Question 10

Explanation

We will find the value of $x$ that maximizes the surface area of the box, and then calculate that surface area.

Determining Surface Area Formula

The surface area of a cuboid ($L$) can be calculated with the formula:

From previous questions, we know the box dimensions in terms of $x$:

• Length ($p$) = $5 - x$
• Width ($l$) = $4x - 5$
• Base Perimeter ($K$) = $6x$
• Height ($t$) = $5 - \frac{1}{3}x$

Substitute these dimensions into the surface area formula:

Simplifying the Surface Area Function

We expand the equation above:

Base Area Part:

Perimeter times Height Part:

Total Surface Area:

Finding the Maximum Value

The function $L(x)$ is a quadratic function with a negative $x^2$ coefficient, so its parabolic graph opens downwards and has a maximum value. The maximum value is reached when its first derivative equals zero ($L'(x) = 0$).

We check if $x = 4$ satisfies set $A$ (positive dimension condition) found previously, which is $\frac{5}{4} < x < 5$. Since $1.25 < 4 < 5$, the value $x = 4$ is valid.

Calculating Maximum Surface Area

Substitute $x = 4$ into the function $L(x)$:

So, the maximum surface area of the box is $110 \text{ cm}^2$.