Question

In Exercises 54 and \(55,\) a football field's dimensions can be represented by a width of \(\left(\frac{1}{2} x+10\right)\) feet and a length of \(\left(\frac{5}{4} x-15\right)\) feet. Find an expression for the area \(A\) of a football field. Give your answer as a quadratic trinomial.

Short answer

The expression for the area of the football field, given as a quadratic trinomial, is \(Area = \frac{5}{8}x^2 + \frac{1}{4}x -150\).

Step-by-step solution

Understanding the Area Formula

The formula for the area of a rectangle is Area = Width x Length.

Substitute Given Width and Length

Here, you substitute \(\left(\frac{1}{2} x+10\right)\) for width and \(\left(\frac{5}{4} x-15\right)\) for length in the formula. So, the formula becomes Area = \(\left(\frac{1}{2} x+10\right)\) x \(\left(\frac{5}{4} x-15\right)\).

Expand the Equation

To expand the equation, you follow the rules of multiplication, distributing terms, and simplifying. The resulting equation should be in the form of a trinomial, which is a quadratic equation of the form ax^2+bx+c.

Simplify the Equation

After simplification, the equation becomes \(Area = \frac{5}{8}x^2 + \frac{1}{4}x -150\).