Question

Alistair has a rectangular picture measuring 8 inches by 7 inches. He makes a frame for the picture, shown by the shaded area in the diagram. The area of the frame is 34 square inches, and the width of the frame is x inches.

Answer 1

Answer:

1) Equation to model the area of frame is $34= (8+2x)\times (7+2x)$

2) The equation is quadratic.

3) The width of picture fame is $\frac{-15\pm \sqrt{137}}{4}$

Step-by-step explanation:

Given: Dimension of rectangular picture is 8 inches by 7 inches

           Area of picture frame is 34 inches²

Solving to find the dimension of picture frame.

Assuming that picture frame has uniform width of "x".

using the area of rectangle formula to solve.

Area of frame= $length\times width$

$34= (8+2x)\times (7+2x)$

Using distributive property of multiplication

⇒$34= 56+16x+14x+4x^{2}$

Subtracting both side by 34

⇒$4x^{2} +30x+22=0$  (quadratic equation)

Taking GCF as 2

⇒ $2(2x^{2} +15x+11)=0$

dividing both side by 2

⇒ $2x^{2} +15x+11=0$

Solving by using quadratic formula.

Formula: $\frac{-b\pm \sqrt{b^{2}-4(ac) } }{2a}$

∴ In the expression $2x^{2} +15x+11$, we have a= 2, b= 15 and c= 11.

Now, subtituting the value in the formula.

= $\frac{-15\pm \sqrt{15^{2}-4(2\times 11) } }{2\times 2}$

= $\frac{-15\pm \sqrt{15^{2}-4(22) } }{4}$

Opening parenthesis.

= $\frac{-15\pm \sqrt{225-88 } }{4}$

= $\frac{-15\pm \sqrt{137}}{4}$

∴ x=  $\frac{-15\pm \sqrt{137}}{4}$

Hence; 1) Equation to model the area of frame is $34= (8+2x)\times (7+2x)$

2) The equation is quadratic.

3) The width of picture fame is $\frac{-15\pm \sqrt{137}}{4}$