Question

A picture measures x cm by (x + 4) cm. It is enlarged by a scale factor of 3 to form a poster. The area of the enlarged poster is 1053 cm².

(i) Pauline made a wild guess, without making any prior calculation, that the value of x is 31. Without finding the actual value of x, verify whether her claim is correct.
(ii) Form an expression in terms of x and show that it reduces to x² + 4x - 117 = 0.
(iii) Calculate the perimeter of the original picture before the enlargement.

pls help me I need it by 25mins​

Answer 1

i) Her claim is incorrect

ii) see below

iii) 44 cm

Explanation:

Given:

• width of picture = x cm
• length of picture = (x + 4) cm

If the picture is enlarged by a scale factor of 3 to form a poster, then the dimensions of the poster will be:

• width of poster = 3x cm
• length of poster = 3(x + 4) cm

Area of rectangle = width × length

⇒ Area of poster = (3x) × 3(x + 4)

= 9x(x + 4)

= 9x² + 36x cm²

Part (i)

Inputting x = 31 into the area of poster formula found above:

⇒ 9(31)² + 36(31)

= 8649 + 1116

= 9765 cm²

Therefore, as 9765 ≠ 1053 her claim in incorrect.

Part (ii)

Using the formula for the area of poster found in part (i).

Area of poster = 1053 cm²

⇒ 9x² + 36x = 1053

⇒ 9x² + 36x - 1053 = 0

⇒ 9(x² + 4x - 117) = 0

⇒ x² + 4x - 117 = 0

Part (iii)

First find x by solving the equation from part (ii):

⇒ x² + 4x - 117 = 0

⇒ x² + 13x - 9x - 117 = 0

⇒ x(x + 13) - 9(x + 13) = 0

⇒ (x - 9)(x + 13) = 0

⇒ x = 9, x = -13

As width cannot be negative, x = 9 only

Perimeter of picture = 2(width + length)

= 2(x + x + 4)

= 2(2x + 4)

= 4x + 8

Input found value of x into the expression for the perimeter:

⇒ Perimeter of picture = 4(9) + 8

= 36 + 8

= 44 cm