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In the redesign of a machine, a metal cubical part has each of its dimensions tripled. By what factor do its surface area and volume change?

$3^{2}=9$

Physics 101 Mechanics

Chapter 2

Motion along a Straight Line

Physics Basics

Motion Along a Straight Line

Motion in 2d or 3d

Newton's Laws of Motion

University of Michigan - Ann Arbor

University of Washington

McMaster University

Lectures

03:28

Newton's Laws of Motion are three physical laws that, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. These three laws have been expressed in several ways, over nearly three centuries, and can be summarised as follows: In his 1687 "Philosophiæ Naturalis Principia Mathematica" ("Mathematical Principles of Natural Philosophy"), Isaac Newton set out three laws of motion. The first law defines the force F, the second law defines the mass m, and the third law defines the acceleration a. The first law states that if the net force acting upon a body is zero, its velocity will not change; the second law states that the acceleration of a body is proportional to the net force acting upon it, and the third law states that for every action there is an equal and opposite reaction.

04:16

In mathematics, a proof is a sequence of statements given to explain how a conclusion is derived from premises known or assumed to be true. The proof attempts to demonstrate that the conclusion is a logical consequence of the premises, and is one of the most important goals of mathematics.

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so we can say that the ah, that thesis surface area is that the surface area of a cube is equaling six times the area of one face, so that would be equal to six times l squared because again, a cube has six faces and the area of one face is equal to l squared. We also know that the velocity of a farce are the volume of a Q equals l cute. And so we can say that Oh, so two equals three times else of one. So essentially the dimensions of the cube, the length of the cube is tripled. So we can say that for the we know that the surface area of the cube is going to be directly proportional to the length squared. So we can say that the surface area of surface area someone divided by else of one squared is going to be equal to the surface area sub two, divided by else of two squared. And we can say that surface area to equal surface area someone Time's Elsa to divide by else of one squared. We know that Elsa two equals three times else of one. So s a someone will equal three else of one divided by else of one squared. And so we know that the surface area is increased by a factor of nine, if the dimensions or tripled. So this would be the new surface area nine times the original surface area for Part B. We know that the velocity of the cute, the volume of the Cube. Rather, my apologies again is directly proportional to the cube to the length cubes, so we can say that velocity of one divided by the length of one cube you equals of the last cities. So the volumes up to my apologies divided by the length sub too cubed. And so we can say that the volumes of two equals the volume someone times the lengths of two divided by the length of one, and this would be cubed. And again we know that this is going to equal the volumes of one times three times the volume sub one divided by the length of the lengths of one divided by the likes of one. And this would be cute as well, so the volume is increased by a factor of 27 so three cubed would be 27 times in the sub one. So if the dimensions air tripled, that means that the volume is going to increase by a factor of 27. That is the end of the solution. Thank you for watching.

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