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Problem 65

Business and Economics

Packaging The manufacturer of a fruit juice drink has decided to try innovative packaging in order to revitalize sagging sales. The fruit juice drink is to be packaged in containers in the shapeof tetrahedra in which three edges are perpendicular, as shown in the figure below. Two of the perpendicular edges will be 3 in. long, and the third edge will be 6 in. long. Find the volume of the container. (Hint: The equation of the plane shown in the figure is $z=f(x, y)=6-2 x-2 y$ .

Answer

9 $\mathrm{in}^{3}$

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## Recommended Questions

Packaging The manufacturer of a fruit juice drink has decided to try innovative packaging in order to revitalize sagging sales. The fruit juice drink is to be packaged in containers in the shape of tetrahedra in which three edges are perpendicular, as shown in the figure on the next page. Two of the perpendicular edges will be 3 in. long, and the third edge will be 6 in. long. Find the volume of the container. (Hint: The equation of the plane shown in the figure is $z=f(x, y)=6-2 x-2 y . )$

A 3D printer is used to create a plastic drinking glass. The

equations given to the printer for the inside of the glass are

$$x=\left(\frac{y}{4}\right)^{1 / 32} \text { and } y=5$$

where $x$ and $y$ are

measured in inches.

What is the total volume

that the drinking glass

can hold when the region

bounded by the graphs of

the equations is revolved

about the $y$ -axis?

Find the volume of each composite figure.

This problem will prepare you for the Concept Connection on page 724

A cylindrical juice container with a 3 in. diameter has a hole for a straw that is 1 in. from the side. Up to 5 in. of a straw can be inserted.

a. Find the height $h$ of the container to the nearest tenth.

b. Find the volume of the container to the nearest tenth.

c. How many ounces of juice does the container hold? (Hint: 1 in $^{3} \approx 0.55$ oz)

(IMAGE CAN'T COPY)

The volume of a rectangular package is 2304 cubic inches. The length of the package is 3 times its width, and the height is $1 \frac{1}{2}$ times its width.

(a) Draw a diagram that illustrates the problem. Label the height, width, and length accordingly.

(b) Find the dimensions of the package. Use a graphing utility to verify your result.

Use a triple integral to find the volume of the following solids. (FIGURE CAN'T COPY)

The solid in the first octant formed when the cylinder $z=\sin y,$ for $0 \leq y \leq \pi$ is sliced by the planes $y=x$ and $x=0$.

Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane $ x + 2y + 3z = 6 $.

Geometry The volume $V$ of a container is modeled by the function $V(x)=x^{3}-3 x^{2}-4 x \cdot$ Let $x, x+1,$ and $x-4$ represent the width, the length, and the height respectively. The container has a volume of 70 $\mathrm{ft}^{3}$ . Find the container's dimensions.

Use a triple integral to find the volume of the following solids. (FIGURE CAN'T COPY)

The wedge of the cylinder $x^{2}+4 z^{2}=4$ created by the planes $y=3-x$ and $y=x-3$.

A rectangular package to be sent by a delivery service (see figure) can have a maximum combined length and girth (perimeter of a cross section) of $ 120 $ inches.

(a) Write a function $ V(x) $ that represents the volume of the package.

(b) Use a graphing utility to graph the function and approximate the dimensions of the package that will yield a maximum volume.

(c) Find values of $ x $ such that $ V = 13,500 $. Which of these values is a physical impossibility in the construction of the package? Explain.

Packaging Design The diagram at the right shows a cologne bottle that consists of a cylindrical base and a hemispherical top.

a. Write an expression for the cylinder's volume.

b. Write an expression for the volume of the hemispherical top.

c. Write a polynomial to represent the total volume.

A rectangular box is inscribed in the region in the first octant

bounded above by the plane with $x$ -intercept $6, y$ -intercept $6,$ and

$z$ -intercept $6 .$

a. Find an equation for the plane.

b. Find the dimensions of the box of maximum volume.

Box with vertex on a plane Find the volume of the largest closed

rectangular box in the first octant having three faces in the coordinate planes and a vertex on the plane $x / a+y / b+z / c=1$

where $a>0, b>0,$ and $c>0$ .

Box with vertex on a plane Find the volume of the largest closed rectangular box in the first octant having three faces in the coordinate planes and a vertex on the plane $x / a+y / b+z / c=1$ where $a>0, b>0,$ and $c>0$

Use a triple integral to find the volume of the following solids. (FIGURE CAN'T COPY)

The solid in the first octant bounded by the plane $2 x+3 y+6 z=12$ and the coordinate planes.

Minimum cost of packaging: Similar to Exercise 54 manufacturers can minimize their costs by shipping merchandise in packages that use a minimum amount of material. After all, rectangular boxes come in different sizes and there are many combinations of length, width, and height that will hold a specified volume.

A maker of packaging materials needs to ship $36 \mathrm{ft}^{3}$ of foam "peanuts" to his customers across the country, using boxes with the dimensions shown. (FIGURE CAN'T COPY)

a. Find a function $S(x, y)$ for the surface area of the box, and a function $V(x, y)$ for the volume of the box.

b. Solve for $y$ in $V(x, y)=36$ (volume is $36 \mathrm{ft}^{3}$ ), and use the result to write the surface area as a function $S(x)$ in terms of $x$ alone (simplify the result).

c. On a graphing calculator, graph the function $S(x)$ using the window $x \in[-10,10] ; y \in[-200,200] .$ Then graph

$y=2 x^{2}+4 x$ on the same screen. How are these two graphs related?

d. Use the graph of $S(x)$ in Quadrant I to determine the dimensions that will minimize the surface area of the box, yet still hold the foam peanuts. Clearly state the values of $x$ and $y,$ in terms of feet and inches, rounded to the nearest $\frac{1}{2}$ in.

A company makes rectangular

shaped bird cages with height $b$ inches and square bottoms. The volume of these cages is given by the function

$$V=b^{3}-6 b^{2}+9 b$$

a) Find an expression for the length of each side of the square bottom by factoring the expression on the right side of the function.

b) Use the function to find the volume of a cage with a height of 18 inches.

c) Use the accompanying graph to estimate the height of a cage for which the volume is $20,000$ cubic inches.

(FIGURE CANNOT COPY)

A box of volume 72 $\mathrm{m}^{3}$ with square bottom and no top is constructed out of two different materials. The cost of the bottom is 40$/ \mathrm{m}^{2}$ and the cost of the sides is $\$ 30 / \mathrm{m}^{2} .$ Find the dimensions of the box that minimize total cost.

A rectangular package sent by a delivery service can have a maximum combined length and girth (perimeter of a cross section) of 120 inches (see figure).

(a) Show that the volume of the package is given by the function $V(x)=4 x^{2}(30-x)$

(b) Use a graphing utility to graph the function and approximate the dimensions of the package that yield a maximum volume.

(c) Find values of $x$ such that $V=13,500 .$ Which of these values is a physical impossibility in the construction of the package? Explain.

Use a triple integral to find the volume of the following solids. (FIGURE CAN'T COPY)

The wedge in the first octant bounded by the cylinder $x=z^{2}$ and the planes $z=2-x, y=2, y=0,$ and $z=0$.

Packaging Design A company wishes to manufacture a box with a volume of 36 $\mathrm{ft}^{3}$ that is open on top and is twice as long as it is wide. Find the dimensions of the box produced from the minimum amount of material.

Use a triple integral to find the volume of the following solids. (FIGURE CAN'T COPY)

The wedge above the $x y$ -plane formed when the cylinder $x^{2}+y^{2}=4$ is cut by the planes $z=0$ and $y=-z$.

(a) sketch the region enclosed by the curves, (b) describe the cross section perpendicular to the $x$ -axis located at $x,$ and (c) find the volume of the solid obtained by rotating the region about the $x$ -axis.

\begin{equation}

y=16-x, \quad y=3 x+12, \quad x=-1

\end{equation}

Use a triple integral to find the volume of the following solids. (FIGURE CAN'T COPY)

The wedge bounded by the parabolic cylinder $y=x^{2}$ and the planes $z=3-y$ and $z=0$.

Let $S$ be the tetrahedron in $\mathbb{R}^{3}$ with vertices at the vectors $\mathbf{0},$

$\mathbf{e}_{1}, \mathbf{e}_{2},$ and $\mathbf{e}_{3},$ and let $S^{\prime}$ be the tetrahedron with vertices at vectors $\mathbf{0}, \mathbf{v}_{1}, \mathbf{v}_{2},$ and $\mathbf{v}_{3} .$ See the figure.

a. Describe a linear transformation that maps $S$ onto $S^{\prime}$ .

b. Find a formula for the volume of the tetrahedron $S^{\prime}$ using the fact that

$$

\{\text { volume of } S\}=(1 / 3) \cdot\{\text { area of base }\} \cdot\{\text { height }\}

$$

Find the maximum volume of a rectangular box with three faces in the coordinate planes and a vertex in the first octant on the line $x+y+z=1$

PERFUME BOTTLES Some perfumes are packaged in square pyramidal containers. The base of one bottle is 3 inches square, and the slant height is 4 inches. A second bottle has the same surface area, but the slant height is 6 inches long. Find the dimensions of the base of the second bottle.

Find the volume of the wedge cut from the first octant by the cylinder $z=12-3 y^{2}$ and the plane $x+y=2$

Find the volume of the wedge cut from the first octant by the cylinder $z=12-3 y^{2}$ and the plane $x+y=2$

Find the volume of the given solid.

Enclosed by the paraboloid $ z = x^2 + y^2 + 1 $ and the planes $ x = 0 $, $ y = 0 $, $ z = 0 $, and $ x + y = 2 $

Optimal box Find the dimensions of the rectangular box with maximum volume in the first octant with one vertex at the origin and the opposite vertex on the ellipsoid $36 x^{2}+4 y^{2}+9 z^{2}=36.$

Volume of a Fish Tank A fish tank in an avant-garde restaurant is in the shape of a parallelepiped with a rectangular base that is $300 \mathrm{cm}$ long and $120 \mathrm{cm}$ wide. The front and

back faces are vertical, but the left and right faces are slanted at $30^{\circ}$ from the vertical and measure $120 \mathrm{cm}$ by $150 \mathrm{cm} .$ (See the figure.)

(a) Let $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ be the three vectors shown in the figure. Find $\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w}) .$ [Hint: Recall that $\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta \text { and }|\mathbf{u} \times \mathbf{v}|=|\mathbf{u}||\mathbf{v}| \sin \theta .]$

(b) What is the capacity of the tank in liters? [Note: $\left.1 \mathrm{L}=1000 \mathrm{cm}^{3} .\right]$

Determine the volume of the parallelepiped of Fig. $3.20 b$ when

$$

\begin{array}{l}{(a) \mathbf{P}=4 \mathbf{i}-3 \mathbf{j}+2 \mathbf{k}, \mathbf{Q}=-2 \mathbf{i}-5 \mathbf{j}+\mathbf{k}, \text { and } \mathbf{S}=7 \mathbf{i}+\mathbf{j}-\mathbf{k}} \\ {(b) \mathbf{P}=5 \mathbf{i}-\mathbf{j}+6 \mathbf{k}, \mathbf{Q}=2 \mathbf{i}+3 \mathbf{j}+\mathbf{k}, \text { and } \mathbf{S}=-3 \mathbf{i}-2 \mathbf{j}+4 \mathbf{k}}\end{array}

$$

Find the volume of the given right tetrahedron. (Hint: Consider slices perpendicular to one of the labeled edges.)

Volume of a Parallelepiped Three vectors $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ are given. (a) Find their scalar triple product $\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w}) .$ (b) Are the vectors coplanar? If not, find the volume of the parallelepiped that they determine.

$$

\mathbf{u}=2 \mathbf{i}-2 \mathbf{j}-3 \mathbf{k}, \quad \mathbf{v}=3 \mathbf{i}-\mathbf{j}-\mathbf{k}, \quad \mathbf{w}=6 \mathbf{i}

$$

Volume of a Fish Tank A fish tank in an avant-garde restaurant is in the shape of a parallelepiped with a rectangular base that is 300 $\mathrm{cm}$ long and 120 $\mathrm{cm}$ wide. The front and back faces are vertical, but the left and right faces are slanted at $30^{\circ}$ from the vertical and measure 120 $\mathrm{cm}$ by $150 \mathrm{cm} .$ (See the figure.)

(a) Let $a, b,$ and $c$ be the three vectors shown in the figure. Find $a \cdot(b \times c) .[\text { Hint: Recall that } \mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta$ and $|\mathbf{u} \times \mathbf{v}|=|\mathbf{u}||\mathbf{v}| \sin \theta . ]$

(b) What is the capacity of the tank in liters? [Note: $1 \mathrm{L}=1000 \mathrm{cm}^{3} .$]

Use a triple integral to find the volume of the following solids. (FIGURE CAN'T COPY)

The solid bounded by the cylinder $y=9-x^{2}$ and the paraboloid $y=2 x^{2}+3 z^{2}$.

Find the volume of the solid bounded by the surface $z=f(x, y)$ and the $x y$-plane. (Check your book to see figure)

$$f(x, y)=\frac{20}{1+x^{2}+y^{2}}-2$$

$21-30=$ Find the volume of the given solid.

Enclosed by the paraboloid $z=x^{2}+3 y^{2}$ and the planes

$x=0, y=1, y=x, z=0$

Find the volume of the solid by subtracting two volumes.

The solid enclosed by the parabolic cylinder $ y = x^2 $ and the planes $ z = 3y $, $ z = 2 + y $

Scalar triple product Another operation with vectors is the scalar triple product, defined to be $\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w}),$ for vectors $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ in $\mathbb{R}^{3}$ Consider the parallelepiped (slanted box) determined by the position vectors $\mathbf{u}, \mathbf{v},$ and $\mathbf{w}$ (see figure). Show that the volume of the parallelepiped is $|\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})|$ CANT COPY THE GRAPH

Find the volume of the given solid.

Under the plane $ 3x + 2y - z = 0 $ and above the region enclosed by the parabolas $ y = x^2 $ and $ x = y^2 $

Solve each problem.

Packing Cheese Workers at the Green Bay Cheese Factory are trying to cover a block of cheese with an 8 in, by 12 in. piece of foil paper as shown in the figure. The ratio of the length and width of the block must be 4 to 3 to accommodate the label. Find a polynomial function that gives the volume of the block of cheese covered in this manner as a function of the thickness $x$. Use a graphing calculator to find the dimensions of the block that will maximize the volume.

(a) sketch the region enclosed by the curves, (b) describe the cross section perpendicular to the $x$ -axis located at $x,$ and (c) find the volume of the solid obtained by rotating the region about the $x$ -axis.

\begin{equation}

y=\sec x, \quad y=0, \quad x=-\frac{\pi}{4}, \quad x=\frac{\pi}{4}

\end{equation}

A box manufactured in the Netherlands was large enough to hold $43,000$ liters of water. It was made from one large sheet of cardboard. If $x$ is 1.2 meters and $y$ is 0.1 meter, use the information in the diagram to write a polynomial that represents the surface area of the box and find the total amount of cardboard used to make the box. $\left(\text {Hint}:(2 x+2 y)(6 x-2 y)=12 x^{2}+8 x y\right.$$\left.-4 y^{2}\right)$

CAN'T COPY THE FIGURE

Volume of a Box An open box is constructed from a 6 in. by 10 in. sheet of cardboard by cutting a square piece from each corner and then folding up the sides, as shown in the figure. The volume of the box is

$$V=x(6-2 x)(10-2 x)$$

a. Explain how the expression for $V$ is obtained.

b. Expand the expression for $V$. What is the degree of the resulting polynomial?

c. Find the volume when $x=1$ and when $x=2$

Use a triple integral to compute the volume of the following regions.

The larger of two solids formed when the parallelepiped (slanted box ) with vertices (0,0,0),(2,0,0),(0,2,0),(2,2,0),(0,1,1)

$(2,1,1),(0,3,1),$ and (2,3,1) is sliced by the plane $y=2$.

(a) sketch the region enclosed by the curves, (b) describe the cross section perpendicular to the $x$ -axis located at $x,$ and (c) find the volume of the solid obtained by rotating the region about the $x$ -axis.

\begin{equation}

y=\sec x, \quad y=0, \quad x=0, \quad x=\frac{\pi}{4}

\end{equation}