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

1. So, the length of the frame is 8 + 2x, and the width of the frame is 7 + 2x.

area of a rectangle = length × width

total area = (8 + 2x) × (7 + 2x)

= 4x2 + 30x + 56 square inches (I)

The length (l) of the picture is 8 inches, and its width (w) is 7 inches. So:

area of the picture = 8 × 7

= 56 square inches(II)

area of the frame = total area − area of the picture

34 = (4x2 + 30x + 56) − 56 (I – II)

34 = 4x2 + 30x

4x2 + 30x − 34 = 0

2x2 + 15x − 17 = 0 (dividing by 2)

The equation 2x2 + 15x − 17 = 0 models the scenario.

2. a quadratic equation

3.2x2 + 15x – 17 = 0

The coefficients of the equation are a = 2, b = 15, and c = -17.

Factor and solve for x:

2x^2+15x-17=0

(2x+17)(x-1)=0

x= -17/2 or x= -1

Width cannot be a negative value, so the width of the frame is 1 inch.

4.The length of a rectangle is twice its width. If the length is increased by 15 units, the area of the resulting rectangle is 17 square units. If the width of the original rectangle is x, find the length and width of the original rectangle.

These are all answers from Edmentum

Answer 2

1) Equation to model the area of frame is
`34= (8+2x)* (7+2x)`

2) The equation is quadratic.

3) The width of picture fame is
`(-15\pm √(137))/(4)`

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* width`

`34= (8+2x)* (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* 11) } }{2* 2}`

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

Opening parenthesis.

=
`(-15\pm √(225-88 ) )/(4)`

=
`(-15\pm √(137))/(4)`

∴ x=
`(-15\pm √(137))/(4)`

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

2) The equation is quadratic.

3) The width of picture fame is
`(-15\pm √(137))/(4)`