EHGF is a rectangle. HE is produced $x^2$ cm to N and EH is produced $x^2$ cm to P. NF produced intersects PG produced at M to form an isosceles triangle MNP with NM = MP. D lies on NP where MD ⊥ NP. NR = 4 cm and MD = 3 cm. 10.1 Show that the area of EFHG is given by $A(x) = 6x^2 - 3x^4$. 10.2 Calculate the maximum area of rectangle EFHG.

NSC Mathematics Calculus: EHGF is a rectangle. HE is produ

EHGF is a rectangle - NSC Mathematics - Question 10 - 2024 - Paper 1

Question 10

EHGF is a rectangle. HE is produced $x^2$ cm to N and EH is produced $x^2$ cm to P. NF produced intersects PG produced at M to form an isosceles triangle MNP with NM... show full transcript

Worked Solution & Example Answer:EHGF is a rectangle - NSC Mathematics - Question 10 - 2024 - Paper 1

Step 1

Show that the area of EFHG is given by $A(x) = 6x^2 - 3x^4$

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Answer

To find the area of rectangle EFHG, we first determine the dimensions of the rectangle. Given that:

NE = $\frac{2}{3} x^2$ and EF = $\frac{1}{3} b$ .

From the rectangle's dimensions, we have:

The width (EF) is given by:

$EF = b = \frac{3}{2} NE = \frac{3}{2} \left(\frac{2}{3} x^2\right) = x^2$
The height (EH) is calculated as:

$EH = 4 - 2 \cdot \frac{1}{3} x^2 = 4 - \frac{2}{3} x^2$

Thus, the area $A(x)$ can be expressed as:

$A(x) = EF \cdot EH = x^2 \left(4 - \frac{2}{3} x^2\right)$

This simplifies to:

$$A(x) = 6x^2 - 3x^4.$

Step 2

Calculate the maximum area of rectangle EFHG

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Answer

To find the maximum area of the rectangle, we take the derivative of $A(x)$ :

$A'(x) = 12x - 12x^3$

Setting the derivative to zero for maximization:

$12x - 12x^3 = 0\rightarrow 12x(1 - x^2) = 0$

This gives us critical points at $x = 0$ and $x = 1$ . Evaluating at these points to find the maximum:

At $x = 0$ : $A(0) = 0$
At $x = 1$ : $A(1) = 6(1)^2 - 3(1)^4 = 6 - 3 = 3$

Therefore, the maximum area is:

$\text{Maximum Area} = 3 \text{ cm}^2.$

EHGF is a rectangle - NSC Mathematics - Question 10 - 2024 - Paper 1

Worked Solution & Example Answer:EHGF is a rectangle - NSC Mathematics - Question 10 - 2024 - Paper 1

Show that the area of EFHG is given by $A(x) = 6x^2 - 3x^4$

Calculate the maximum area of rectangle EFHG

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