A-Level Edexcel Maths Pure General Binomial Expansion: Figure 4 shows an open-topped wa

Figure 4 shows an open-topped water tank, in the shape of a cuboid, which is made of sheet metal - Edexcel - A-Level Maths Pure - Question 3 - 2008 - Paper 2

Question 3

Figure 4 shows an open-topped water tank, in the shape of a cuboid, which is made of sheet metal. The base of the tank is a rectangle $x$ metres by $y$ metres. The h... show full transcript

Worked Solution & Example Answer:Figure 4 shows an open-topped water tank, in the shape of a cuboid, which is made of sheet metal - Edexcel - A-Level Maths Pure - Question 3 - 2008 - Paper 2

Step 1

Show that the area $A$ m² of the sheet metal used to make the tank is given by

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Answer

The volume of the tank is represented by: $V = x \cdot y \cdot x = x^2y$ Given that the volume is 100 m³, we can express $y$ in terms of $x$ : $y = \frac{100}{x^2}$

The area of the sheet metal used for the open-topped tank is: $A = xy + 2xh$ Substituting $h = x$ and $y = \frac{100}{x^2}$ gives: $A = \frac{100}{x} + 2x^2$ This shows that the area is given by $A = \frac{300}{x} + 2x^2$ .

Step 2

Use calculus to find the value of $x$ for which $A$ is stationary.

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Answer

To find the stationary points, we need to differentiate $A$ with respect to $x$ : $A' = -\frac{300}{x^2} + 4x$ Setting this equal to zero to find the critical points: $0 = -\frac{300}{x^2} + 4x$

Multiplying through by $x^2$ gives: $0 = -300 + 4x^3$ Thus:

\Rightarrow x^3 = 75 \ \Rightarrow x = (75)^{1/3} \approx 4.22.$$

Step 3

Prove that this value of $x$ gives a minimum value of $A$.

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Answer

To confirm that this critical point corresponds to a minimum, we can use the second derivative test. Differentiating $A'$ gives: $A'' = \frac{600}{x^3} + 4$ Since both terms are positive, $A'' > 0$ . This indicates that the function is concave up at this point, confirming a local minimum.

Step 4

Calculate the minimum area of sheet metal needed to make the tank.

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Answer

Substituting $x = (75)^{1/3}$ back into the area formula: $A = \frac{300}{(75)^{1/3}} + 2\left((75)^{1/3}\right)^2$ Computing this will yield: The minimum area $A$ is approximately $A \approx 140.26 \text{ m}^2$ .

Figure 4 shows an open-topped water tank, in the shape of a cuboid, which is made of sheet metal - Edexcel - A-Level Maths Pure - Question 3 - 2008 - Paper 2

Worked Solution & Example Answer:Figure 4 shows an open-topped water tank, in the shape of a cuboid, which is made of sheet metal - Edexcel - A-Level Maths Pure - Question 3 - 2008 - Paper 2

Show that the area $A$ m² of the sheet metal used to make the tank is given by

Use calculus to find the value of $x$ for which $A$ is stationary.

Prove that this value of $x$ gives a minimum value of $A$.

Calculate the minimum area of sheet metal needed to make the tank.

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