Looking at formulas, which 2 shapes describe this relationship: the area of the 2 dimensional shape is one-fourth the surface area of the 3-demensional figure?
Looking at formulas, which two figures describes this relationship: the volume of the first 3-dimensional figure is three times the volume of thue second 3-dimensional figure?
says the hater who is jealous
The first question is asking us to find two shapes that represent the relationship where the area of the 2D shape is one-fourth the surface area of the 3D figure. To determine these shapes, let's consider the formulas for the area of a 2D shape and the surface area of a 3D figure.
1. Area of a Square: A = a^2
2. Surface Area of a Cube: SA = 6a^2
From the given relationship, we can set up an equation:
Area of 2D shape = 1/4 * Surface Area of 3D figure
A = 1/4 * SA
Substituting the formulas, we get:
a^2 = 1/4 * 6a^2
a^2 = 3/2 * a^2
From this equation, we can see that the shape with an area equal to one-fourth the surface area of the 3D figure is a SQUARE.
For the second question, we need to find two figures that represent the relationship where the volume of the first 3D figure is three times the volume of the second 3D figure. Let's consider the formulas for the volume of two different 3D figures.
1. Volume of a Sphere: V = (4/3)πr^3
2. Volume of a Cylinder: V = πr^2h
From the given relationship, we can set up an equation:
Volume of first 3D figure = 3 * Volume of second 3D figure
V1 = 3 * V2
Substituting the formulas, we get:
(4/3)πr1^3 = 3 * (πr2^2h2)
Simplifying, we have:
4r1^3 = 9r2^2h2
From this equation, we can see that the figures with volumes satisfying this relationship are a SPHERE and a CYLINDER.
To determine which two shapes describe the relationship where the area of the 2-dimensional shape is one-fourth the surface area of the 3-dimensional figure, we can start by understanding the formulas for the area and surface area of common shapes.
1. Rectangle:
The area of a rectangle is given by the formula: area = length × width.
The surface area of a cuboid (a 3-dimensional rectangle) is given by the formula: surface area = 2(length × width + length × height + width × height).
2. Circle:
The area of a circle is given by the formula: area = π × radius^2.
The surface area of a sphere (a 3-dimensional circle) is given by the formula: surface area = 4π × radius^2.
To find the shapes that have a relationship where the area of the 2-dimensional shape is one-fourth the surface area of the 3-dimensional figure, we need to find a pair of formulas where the area formula is one-fourth the surface area formula.
For this relationship, the rectangle and cuboid shapes satisfy the condition. The area of a rectangle is indeed one-fourth the surface area of a cuboid when the height of the cuboid is equal to the width and length of the rectangle.
Now, let's address the second question:
To find the two shapes that describe the relationship where the volume of the first 3-dimensional figure is three times the volume of the second 3-dimensional figure, we need to understand the volume formulas for common shapes.
1. Cube:
The volume of a cube is given by the formula: volume = edge length^3.
2. Cylinder:
The volume of a cylinder is given by the formula: volume = π × radius^2 × height.
In this case, the shapes that satisfy the relationship where the volume of the first figure is three times the volume of the second figure are a cube and a cylinder with equal height and radius values. The volume of the cube will be three times the volume of the cylinder if the dimensions are properly chosen.
Remember, understanding the formulas and their relationships allows us to determine which shapes satisfy specific conditions.