Certainly! Let's solve this step-by-step.
1. Define the variables:
Let [tex]\( x \)[/tex] be the length of the shorter base of the trapezoid.
2. Express the length of the longer base in terms of [tex]\( x \)[/tex]:
According to the problem, the length of the longer base is [tex]\( 3x - 12 \)[/tex].
3. Use the formula for the area of a trapezoid:
The area of a trapezoid is given by:
[tex]\[
\text{Area} = \frac{1}{2} \times (\text{Base 1} + \text{Base 2}) \times \text{Height}
\][/tex]
For our trapezoid, this becomes:
[tex]\[
100 = \frac{1}{2} \times (x + (3x - 12)) \times 10
\][/tex]
4. Simplify the equation:
Combine the bases:
[tex]\[
100 = \frac{1}{2} \times (4x - 12) \times 10
\][/tex]
Multiply both sides by 2 to get rid of the fraction:
[tex]\[
200 = (4x - 12) \times 10
\][/tex]
Divide both sides by 10:
[tex]\[
20 = 4x - 12
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
Add 12 to both sides:
[tex]\[
32 = 4x
\][/tex]
Divide both sides by 4:
[tex]\[
x = 8
\][/tex]
6. Conclusion:
The length of the shorter base is [tex]\( 8 \)[/tex] feet.
