A restaurant uses rectangular napkins where the length, [tex]$l$[/tex], is twice as long as the width. The length of the napkin along the diagonal is [tex]$x$[/tex]. What is [tex]$x$[/tex] in terms of [tex]$l$[/tex]?
Answer (1)
The problem involves finding the diagonal of a rectangle given its length and the relationship between its length and width.
Use the Pythagorean theorem to relate the length, width, and diagonal: x 2 = l 2 + w 2 .
Substitute w = 2 l into the Pythagorean theorem: x 2 = l 2 + ( 2 l ) 2 .
Simplify the equation: x 2 = 4 5 l 2 .
Solve for x : x = 2 5 l .
x = 2 5 l
Explanation
Problem Analysis Let's analyze the problem. We have a rectangular napkin with length l and width w . We are given that the length is twice the width, which means l = 2 w . The diagonal of the napkin is x . We need to find x in terms of l . We can use the Pythagorean theorem to relate the length, width, and diagonal of the rectangle.
Applying the Pythagorean Theorem The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In this case, the diagonal x is the hypotenuse, and the length l and width w are the other two sides. So, we have: x 2 = l 2 + w 2
Expressing Width in Terms of Length Since l = 2 w , we can express the width in terms of the length as w = 2 l . Now, substitute this into the Pythagorean theorem: x 2 = l 2 + ( 2 l ) 2
Simplifying the Equation Simplify the equation: x 2 = l 2 + 4 l 2 x 2 = 4 4 l 2 + 4 l 2 x 2 = 4 5 l 2
Solving for x Now, solve for x by taking the square root of both sides: x = 4 5 l 2 x = 2 5 l
Final Answer So, x = 2 5 l . Comparing this with the required format x = b a l , we have a = 5 and b = 2 .
Examples
Understanding the relationship between the sides and diagonal of a rectangle is useful in various real-life scenarios. For example, when designing a rectangular garden, you might know the length and want to determine the length of the diagonal to plan the layout of a path that cuts across the garden. Similarly, in construction, knowing the diagonal of a rectangular structure is crucial for ensuring structural integrity and proper alignment. This principle is also applied in sports, such as determining the distance a player needs to run diagonally across a rectangular field.
