A manufacturer of chocolates is launching a new product in novelty shaped cardboard boxes. The box is a cuboid with a cuboid shaped tunnel through it. The height of the box is $h$ centimetres. The top of the box is a square of side $3x$ centimetres. The end of the tunnel is a square of side $x$ centimetres. The volume of the box is $2000 \, cm^3$. (a) Show that the total surface area, $A \, cm^2$, of the box is given by $$A = 16x^2 + \frac{4000}{x}$$ (b) To minimise the cost of production, the surface area, $A$, of the box should be as small as possible. Find the minimum value of $A$.

Scottish Highers Maths Differentiation: A manufacturer of chocolates is

A manufacturer of chocolates is launching a new product in novelty shaped cardboard boxes - Scottish Highers Maths - Question 11 - 2019

Question 11

A manufacturer of chocolates is launching a new product in novelty shaped cardboard boxes. The box is a cuboid with a cuboid shaped tunnel through it. The height o... show full transcript

Worked Solution & Example Answer:A manufacturer of chocolates is launching a new product in novelty shaped cardboard boxes - Scottish Highers Maths - Question 11 - 2019

Step 1

Show that the total surface area, $A \, cm^2$, of the box is given by

96%

114 rated

Answer

To find the total surface area $A$ of the box, we will first identify its components:

The surface area of the sides of the box excluding the tunnel:
- The box has a height $h$ and a square top of side $3x$ . Thus, the top surface area is:
  
  $S_{top} = (3x)^2 = 9x^2$
- The lateral surface area consists of the four sides and the bottom:
  - Each side (height $h$ , width $3x$ ) contributes:
    
    $S_{sides} = 4h(3x) = 12hx$
- Finally, we need to subtract the area of the tunnel's entrance and exit:
  - The exit area (top) adds:
    
    $S_{tunnel} = (3x - x)^2 = (2x)^2 = 4x^2$
Therefore, the total surface area becomes:

$A = S_{top} + S_{sides} - S_{tunnel} = 9x^2 + 12hx - 4x^2 = 5x^2 + 12hx$
To express $h$ in terms of $x$ , use the volume formula:

$V = (3x)(3x)(h) - (x)(x)(h) = 2000 \, cm^3$

Simplifying this:

$V = 9hx - hx = 8hx = 2000 \, cm^3$

Therefore, we find:

$h = \frac{2000}{8x} = \frac{250}{x}$
Substituting for $h$ in the total surface area formula, we have:

$A = 5x^2 + 12 \left( \frac{250}{x} \right)x = 5x^2 + 3000$
Collecting all terms gives:

$A = 16x^2 + \frac{4000}{x}$

Thus, we have shown that the total surface area is given by the equation.

Step 2

To minimise the cost of production, the surface area, $A$, of the box should be as small as possible. Find the minimum value of $A$.

99%

104 rated

Answer

To minimize the surface area $A$ , we need to take the derivative of $A$ with respect to $x$ and set it to zero:

Start with: $A = 16x^2 + \frac{4000}{x}$
Differentiate with respect to $x$ :

$\frac{dA}{dx} = 32x - \frac{4000}{x^2}$
Set the derivative equal to zero for critical points:

$32x - \frac{4000}{x^2} = 0$

Rearranging gives:

$32x = \frac{4000}{x^2}$

Multiplying both sides by $x^2$ results in:

$32x^3 = 4000$

This simplifies to:

$x^3 = \frac{4000}{32} = 125 \Rightarrow x = 5$
To confirm that this is a minimum, we can use the second derivative test:

$\frac{d^2A}{dx^2} = 32 + \frac{8000}{x^3}$

This is positive for all $x > 0$ , confirming that $A$ has a local minimum at $x = 5$ .
Finally, substitute $x = 5$ back into the original area equation to find the minimum value:

$A = 16(5)^2 + \frac{4000}{5} = 400 + 800 = 1200 \, cm^2$

Thus, the minimum surface area is $1200 \, cm^2$ .

A manufacturer of chocolates is launching a new product in novelty shaped cardboard boxes - Scottish Highers Maths - Question 11 - 2019

Worked Solution & Example Answer:A manufacturer of chocolates is launching a new product in novelty shaped cardboard boxes - Scottish Highers Maths - Question 11 - 2019

Show that the total surface area, $A \, cm^2$, of the box is given by

To minimise the cost of production, the surface area, $A$, of the box should be as small as possible. Find the minimum value of $A$.

Join the Scottish Highers students using SimpleStudy...