A company has to design a rectangular box for a new range of jellybeans - Leaving Cert Mathematics - Question 7 - 2012

The capacity of the box can be calculated using the formula:

Capacity = length × width × height.

Substituting the expressions from part (a), we get:

Capacity = (29 - h)(20 - h)h = h[(29 - h)(20 - h)] = h(580 - 49h + h²) = -h³ + 49h² - 580h.

Thus, the expression for capacity in terms of h is:

Capacity = -h³ + 49h² - 580h.

To create a box with a square bottom, we set w = l. From the equations derived:

1. We have w = 20 - h and l = 29 - h,

2. Setting these equal gives us: 20 - h = 29 - h. This simplifies to h - 20 = h - 29. Therefore, h = 5 cm.

3. We substitute h = 5 back into the capacity expression:

   • Length = 29 - 5 = 24 cm,
   • Width = 20 - 5 = 15 cm,
   • Therefore, Volume = 24 * 15 * 5 = 1800 cm³.

Thus, when h = 5, the box dimensions are appropriate, providing the correct capacity.

We can solve for the other values of h by solving:

$2h^3 - 50h^2 + 300h = 500$ This simplifies to: $2h^3 - 50h^2 + 300h - 500 = 0$ Using the quadratic formula:

1. Recognizing a factor of (h - 5), we further solve:
   • We can replace the quadratic equation as follows to apply the formula which leads to: $h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
2. After calculations, we find:
   • The values become approx h ≈ 17.1 and h ≈ 2.9.
3. Thus, the other possible value of h is 2.9.

From the graph, we observe the capacity curve shown:

1. The horizontal line representing Capacity = 550 cm³ crosses the curve at one point,
2. Based on our derived limits, h must be less than 15 cm for construction constraints, as denoted.
3. Therefore, since the height must be less than 15, it indicates that constructing a larger box with a volume of 550 cm³ from the same size cardboard cannot be achieved.