answer:1. **Define the problem and variables:** We consider a pyramid with its base triangle ABC at a height ( H ) from the apex ( S ). The pyramid is intersected by a plane parallel to its base, which creates a smaller, similar triangle in the cross-section at height ( h ) from the apex ( S ) to the plane of the cross-section. Let ( S_{triangle ABC} ) be the area of the base triangle and ( S_{triangle A_1 B_1 C_1} ) be the area of the cross-sectional triangle. 2. **Use the similarity of triangles:** Since the intersecting plane is parallel to the base of the pyramid, (triangle A_1 B_1 C_1) is similar to (triangle ABC). The ratio of the areas of these similar triangles is equal to the square of the ratio of their corresponding heights from the apex ( S ). This can be stated as: [ frac{S_{triangle A_1 B_1 C_1}}{S_{triangle ABC}} = left( frac{h}{H} right)^2 ] 3. **Express the area of the cross-section in terms of base area and heights:** Rearranging the above equation to express ( S_{triangle A_1 B_1 C_1} ): [ S_{triangle A_1 B_1 C_1} = S_{triangle ABC} left( frac{h}{H} right)^2 ] 4. **Identify constants and variables:** In this context: - ( S_{triangle ABC} ) and ( H ) are constants. - ( h ) is the variable representing the distance from the apex to the intersecting plane. 5. **Determine the function form:** The expression ( S_{triangle A_1 B_1 C_1} = S_{triangle ABC} left( frac{h}{H} right)^2 ) shows that the relationship between the area of the cross-section and the height ( h ) is quadratic. 6. **Express the function with given parameters:** Identify the constant ( k ) in the quadratic form ( f(x) = kx^2 ): [ k = frac{S_{triangle ABC}}{H^2} ] Hence, the function expressing the dependence of the area of the cross-section on the distance ( h ) from the apex to the intersecting plane is: [ f(h) = left( frac{S_{triangle ABC}}{H^2} right) h^2 ] # Conclusion: The quadratic function that expresses the relationship between the area of the cross-section and the distance from the apex to the intersecting plane is given by: [ boxed{f(h) = k h^2 text{ where } k = frac{S_{triangle ABC}}{H^2}} ]

answer:To find out how many hectares have been reserved for future expansion, we need to subtract the hectares used for the house, farm machinery, cattle rearing, and crop production from the total hectares of land. Total land = 150 hectares House and farm machinery = 25 hectares Cattle rearing = 40 hectares Crop production = 70 hectares Reserved for future expansion = Total land - (House and farm machinery + Cattle rearing + Crop production) Reserved for future expansion = 150 - (25 + 40 + 70) Reserved for future expansion = 150 - 135 Reserved for future expansion = 15 hectares So, boxed{15} hectares have been reserved for future expansion.

answer:1. **Calculate the total area of the large square**: The total area is: [ 144 times 144 = 20,736 text{ square inches} ] 2. **Total area occupied by the L-shaped regions**: Each L-shaped region now occupies 1/9 of the total area. For four regions, the total area they occupy is: [ 4 times frac{1}{9} times 20,736 = frac{4}{9} times 20,736 = 9,216 text{ square inches} ] 3. **Area of the center square**: The remaining area is for the center square: [ 20,736 - 9,216 = 11,520 text{ square inches} ] 4. **Side length of the center square**: Setting up the equation for the area: [ s^2 = 11,520 ] Solving for (s): [ s = sqrt{11,520} approx 107.34 text{ inches} ] To keep the values simple and since our problem setup might dictate the center square should have a whole number side length, the perfect square closest to 11520 under reasonable adjustment would be 11025 (which is (105^2)). This minor discrepancy suggests there might be a real-world mishap or rounding. However, with ideal mathematical treatment, we assume (107.34) inches. 5. **Conclusion**: Thus, the side length of the center square is 107 text{ inches}. The final answer is boxed{C) 107 inches}

answer:1. **Define the operation star:** Given x star y = x^3 - xy. 2. **Calculate j star j:** Using the new definition, substitute x = j and y = j: [ j star j = j^3 - jj = j^3 - j^2. ] 3. **Calculate j star (j star j):** Now, substitute x = j and y = j star j into the new definition: [ j star (j star j) = j star (j^3 - j^2). ] Using the definition again with x = j and y = j^3 - j^2, we get: [ j star (j^3 - j^2) = j^3 - j(j^3 - j^2). ] Simplifying the expression: [ j^3 - j(j^3 - j^2) = j^3 - j^4 + j^3 = 2j^3 - j^4. ] 4. **Conclusion:** Therefore, j star (j star j) = 2j^3 - j^4. [ 2j^3 - j^4 ] The final answer is - boxed{textbf{(D)} 2j^3 - j^4}