# Questions tagged [cross-sections]

A cross section is the intersection of a body in three-dimensional space with a plane, or the analog in higher-dimensional space. Cutting an object into slices creates many parallel cross sections.

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### Find the formula for a function $f(x,y)$ with two given cross sections

I have done problems where I am given a contour map and I am asked to find a formula of a linear function but I am having trouble with this one. I found the change in $x$ over the change in $z$ but I ...
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### sketch of a solid for calculating volume using slicing method

In reference to the textbook: Thomas’ Calculus, 14th edition, Heil, Weir & Hass In chapter 6, section 1, Exercise 1: We need to find the volume of a solid using the slicing method It says that the ...
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### Let $R$ be a region bounded by $y=x^2$, $y=2x$, and $y=3$. Solid $S$ is obtained by rotating region $R$ about $x=-1$.

Let $R$ be a region bounded by $y=x^2$, $y=2x$, and $y=3$. Solid $S$ is obtained by rotating region $R$ about $x=-1$. UPDATE: the region R is the region bounded below the line y=3. Write the integral ...
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### Volumes with Cross Sections, Circle Base and Square Cross Section

The base of a solid is a circle centered at the origin with a radius r, and every plane section perpendicular to a diameter is a square. What is the volume of the solid? So this would usually be easy ...
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### Can the cross section of parallelepiped be a regular pentagon

Came across this question in a children's recreational mathematics book. Apparently, the cross section of a cube cannot be a regular pentagon. It could be a irregular pentagon though. But if we ...
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### Pre-Calculus Question: How to find the line of sight that someone cannot see?

If Dave is standing next to a silo of cross-sectional radius r=9 feet at the indicated position, his vision will be partially obstructed. Find the portion of the y-axis that Dave cannot see. (Hint: ...
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### Finding volume of solid by cross sections?

Find the volume of the solid whose base is the region bounded by 𝑦 = 𝑥² and the line 𝑦 = 1, and whose cross sections perpendicular to the base and parallel to the x-axis are squares. I'm not ...
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### Find axial intercepts of a perpendicular plane of vector [xyz], the plane passing through its position [x,y,z] [closed]

I tried to make the question as brief as possible in the title without omitting critical aspects. There are three elements to it. 1) A position vector [xyz] is given first and then 2) Its ...
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### Volume of a solid with base bounded by a parabola and a straight line

The base of a solid is the region bounded by $y=0,25x^2$ and $y=3-x$. Cross sections perpendicular to the $y$ axis are rectangles with base in the XY plane and its height is twice its length. Find the ...
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### Vertical cross-section of $S : r(u,v) = (u+v,uv,u^2v)$.

Exercise : Consider the surface $S : r(u,v) = (u+v,uv,u^2v), \; (u,v) \in \mathbb R^2$. Express the vertical cross section $c$ of the surface at the point $(2,1,1)$ with direction $(2,1)$ and ...
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### Find area of cross section of cylinder by the plane $x$

I am working on my scholarship exam practice (assume high school/pre-university math background) and I think I got half way through but I am not sure how I could continue. Let $r$ be a positive ...
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### Why do early math courses focus on the cross sections of a cone and not on other 3D objects?

Conic sections seem to get special attention in early math classes. My question is why do these cross sections of cones deserve more attention than those of, say, a rectangular prism, a cube, or some ...
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### Do all functions represent a section of an n dimensional object by another object of n-1 dimension?

If $x^2 + y^2 = 4$ represents the section of a cone by a plane horizontal to the cone, a circle, and $y^2 - x^4 = 4$ represents the section of a cone by a plane vertical to the cone, a hyperbola, what ...
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### I have no idea how to solve this problem using areas of known cross section

The problem involving cross sections I am so confused on how to find volume using known cross sections. I've never understood it. This problem that I've encountered is very difficult, and I tried ...
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### What is the terminology for a subset of a product of sets that is the product of its cross-sections?

Let $X$ and $Y$ be non-empty sets. For every $x \in X$, let $S_x$ be a non-empty subset of $Y$. Define $S := \prod_{x \in X}S_x$. $S$ is a subset of $Y^X$. I think I once saw a name given to this kind ...
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### cross section of torus

Is there any particular name for the plane revolved about an external axis to form a torus? I was thinking of "cross section," but that could be taken as a vertical plane cutting the whole torus in ...
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### In what sense are "projecting" and "taking sections" of polytopes dual operations?

It seems to be folklore that projecting and taking sections of polytopes are somehow "dual operations" (e.g. explicitly noted in the abstract of this paper, or suggested by this answer to an MO ...
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### cross section for y=1

My class has just started Multivariable, and I'd just gotten back my quiz results. However, I don't understand why for B), the correct graph is apparently in the other direction I thought it would be —...
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### Cross sections of randomly distributed defects

A $10 \times 10 \times 10$ mm cube has $1000000$ 2-micron spheres randomly distributed thru out it. If a random cross-section was done. What would be the area of the cross-sectioned spheres and how ...
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### Cylindric sections

When a cylinder intersects with a plane, what are the resultant shapes and curves? I think that the curves are hyperbola, parabola and line, and the shapes are circle, ellipse, rectangle and trapezium....
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### Finding the cross section of a prolate spheroid at a given rotation [theta (x,y), theta (x,z), theta (y,z)] on a central plane

I currently have an assignment in which I have to model the drag forces acting on a rugby ball as it rotates through the air. One of the variables in the drag force equation is the cross section of ...
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### Volume of a solid formed by a triangle base with square cross sections parallel to a line

Reviewing for a test, I was given this problem. "The base of a solid is the region in the first quadrant bounded by the line x = -2y + 6 and the coordinate axes. What is the volume of the solid if ...
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### Cross sections of a cube

Suppose we take the set of all cross sections of a cube and construct from them a set $A$ whose elements are sets of vertices of the cube as follows. If there exists a cross section of the cube which ...
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### Finding the volume of a solid s using cross sections

I am a given a problem that reads "The base of $S$ is a region enclosed by $y = 2-x^2$ and the $x$-axis. Cross-sections perpendicular to the $y$-axis are quarter circles." The instructions are "Find ...
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### Cross-Sections of Solids

I know already that conic sections (or conics) have been widely explored and many things about them are already known. I was wondering if this sort of exploration has taken place for any other sorts ...
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### 3D Ellipsoid Average Value Integrals

An ellipsoidal meteor is careening down from outer space. The comet's exterior takes the shape of $\frac{(x-30)^2}{36}+\frac{(y-70)^2}{9}+\frac{(z-40)^2}{25}=1$. When the meteor hits the Martian ...
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### Why are the middle cross sections of dual polyhedrea the same?

A tetrahedron is self-dual, so it is no surprise that in both the tetrahedron and its dual the middle cross section is the same shape (a square). A cube and an octahedron are dual, and the middle ...
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### Single-Variable Integration Volume Problem. "When lengths change linearly, areas change quadratically"?

Let $C$ be a cone with base any shape (not necessarily a disk or an ellipse) having area $A$, and height $h$ (that is, the distance from the apex to the plane containing the base is $h$). Note that a ...
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### An inverse problem for cross sections

I apologize beforehand for the vague title and the length of the description I am using to setup my question; I can't seem to be more concise without sacrificing clarity. Call a region in the plane "...
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### Island Sketching

"The Island Euleria sank over the course of a year. A helicopter captured 11 photographs of the top of the island. It took a picture every time the island had sunk three feet, and one of the island ...
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