To find the lengths of the two bases of the trapezoidal banner, we can start by using the formula for the area of a trapezoid:
\[ A = \frac{1}{2}(b_1 + b_2)h \]
We are given the area \( A = 120 , \text{m}^2 \) and the height \( h = 10 , \text{m} \). We can plug these values into the formula:
\[ 120 = \frac{1}{2}(b_1 + b_2) \cdot 10 \]
This simplifies to:
\[ 120 = 5(b_1 + b_2) \]
Now, divide both sides by 5:
\[ b_1 + b_2 = 24 \quad (1) \]
Next, we know that base 1 \( b_1 \) is 16 m shorter than base 2 \( b_2 \). We can express this relationship as:
\[ b_1 = b_2 - 16 \quad (2) \]
Now, we can substitute equation (2) into equation (1):
\[ (b_2 - 16) + b_2 = 24 \]
Combine like terms:
\[ 2b_2 - 16 = 24 \]
Add 16 to both sides:
\[ 2b_2 = 40 \]
Now, divide by 2:
\[ b_2 = 20 \quad (base , 2) \]
Now, we can find \( b_1 \) using equation (2):
\[ b_1 = b_2 - 16 = 20 - 16 = 4 \quad (base , 1) \]
Thus, the lengths of the two bases are:
- Base 1 (\( b_1 \)) = 4 m
- Base 2 (\( b_2 \)) = 20 m