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A cedar chest (closed box) is to be built, but to reduce costs, the base and the back of the chest will be made of pine. The cost of the cedar is
$8//m^(2) and the cost of the pine is
$4//m^(2). The ends of the chest are to be square.
a) Find the dimensions of the least expensive chest that can be built if the capacity must be
2m^(3).
b) Find the dimensions of the largest chest that can be built for
$1200.

$23$ . A cedar chest (closed box) is to be built, but to reduce costs, the base and the back of the chest will be made of pine. The cost of the cedar is $\$ 8 / \mathrm{m}^{2}$ and the cost of the pine is $\$ 4 / \mathrm{m}^{2}$ . The ends of the chest are to be square. $\newline$ a) Find the dimensions of the least expensive chest that can be built if the capacity must be $2 \mathrm{~m}^{3}$ . $\newline$ b) Find the dimensions of the largest chest that can be built for $\$ 1200$ .

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Volume of cubes and rectangular prisms: word problems

Full solution

Q.

23

. A cedar chest (closed box) is to be built, but to reduce costs, the base and the back of the chest will be made of pine. The cost of the cedar is

\$ 8 / \mathrm{m}^{2}

and the cost of the pine is

\$ 4 / \mathrm{m}^{2}

. The ends of the chest are to be square.

\newline

a) Find the dimensions of the least expensive chest that can be built if the capacity must be

2 \mathrm{~m}^{3}

.

\newline

b) Find the dimensions of the largest chest that can be built for

\$ 1200

.

Denote dimensions and volume: Let's denote the height of the chest as $h$ meters, the width as $w$ meters, and the depth as $d$ meters. Since the ends are square, $h = w$ .
Calculate volume and rewrite equation: The volume of the chest is given by $V = h \times w \times d$ . We know $V = 2\,\text{m}^3$ , so $h \times h \times d = 2\,\text{m}^3$ .
Minimize cost by minimizing surface area: Since $h = w$ , we can rewrite the volume equation as $h^2 \cdot d = 2m^3$ . To minimize cost, we want to minimize the surface area because the cost is based on the area of the material used.
Calculate surface area of the chest: The surface area $S$ of the chest is $S = 2 \times (h \times d) + 2 \times (h \times h) + (d \times h)$ , where the first term is for the pine back and base, and the other terms are for the cedar sides and front.
Substitute variables to simplify equation: Substitute $h^2$ for $\frac{2}{d}$ into the surface area equation to get $S$ in terms of a single variable, $d$ .

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