Let's solve the problem step-by-step.
1. Define the variables:
- Let's denote the length of the shorter side of the rectangle as [tex]\( x \)[/tex] meters.
- The length of the longer side is given as 16 meters longer than four times the shorter side.
2. Express the longer side in terms of the shorter side:
- The longer side of the rectangle can be written as [tex]\( 4x + 16 \)[/tex] meters.
3. Write the equation for the area of the rectangle:
- The area of a rectangle is given by the product of its length and width. So, the area can be written as:
[tex]\[
x \times (4x + 16) = 128
\][/tex]
- To simplify, distribute [tex]\( x \)[/tex] on the left side:
[tex]\[
4x^2 + 16x = 128
\][/tex]
4. Rearrange the equation to standard quadratic form:
- Bring all terms to one side to set the equation to zero:
[tex]\[
4x^2 + 16x - 128 = 0
\][/tex]
5. Solve the quadratic equation:
- We can solve for [tex]\( x \)[/tex] using the quadratic formula where [tex]\( ax^2 + bx + c = 0 \)[/tex]:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
- Here, [tex]\( a = 4 \)[/tex], [tex]\( b = 16 \)[/tex], and [tex]\( c = -128 \)[/tex].
6. Calculate the discriminant [tex]\( \Delta \)[/tex]:
[tex]\[
\Delta = b^2 - 4ac = 16^2 - 4 \cdot 4 \cdot (-128) = 256 + 2048 = 2304
\][/tex]
7. Find the roots using the quadratic formula:
[tex]\[
x = \frac{-16 \pm \sqrt{2304}}{2 \cdot 4}
\][/tex]
[tex]\[
x = \frac{-16 \pm 48}{8}
\][/tex]
- This gives us two solutions:
1. [tex]\( x = \frac{-16 + 48}{8} = \frac{32}{8} = 4 \)[/tex]
2. [tex]\( x = \frac{-16 - 48}{8} = \frac{-64}{8} = -8 \)[/tex]
8. Select the valid solution:
- Since lengths cannot be negative, we discard [tex]\( x = -8 \)[/tex].
Therefore, the length of the shorter side of the rectangle is [tex]\( 4 \)[/tex] meters.
