Final Answer:
The length of the rectangle is 19 miles and the width is 13 miles.
Step-by-step explanation:
Let x be the width of the rectangle.
According to the problem, the length of the rectangle is 6 miles longer than the width, so the length can be represented as x+6.
The area of the rectangle is given as 247 square miles. The formula for the area of a rectangle is length × width, so we can set up an equation to represent this information:
(x+6) × x = 247
Expanding the left side of the equation, we get:
x² + 6x = 247
Simplifying the equation, we get:
x² + 6x - 247 = 0
Factoring the equation, we get:
(x - 13)(x + 19) = 0
Since the product of two numbers is equal to zero only if one or both of the numbers is equal to zero, we can set each factor to zero and solve for x:
x - 13 = 0 or x + 19 = 0
Solving for x in the first equation, we get:
x = 13
Solving for x in the second equation, we get:
x = -19
Since the width of a rectangle cannot be negative, we disregard the solution x = -19. Therefore, the width of the rectangle is 13 miles.
Substituting x = 13 into the expression x+6 for the length, we get:
13 + 6 = 19
Therefore, the length of the rectangle is 19 miles.
In conclusion, the dimensions of the rectangle are 19 miles by 13 miles.