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I have a regular triangular pyramid. I want to cut it with a horizontal plane so...
Jan 29, 2024
I have a regular triangular pyramid. I want to cut it with a horizontal plane so that the resulting two solids have the same volume. How far from the apex is the plane?
Solution by Steps
step 1
To find the height at which to cut the regular triangular pyramid, we need to use the formula for the volume of a pyramid: V=13BhV = \frac{1}{3}Bh, where BB is the area of the base and hh is the height
step 2
Since the pyramid is regular, the base is an equilateral triangle. If the side of the base is ss, the area of the base BB can be calculated using the formula B=34s2B = \frac{\sqrt{3}}{4}s^2
step 3
Let the total height of the pyramid be HH. We want to find the height hh from the apex where the pyramid should be cut so that the volume of the upper part is equal to the volume of the lower part
step 4
Let the height of the upper smaller pyramid be hh and the height of the lower frustum be H−hH - h. The volume of the upper pyramid is V1=13B1hV_1 = \frac{1}{3}B_1h and the volume of the lower frustum can be expressed as V2=V−V1V_2 = V - V_1, where VV is the volume of the original pyramid
step 5
Since V1=V2V_1 = V_2, we have 13B1h=13B(H−h)\frac{1}{3}B_1h = \frac{1}{3}B(H - h). The area of the base of the upper pyramid B1B_1 will be smaller than BB in proportion to the square of the ratio of their heights
step 6
The ratio of the areas of the bases is equal to the square of the ratio of their heights: (hH)2=B1B\left(\frac{h}{H}\right)^2 = \frac{B_1}{B}
step 7
Substituting B1=B(hH)2B_1 = B\left(\frac{h}{H}\right)^2 into the volume equation, we get 13B(hH)2h=13B(H−h)\frac{1}{3}B\left(\frac{h}{H}\right)^2h = \frac{1}{3}B(H - h)
step 8
Simplifying the equation, we get h3=H2(H−h)h^3 = H^2(H - h)
step 9
This is a cubic equation in hh: h3+hH2−H3=0h^3 + hH^2 - H^3 = 0
step 10
To solve for hh, we can use numerical methods or algebraic manipulation. One real solution to this equation will give us the height hh at which to cut the pyramid
Answer
The height hh from the apex where the pyramid should be cut can be found by solving the cubic equation h3+hH2−H3=0h^3 + hH^2 - H^3 = 0.
Key Concept
Volume ratio of similar pyramids
Explanation
The volume of a pyramid is proportional to the cube of its linear dimensions. When cutting a pyramid to create two solids of equal volume, the height at which the cut is made can be found by setting the volume of the smaller pyramid equal to the volume of the frustum formed by the cut and solving for the height.
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