Question

geometry the area of a rectangle is $2x^2 - 11x + 15$ square feet. the length of the rectangle is $2x - 5$ feet. what is the width of the rectangle?
$\bigcirc$ $4x^3 - 12x^2 - 6x + 75$ ft
$\bigcirc$ $x - 3$ ft
$\bigcirc$ $2x^2 - 11x + 15$ ft
$\bigcirc$ $2x - 5$ ft

Explanation:

Step1: Recall the formula for the area of a rectangle

The area \( A \) of a rectangle is given by \( A = \text{length} \times \text{width} \). So, to find the width \( w \), we can rearrange the formula as \( w=\frac{A}{\text{length}} \).

Step2: Substitute the given area and length into the formula

We are given that the area \( A = 2x^{2}-11x + 15 \) and the length \( l=2x - 5 \). So we need to perform the division \( \frac{2x^{2}-11x + 15}{2x - 5} \).

We can factor the numerator or use polynomial long division. Let's try factoring the numerator. We need to find two numbers that multiply to \( 2\times15 = 30 \) and add up to \( - 11 \). The numbers are \( -6 \) and \( -5 \).

So we rewrite the middle term:
\( 2x^{2}-6x-5x + 15 \)

Now, group the terms:
\( (2x^{2}-6x)+(-5x + 15) \)

Factor out the common factors from each group:
\( 2x(x - 3)-5(x - 3) \)

Now, we can factor out \( (x - 3) \):
\( (2x - 5)(x - 3) \)

So the area \( 2x^{2}-11x + 15=(2x - 5)(x - 3) \)

Now, divide by the length \( 2x - 5 \):
\( \frac{(2x - 5)(x - 3)}{2x - 5} \)

Assuming \( 2x-5
eq0 \), we can cancel out \( 2x - 5 \) from the numerator and the denominator, and we get \( x - 3 \).

Answer:

\( x - 3 \) ft (which corresponds to the option "x - 3 ft")