EXAMPLE 3 Maximizing Volume A supplier of bolts wants to create open boxes for the bolts by cutting a square from each corner of a 12-in. by 12-in. piece of metal and then folding up the sides. What size square should be cut from each corner to produce a box of maximum volume? 0001 APPLY IT SOLUTION Let x represent the length of a side of the square that is cut from each corner, as shown in Figure 8(a). The width of the box is 12 2x, with the length also 12 As shown in Figure 8(b), the depth of the box will be x inches. The volume of the box is given by the product of the length, width, and height. In this example, the volume, V(x), depends on x: 2x. V(x) = x(12 - 2x)(12 - 2x) = 144x – 48x2 + 4x³. %3D %3D Clearly, 0 < x, and since neither the length nor the width can be negative, 0 < 12 – 2x, so x < 6. Thus, the domain of V is the interval 0, 6]. ninmob sdi o1duovitoged Jer ods 0 21 12-2х — 0000 x = depth 002 12 2x 12 2x (a) (b) FIGURE 8 766 CHAPTER 14 Applications of the Derivative Extrema Candidates The derivative is V'(x) = 144 - 96x + 12x². Set this derivative equal to 0. %3D х V(x) 12x2 - 96x + 144 = 0 12(x² - 8x + 12) = 0 12(x - 2)(x - 6) =D 0 %3D 128 Maximum 6. 0. x - 2 = 0 x – 6 = 0 %3D or %3D %3D YOUR TURN 3 Repeat Example 3 using an 8-m by 8-m piece of metal. Find V(x) for x equal to 0, 2, and 6 to find the depth that will maximize the volume. The table indicates that the box will have maximum volume when x = 2 and that the maximum volume will be 128 in. %3D 3 TRY YOUR TURN 3
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
Your turn 3
![EXAMPLE 3
Maximizing Volume
A supplier of bolts wants to create open boxes for the bolts by cutting a square from each
corner of a 12-in. by 12-in. piece of metal and then folding up the sides. What size square
should be cut from each corner to produce a box of maximum volume?
0001
APPLY IT SOLUTION Let x represent the length of a side of the square that is cut from each corner,
as shown in Figure 8(a). The width of the box is 12 2x, with the length also 12
As shown in Figure 8(b), the depth of the box will be x inches. The volume of the box is
given by the product of the length, width, and height. In this example, the volume, V(x),
depends on x:
2x.
V(x) = x(12 - 2x)(12 - 2x) = 144x – 48x2 + 4x³.
%3D
%3D
Clearly, 0 < x, and since neither the length nor the width can be negative, 0 < 12 – 2x, so
x < 6. Thus, the domain of V is the interval 0, 6].
ninmob sdi o1duovitoged
Jer ods
0 21
12-2х —
0000
x = depth
002
12 2x
12 2x
(a)
(b)
FIGURE 8](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3f84cbe0-b26c-48b0-933d-c65044943783%2Fd3697c53-3a7d-4926-9faf-dfaf8feb2550%2Fnnzfnn.jpeg&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 4 images
