Question

Make a sketch and write a quadratic equation to model the situation. Then solve the equation. A rectangle is \(2 x\) feet long and \(x+5\) feet wide. The area is 600 square feet. What are the dimensions of the rectangle?

Short answer

The dimensions of the rectangle are 40 feet (length) and 25 feet (width).

Step-by-step solution

Write the equation

Write down the equation from the problem: \(2x(x + 5) = 600\)

Simplify the equation

Expand the equation to get \(2x^2 + 10x - 600 = 0\). Now the equation is in the standard form of a quadratic equation \(ax^2 + bx + c = 0\)

Solve the quadratic equation

Use the quadratic formula \[x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}\] to find the solutions. After substituting a=2, b=10 and c=-600, we get the roots as \[x = \frac{-10 ± \sqrt{(10)^2 - 4*2*(-600)}}{2*2}\]

Find the value of x

Solving the equation we get two roots. One is \(x = 20\) and the other is \(x = -30\). But a dimension cannot be negative, so the valid value for \(x\) is 20.

Substitute x in dimensions

Substitute \(x = 20\) back into the expressions for the length and width to find the dimensions: The length is \(2x = 2*20 = 40\) feet and the width is \(x + 5 = 20 + 5 = 25\) feet.