Problem 30 - Algebra 2 and Trigonometry

Question

The area of a rectangle is 25\(\sqrt{35}\) square feet and the width is 10\(\sqrt{5}\) feet. Find the length of the rectangle in simplest radical form.

Step-by-Step Solution

Answer

The length of the rectangle is \( \frac{5\sqrt{7}}{2} \) feet.

Step 1: Understand the formula

The formula for the area of a rectangle is given by \( \text{Area} = \text{Length} \times \text{Width} \). Here, the area of the rectangle is 25\( \sqrt{35} \) square feet and the width is 10\( \sqrt{5} \) feet.

Step 2: Set up the equation

With the known values, our equation becomes \( 25\sqrt{35} = \text{Length} \times 10\sqrt{5} \).

Step 3: Isolate the length

To find the length, divide both sides of the equation by the width (10\( \sqrt{5} \)): \( \text{Length} = \frac{25\sqrt{35}}{10\sqrt{5}} \).

Step 4: Simplify the expression

Simplify the fraction \( \frac{25\sqrt{35}}{10\sqrt{5}} \):- Divide \( 25 \) by \( 10 \) to get \( 2.5 \).- Simplify the square roots: divide \( \sqrt{35} \) by \( \sqrt{5} \) to get \( \sqrt{7} \) because \( \frac{\sqrt{35}}{\sqrt{5}} = \sqrt{\frac{35}{5}} = \sqrt{7} \).- Thus, \( \text{Length} = 2.5 \times \sqrt{7} \).

Step 5: Express in simplest radical form

Multiply 2.5 by \( \sqrt{7} \) to express the length in simplest radical form: \( \text{Length} = \frac{5}{2} \times \sqrt{7} \). This simplifies further to \( \frac{5\sqrt{7}}{2} \).

Key Concepts

Rectangle Area

The area of a rectangle is calculated by multiplying its length by its width. This basic formula is very important, especially when dealing with expressions involving radicals. In our problem, the area is given as \( 25\sqrt{35} \) square feet.
Understanding that the area represents the total size of the rectangle's surface helps us set up equations to solve for any missing dimensions. In our case, knowing the area allows us to determine the length of the rectangle from the width. Remember, every time numbers involve radicals, you have to follow the rules of radical operations which might include simplification.

Simplifying Radicals

Simplifying radicals is about rewriting a radical so that no perfect square factors remain under the square root. It's an essential skill when working with radical expressions. In this exercise, we simplified \( \frac{25\sqrt{35}}{10\sqrt{5}} \) by:

• Dividing the coefficients (numbers in front of the radicals), which gave us \( 2.5 \).
• Simplifying the radicals \( \sqrt{35} \) and \( \sqrt{5} \) by dividing them. This works because \( \frac{\sqrt{35}}{\sqrt{5}} = \sqrt{7} \).

Finally, multiplying \( 2.5 \) by \( \sqrt{7} \) and converting it into a rational expression in simplest form gives \( \frac{5\sqrt{7}}{2} \). Notice how simplifying makes the expression cleaner and easier to work with.

Rectangle Dimensions

The dimensions of a rectangle are the measurements of its length and width. These are essential parameters that help define the rectangle's shape and size. When one or both dimensions involve radicals, extra steps are needed to manage these expressions properly.
In this exercise, the width was given as \( 10\sqrt{5} \) feet, and the task was to find the length. By using the formula for the area and simplifying the radicals, we managed to deduce that the length is \( \frac{5\sqrt{7}}{2} \) feet.
Understanding how to manipulate and simplify radical expressions is crucial for accurately determining a rectangle's dimensions when dealing with such problems. This makes it easier to visualize and utilize the rectangle in real-world applications.