Answer: Let's assume the width of the picture is "x" inches.
According to the problem, the length of the picture is 6 inches longer than its width, so the length is "x + 6" inches.
The formula for the area of a rectangle is:
Area = Length × Width
We are given that the area is 167 square inches, so we can set up the equation:
167 = (x + 6) × x
Now, let's solve for "x":
167 = x^2 + 6x
Move all terms to one side to form a quadratic equation:
x^2 + 6x - 167 = 0
Now, we can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
where a = 1, b = 6, and c = -167.
x = (-(6) ± √(6^2 - 4 × 1 × (-167))) / 2 × 1
x = (-6 ± √(36 + 668)) / 2
x = (-6 ± √704) / 2
Now, calculate the two possible values for "x":
x₁ = (-6 + √704) / 2 ≈ 10.39 (rounded to two decimal places)
x₂ = (-6 - √704) / 2 ≈ -16.39 (rounded to two decimal places)
Since dimensions cannot be negative, we discard the negative value. Therefore, the width of the picture is approximately 10.39 inches.
Now, find the length using the formula for the length:
Length = Width + 6 = 10.39 + 6 ≈ 16.39 (rounded to two decimal places)
So, the dimensions of the picture are approximately 10.39 inches (width) and 16.39 inches (length).