂
V₂ = 300 m³
Therefore, the volume of the larger pyramid is 300 cubic meters.
To find the volume of the larger pyramid, we need to use the concept of similarity between the two pyramids.
Let's denote the slant heights of the smaller and larger pyramids as h1 and h2, respectively. We are given that the ratio of the slant heights is 1:5, so we can write:
h1 : h2 = 1 : 5
Now, recall that the ratio of the volumes of two similar three-dimensional objects is equal to the ratio of the cubes of their corresponding linear dimensions. In this case, we can use the slant height ratio to find the ratio of their heights.
Since the height of a pyramid is perpendicular to the base, we can use the Pythagorean theorem to relate the height and the slant height. Let's denote the height of the smaller pyramid as H1 and the height of the larger pyramid as H2.
For the smaller pyramid, by applying the Pythagorean theorem, we have:
H1^2 + (1/2)^2 = h1^2
Similarly, for the larger pyramid:
H2^2 + (1/2)^2 = h2^2
Now, we are given that the volume of the smaller pyramid is 60 m³. The volume of a pyramid is calculated using the formula: V = (1/3) * base area * height.
Since the base area is not given, we can ignore it for now and focus on finding the ratio of the heights.
Using the given ratio of the slant heights:
h1 : h2 = 1 : 5
Let's substitute the values we have:
(H1^2 + (1/2)^2) : (H2^2 + (1/2)^2) = 1 : 5
Now, solve this ratio equation to find the ratio of the heights.
H1^2 + (1/2)^2 = (1/5)(H2^2 + (1/2)^2)
5H1^2 + 5(1/4) = H2^2 + (1/4)
Multiply both sides of the equation by 4 to eliminate the denominators:
20H1^2 + 5 = 4H2^2 + 1
Rearranging the equation:
20H1^2 = 4H2^2 - 4
Divide both sides by 4 to simplify:
5H1^2 = H2^2 - 1
Substituting the volume of the smaller pyramid into the volume formula:
V1 = (1/3) * A1 * H1 = 60
To find the volume of the larger pyramid, we need to find the base area of the larger pyramid. We can use the formula for the ratio of the volumes of similar objects:
(V1/V2) = (H1^3/H2^3)
Substituting the given and derived values:
60/V2 = (H1^3)/(H2^3) = (H1/H2)^3
Using the ratio of heights we calculated earlier, we can substitute:
60/V2 = (H1/H2)^3 = (1/5)^3 = 1/125
Now, solve this equation for V2 to find the volume of the larger pyramid:
V2 = 60/(1/125)
V2 = 60 * 125
V2 = 7500
Therefore, the volume of the larger pyramid is 7500 m³.