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. NM = 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 Euclidean Geometry: 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 identify the lengths of sides NE and EH. From the diagram, we can see that:

The ratio of NE to EF is given by:

$NE = \frac{x^2}{3} \times 2$
Hence, we can express EF as:

$EF = \frac{2}{3} x^2$
Given EH = 4 - 2 * NE, we have:

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

The area A of the rectangle EFHG can now be calculated by:

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

After simplifying, we will derive:

$A(x) = \frac{2}{3} x^2 (4 - \frac{2x^2}{3}) = 6x^2 - 3x^4$

Step 2

Calculate the maximum area of rectangle EFHG

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Answer

To determine the maximum area, we first set:

$A'(x) = 6 - 12x^2$

Setting the derivative equal to zero to find critical points:

$12 - 12x^2 = 0\implies x^2 = 1 \implies x = 1$

We then evaluate the area at this critical point:

$A(1) = 6(1)^2 - 3(1)^4 = 6 - 3 = 3$

Thus, the maximum area of rectangle EFHG is $3 cm^2$ .

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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