A rectangle has a width of 9 units and a length of 40 units. What is the length of a diagonal?
The problem asks for the length of the diagonal of a rectangle given its width and length. Let the width be \(w = 9\) units and the length be \(l = 40\) units. The diagonal \(d\) of a rectangle splits the rectangle into two congruent right-angled triangles, where the width and length act as the legs of the triangles and the diagonal is the hypotenuse.
To find the length of the diagonal, we use the Pythagorean theorem, which states that for any right-angled triangle with legs \(a\) and \(b\) and hypotenuse \(c\), the relationship is \(a^2 + b^2 = c^2\). In this case, \(a = 9\) and \(b = 40\), so the diagonal \(d\) satisfies the equation:
\[d^2 = 9^2 + 40^2\]
First, calculate the squares of the sides:
\(9^2 = 81\)
\(40^2 = 1600\)
Next, sum these values to find the square of the diagonal:
\(d^2 = 81 + 1600 = 1681\)
Finally, find the length of the diagonal by taking the square root:
\(d = \sqrt{1681}\)
Since \(41 \times 41 = 1681\), the length of the diagonal is \(d = 41\).
Comparing our result \(d = 41\) units with the given options:
The correct option is c. Therefore, the length of the diagonal is \(\boxed{41}\) units.
In real-life scenarios, calculating the diagonal of a rectangle is useful for tasks such as determining the size of a screen. For example, if a television screen has a width of 9 inches and a length of 40 inches, calculating the diagonal helps you understand its size designation (e.g., a "41-inch" TV). This application extends to carpentry, where measuring diagonals ensures that a wooden frame is perfectly square.