Chapter 2.6, Problem 26AYU

26. Constructing an Open Box An open box with a square base is required to have a volume of 10 cubic feet.

(a) Express the amount A of material used to make such a box as a function of the length x of a side of the square base.

(b) How much material is required for a base 1 foot by 1 foot?

(c) How much material is required for a base 2 feet by 2 feet?

(d) Use a graphing utility to graph $A\text{ = }A\left(x\right)$. For what value of x is A smallest?

To determine

To find:

a. To expresses the Amount $A$ of material used to make a open box with square base as a function of length $x$ of the side of the square base.

Answer to Problem 26AYU

Solution:

a. $A\left(x\right)=x^2+\frac{40}{x}$

Explanation of Solution

Given:

An open box with a square base is required to have a volume to 10 cubic feet.

Calculation:

a. To expresses the Amount $A$ of material used to make a open box with square base as a function of length $x$ of the side of the square base.

Let $x$ represent the side of the square base.

Let $h$ represent the height of the box.

Let $A$ represent the amount of material required for the box.

The volume $V$ of the box $h\times x\times x$, where $V=10{\text{ feet}}^3$

Hence height $h=\frac{10}{x^2}$.

$A=\text{Area of the base}+4\times \text{ Area of side wall}$

Area of the base $A_1=x^2$.

Area of side wall $A_2=x\times h$.

$=x \times \frac{10}{x^2}$

$= \frac{10}{x}$

$A\left(x\right)=x^2+4\times \frac{10}{x}$

$=x^2+\frac{40}{x}$

To determine

To find:

b. To find how much material is required for a base of 1 foot by 1 foot.

Answer to Problem 26AYU

Solution:

b. 41 sq feet

Explanation of Solution

Given:

An open box with a square base is required to have a volume to 10 cubic feet.

Calculation:

b. To find how much material is required for a base of 1 foot by 1 foot.

$A\left(1\right)=1^2+\frac{40}{1}$

$=41\text{ sq feet}$

To determine

To find:

c. To find how much material is required for a base of 2 feet by 2 feet.

Answer to Problem 26AYU

Solution:

c. 24 sq feet

Explanation of Solution

Given:

An open box with a square base is required to have a volume to 10 cubic feet.

Calculation:

c. To find how much material is required for a base of 2 feet by 2 feet.

$A\left(2\right)=2^2+\frac{40}{2}$

$=\text{ 24 sq feet}$

To determine

To find:

d. To graph $A=A(x)$ and find value of $x$ where $A$ is smallest.

Answer to Problem 26AYU

Solution:

d. $x=2.7\text{ feet}$

Explanation of Solution

Given:

An open box with a square base is required to have a volume to 10 cubic feet.

Calculation:

d. To graph $A=A(x)$ and find value of $x$ where $A$ is minimum.

From the graph, we can see that the $A(x)$ is minimum $\text{(22}\text{.1 sq feet)}$ when $x=2.7\text{ feet}$.