Chapter 2: Q. 40 (page 125)

Constructing a Closed Box A closed box with a square base is required to have a volume of $10$ cubic feet.
(a) Build a model that expresses 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) Graph $A = A (x)$ . For what value of $x$ is $A$ smallest?

Short Answer

Expert verified

(a) The amount $A$ of material used to make such a box is $A (x) = 2 x^{2} + \frac{40}{x}$

(b) Material required for a base $1$ foot by

$1$ foot is $42$ foot².

$2$ foot is $28$ foot².

(d) The value of $A$ is smallest at $\sqrt[3]{10}$ feet.

Step by step solution

Step 1. Part (a) Build a model that expresses 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.

Let $h$ be the height of the box and $x$ be the side of a square base.

Volume $=$ base area $\times$ height

$V = x^{2} h h = \frac{V}{x^{2}}$

Since $V = 10$ , we get

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

Now the amount $A$ of material used to make such a box is

$A = 2 \times a r e a o f t h e b a s e + 4 \times a r e a o f t h e s i d e A = 2 x^{2} + 4 h x$

Since $h = \frac{10}{x^{2}}$ ,

$A (x) = 2 x^{2} + 4 (\frac{10}{x^{2}}) x A (x) = 2 x^{2} + \frac{40}{x}$

Step 2. Part (b) Find material required for the box of base 1 foot by 1 foot.

$A (1) = 2 {(1)}^{2} + \frac{40}{1} A (1) = 2 + 40 A (1) = 42 f o o t^{2}$

Step 3. Part (c) Find material required for the box of base 2 foot by 2 foot.

$A (2) = 2 {(2)}^{2} + \frac{40}{2} A (2) = 8 + 20 A (2) = 28 f o o t^{2}$

Step 4. Part (d) Draw the graph of A(x) and find the value of x where A is smallest.

The graph is

The value of $x$ where $A$ is smallest is $x = 2.15$ feet(accurate value is $\sqrt[3]{10}$ feet)

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Short Answer

Step by step solution

Step 1. Part (a) Build a model that expresses 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.

Step 2. Part (b) Find material required for the box of base 1 foot by 1 foot.

Step 3. Part (c) Find material required for the box of base 2 foot by 2 foot.

Step 4. Part (d) Draw the graph of A(x) and find the value of x where A is smallest.

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