Question

that postal requirements specify that parcels length plus girth at most 84 inches. consider the problem of finding the dimensions of the square - ended rectangular package of greatest volume that is mailable. each square end is labeled with x and the remaining dimension is labeled with h. (a) express the length plus the girth in terms of x and h. 4x + h (b) determine the objective and constraint equations. what is the constraint equation? 4x + h = 84 what is the objective equation? the volume is v = x²h. (c) express the quantity to be maximized as a function of x. v = 84x² - 4x³ (d) find the optimal values of x and h. x = inches and h = inches.

Explanation:

Step1: Differentiate volume function

We have $V(x)=84x^{2}-4x^{3}$. Using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we get $V'(x)=168x-12x^{2}$.

Step2: Set derivative equal to zero

To find critical points, set $V'(x) = 0$. So, $168x-12x^{2}=0$. Factor out $12x$: $12x(14 - x)=0$. This gives two solutions: $x = 0$ and $x=14$.

Step3: Determine the valid critical point

The value $x = 0$ will give a volume of $V=0$. We discard it. For $x = 14$.

Step4: Find the value of h

Since $4x+h=84$ and $x = 14$, we substitute $x$ into the constraint equation: $4\times14+h=84$. Then $56+h=84$, and $h=84 - 56=28$.

Answer:

$x = 14$ inches and $h = 28$ inches