The base of a solid $S$ is the region bounded by the parabola $x^{2}=8 y$ and the line $y=4$.

Cross-sections perpendicular to the $y$-axis are isosceles right triangles with the hypotenuse lying in the base.

Determine the exact volume of solid $S$.

Question

The base of a solid $S$ is the region bounded by the parabola $x^{2}=8 y$ and the line $y=4$.

Cross-sections perpendicular to the $y$-axis are isosceles right triangles with the hypotenuse lying in the base.

Determine the exact volume of solid $S$.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ The base of the solid $S$ is the region bounded by the parabola $x^2 = 8y$ and the line $y = 4$. We need to find the volume of the solid with cross-sections perpendicular to the $y$-axis that are isosceles right triangles with the hypotenuse lying in the base. ‖ ‖ step 2 ⋮ The length of the hypotenuse of each isosceles right triangle is the distance between the points where the parabola intersects the line $y = 4$. Solving $x^2 = 8 \cdot 4$, we get $x^2 = 32$, so $x = \pm \sqrt{32} = \pm 4\sqrt{2}$. Thus, the length of the hypotenuse is $2 \cdot 4\sqrt{2} = 8\sqrt{2}$. ‖ ‖ step 3 ⋮ For an isosceles right triangle, the relationship between the hypotenuse $h$ and the legs $a$ is $h = a\sqrt{2}$. Therefore, $a = \frac{h}{\sqrt{2}} = \frac{8\sqrt{2}}{\sqrt{2}} = 8$. ‖ ‖ step 4 ⋮ The area of each cross-sectional triangle is $\frac{1}{2} \cdot a \cdot a = \frac{1}{2} \cdot 8 \cdot 8 = 32$. ‖ ‖ step 5 ⋮ To find the volume of the solid, we integrate the area of the cross-sections along the $y$-axis from $y = 0$ to $y = 4$: $\int_0^4 32 \, dy$. ‖ ‖ step 6 ⋮ Evaluating the integral, we get $32 \int_0^4 dy = 32 [y]_0^4 = 32 \cdot (4 - 0) = 128$. ‖ ∻Answer∻ ⚹ The exact volume of the solid $S$ is $128$ cubic units. ⚹ ∻Key Concept∻ ⚹Volume of a solid with known cross-sectional area⚹ ∻Explanation∻ ⚹To find the volume of a solid with a known cross-sectional area, integrate the area of the cross-sections along the axis perpendicular to the cross-sections.⚹◊What is the formula for determining the volume of a solid with isosceles right triangles as cross-sections perpendicular to the $y$-axis?⍭ Generate me a similar question◊

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