A rectangular picture frame is 30 cm long and 20 cm wide. A picture of area 459 cm2 is centrally placed inside the frame leaving a uniform margin of width of x cm. Find the dimensions of the picture

multipleQuestions newQuestion Find the dimensions of the picture newAnswer The dimensions of the picture are 27 cm by 17 cm.

Steps:

1. Define variables:

   • Let the length of the frame be $L_f = 30$ cm.
   • Let the width of the frame be $W_f = 20$ cm.
   • Let the area of the picture be $A_p = 459$ cm$^2$.
   • Let the uniform margin width be $x$ cm.
   • Let the length of the picture be $L_p$.
   • Let the width of the picture be $W_p$.

2. Express picture dimensions in terms of frame dimensions and margin: Since the picture is centrally placed with a uniform margin of $x$ cm, the dimensions of the picture are: $L_p = L_f - 2x = 30 - 2x$ $W_p = W_f - 2x = 20 - 2x$

3. Formulate an equation using the picture's area: The area of the picture is $A_p = L_p \times W_p$. $459 = (30 - 2x)(20 - 2x)$

4. Solve the quadratic equation for x: Expand the equation: $459 = 600 - 60x - 40x + 4x^2$ $459 = 4x^2 - 100x + 600$ Rearrange into standard quadratic form ($ax^2 + bx + c = 0$): $4x^2 - 100x + 600 - 459 = 0$ $4x^2 - 100x + 141 = 0$

   Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$: Here, $a=4$, $b=-100$, $c=141$. $x = \frac{-(-100) \pm \sqrt{(-100)^2 - 4(4)(141)}}{2(4)}$ $x = \frac{100 \pm \sqrt{10000 - 2256}}{8}$ $x = \frac{100 \pm \sqrt{7744}}{8}$ $x = \frac{100 \pm 88}{8}$

   Two possible values for $x$: $x_1 = \frac{100 + 88}{8} = \frac{188}{8} = 23.5$ $x_2 = \frac{100 - 88}{8} = \frac{12}{8} = 1.5$

5. Determine the valid value for x: The dimensions of the picture must be positive. If $x = 23.5$ cm: $L_p = 30 - 2(23.5) = 30 - 47 = -17$ cm (Not physically possible)

   If $x = 1.5$ cm: $L_p = 30 - 2(1.5) = 30 - 3 = 27$ cm $W_p = 20 - 2(1.5) = 20 - 3 = 17$ cm These dimensions are positive, so $x = 1.5$ cm is the correct margin width.

6. State the dimensions of the picture: The length of the picture is 27 cm. The width of the picture is 17 cm.

   Verification: Area = $27 \text{ cm} \times 17 \text{ cm} = 459 \text{ cm}^2$, which matches the given area.