Problem 49 - Algebra and Trigonometry

Question

The perimeter of a rectangle is 70 and its diagonal is \(25 .\) Find its length and width.

Step-by-Step Solution

Answer

The dimensions of the rectangle are 20 and 15.

Step 1: Understand the problem

We need to find the length and the width of a rectangle. We know that its perimeter is 70, and the diagonal is 25. The perimeter formula for a rectangle is given by \(P = 2(l + w)\). The diagonal formula, by the Pythagorean theorem, is \(d = \sqrt{l^2 + w^2}\).

Step 2: Set up the perimeter equation

Using the perimeter formula, we have: \(2(l + w) = 70\). Simplifying this gives us \(l + w = 35\).

Step 3: Set up the diagonal equation

Using the diagonal formula, we write: \(\sqrt{l^2 + w^2} = 25\). Squaring both sides gives \(l^2 + w^2 = 625\).

Step 4: Express one variable in terms of the other

Using the equation from Step 2, express \(w\) in terms of \(l\): \(w = 35 - l\).

Step 5: Substitute into the diagonal equation

Substitute \(w = 35 - l\) into the diagonal equation \(l^2 + w^2 = 625\):\[ l^2 + (35 - l)^2 = 625 \]

Step 6: Expand and simplify the equation

Expand \((35 - l)^2\) to get:\[ l^2 + (35^2 - 2\cdot 35 \cdot l + l^2) = 625 \]This simplifies to:\[ 2l^2 - 70l + 1225 = 625 \]

Step 7: Rearrange into a quadratic equation

Bring all terms to one side:\[ 2l^2 - 70l + 600 = 0 \]Simplify by dividing the entire equation by 2:\[ l^2 - 35l + 300 = 0 \]

Step 8: Use the quadratic formula to solve for l

Use the quadratic formula \(l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a = 1\), \(b = -35\), \(c = 300\):\[ l = \frac{35 \pm \sqrt{35^2 - 4 \cdot 1 \cdot 300}}{2} \]Calculate inside the square root:\(35^2 - 1200 = 25\).Hence \(l = \frac{35 \pm 5}{2}\) gives solutions \(20\) or \(15\).

Step 9: Determine both dimensions

If \(l = 20\), then \(w = 35 - 20 = 15\). If \(l = 15\), then \(w = 35 - 15 = 20\). So the dimensions are either length 20 and width 15 or length 15 and width 20.

Key Concepts

Rectangle Dimensions

When dealing with rectangles, dimensions refer to the length and width, which are the two sides of the rectangle. These dimensions are important because they determine other properties of the rectangle, such as the perimeter and area. The perimeter is the total distance around the rectangle, calculated by adding up the lengths of all four sides. The formula for the perimeter (P) of a rectangle is:

• \( P = 2(l + w) \)

where \( l \) is the length and \( w \) is the width.To solve rectangle problems, it's helpful to know some given values, such as the perimeter, and use these in the formulas to compute unknown dimensions. Knowing either the perimeter or one of the dimensions allows you to find the missing values easily.

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of a quadratic equation. Quadratic equations are in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants.Here's the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Using this formula, you can find two possible solutions for \( x \), which could represent important values such as the length or width in a rectangle problem.The part under the square root, \( b^2 - 4ac \), is known as the discriminant. It helps us determine the nature of the roots:

• Positive discriminant: Two real and distinct solutions
• Zero discriminant: One real solution (repeated)
• Negative discriminant: No real solutions (complex numbers)

Understanding and applying the quadratic formula is crucial in solving problems where expressions are not linearly related.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle used to relate the sides of right triangles. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.The theorem is represented by the equation:

• \( a^2 + b^2 = c^2 \)

where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.In the context of a rectangle, if the width and length form the two shorter sides of a right triangle and the diagonal serves as the hypotenuse, we can apply the Pythagorean theorem to solve for unknown dimensions. For rectangles, knowing just the diagonal and either the length or width, allows you to find the missing dimension using this theorem.

Diagonal of a Rectangle

The diagonal of a rectangle connects opposite corners and divides the rectangle into two congruent right triangles. It's an important feature because Diagonals in geometry often help to reveal properties of shapes that aren't immediately obvious.Calculating the length of the diagonal involves using the Pythagorean theorem, as each diagonal forms the hypotenuse of a right triangle:

• \( d = \sqrt{l^2 + w^2} \)

where \( l \) is the length, \( w \) is the width, and \( d \) is the diagonal.Knowing the diagonal is useful when trying to determine the dimensions of a rectangle when some information is missing. For instance, if you have the diagonal and the perimeter, you can solve for the other dimensions effectively, as each piece of information complements the other.