Figure 4 shows a solid brick in the shape of a cuboid measuring 2x cm by x cm by y cm. The total surface area of the brick is 600 cm². (a) Show that the volume, V cm³, of the brick is given by V = \frac{200x - \frac{4x^3}{3}}{} (b) Given that x can vary, use calculus to find the maximum value of V, giving your answer to the nearest cm³. (c) Justify that the value of V you have found is a maximum.

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Figure 4 shows a solid brick in the shape of a cuboid measuring 2x cm by x cm by y cm - Edexcel - A-Level Maths Pure - Question 1 - 2007 - Paper 2

Question 1

Figure 4 shows a solid brick in the shape of a cuboid measuring 2x cm by x cm by y cm. The total surface area of the brick is 600 cm². (a) Show that the volume, V ... show full transcript

Worked Solution & Example Answer:Figure 4 shows a solid brick in the shape of a cuboid measuring 2x cm by x cm by y cm - Edexcel - A-Level Maths Pure - Question 1 - 2007 - Paper 2

Step 1

Show that the volume, V cm³, of the brick is given by

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Answer

To find the volume V of the brick, we start with the dimensions of the cuboid:

Length = 2x cm
Width = x cm
Height = y cm

The formula for the total surface area A of a cuboid is:

$A = 2(lw + lh + wh)$

after substituting the lengths:

$600 = 2(2x \cdot x + 2x \cdot y + x \cdot y)$

This simplifies to:

$600 = 2(2x^2 + 2xy + xy)$

So, we can further simplify:

$600 = 2(2x^2 + 3xy)$

Dividing through by 2 gives:

$300 = 2x^2 + 3xy$

Now solve for y:

$y = \frac{300 - 2x^2}{3x}$

Now, substituting y into the volume formula:

$V = 2x \cdot x \cdot y = 2x^2 \cdot \frac{300 - 2x^2}{3x} = \frac{200x - \frac{4x^3}{3}}{}$

We have effectively shown the volume equation.

Step 2

use calculus to find the maximum value of V, giving your answer to the nearest cm³.

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Answer

To find the maximum value of V, we first need to compute the derivative of V:

$\frac{dV}{dx} = 200 - 4x$

Setting the derivative to zero to find critical points:

$200 - 4x = 0 \Rightarrow x = 50$

Now we evaluate V at this critical point:

$V(50) = \frac{200 \cdot 50 - \frac{4 \cdot (50)^3}{3}}{}$

Calculating this:

$V(50) = \frac{200 \cdot 50 - \frac{4 \cdot 125000}{3}}{} = \frac{10000 - 166667/3}{1} \approx 943 \text{ cm}^3$

Thus, the maximum value of V is approximately 943 cm³.

Step 3

Justify that the value of V you have found is a maximum.

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Answer

To justify that the value of V found is a maximum, we need to check the second derivative:

$\frac{d^2V}{dx^2} = -4$

Since this second derivative is negative, it indicates that the function V is concave down at x = 50, confirming that we have a maximum.

Therefore, the value of V = 943 cm³ is indeed a maximum.

Figure 4 shows a solid brick in the shape of a cuboid measuring 2x cm by x cm by y cm - Edexcel - A-Level Maths Pure - Question 1 - 2007 - Paper 2

Worked Solution & Example Answer:Figure 4 shows a solid brick in the shape of a cuboid measuring 2x cm by x cm by y cm - Edexcel - A-Level Maths Pure - Question 1 - 2007 - Paper 2

Show that the volume, V cm³, of the brick is given by

use calculus to find the maximum value of V, giving your answer to the nearest cm³.

Justify that the value of V you have found is a maximum.

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