To solve this problem, let's go through the steps to understand how a 30% increase in the base and a 30% decrease in the height affect the area of a triangle.
1. Understand the Initial Conditions:
- Assume the initial base of the triangle [tex]\(b_0\)[/tex] is 100 units and the initial height [tex]\(h_0\)[/tex] is 100 units. This simplifies calculations and understanding.
2. Calculate the Initial Area:
- The formula for the area of a triangle is [tex]\(\frac{1}{2} \times \text{base} \times \text{height}\)[/tex].
- Substituting the initial values:
[tex]\[
\text{Initial Area} = \frac{1}{2} \times 100 \times 100 = 5000 \text{ square units}
\][/tex]
3. Find the New Base and Height:
- The base is increased by 30%, so the new base [tex]\( b_1 \)[/tex] is:
[tex]\[
b_1 = 100 + (0.30 \times 100) = 100 + 30 = 130 \text{ units}
\][/tex]
- The height is decreased by 30%, so the new height [tex]\( h_1 \)[/tex] is:
[tex]\[
h_1 = 100 - (0.30 \times 100) = 100 - 30 = 70 \text{ units}
\][/tex]
4. Calculate the New Area:
- Using the new base and height:
[tex]\[
\text{New Area} = \frac{1}{2} \times 130 \times 70 = \frac{1}{2} \times 9100 = 4550 \text{ square units}
\][/tex]
5. Determine the Percentage Change in Area:
- The percentage change in area is calculated as:
[tex]\[
\text{Percentage Change} = \left( \frac{\text{New Area} - \text{Initial Area}}{\text{Initial Area}} \right) \times 100
\][/tex]
- Substituting the areas:
[tex]\[
\text{Percentage Change} = \left( \frac{4550 - 5000}{5000} \right) \times 100 = \left( \frac{-450}{5000} \right) \times 100 = -9\%
\][/tex]
Therefore, the area of the triangle decreases by 9%.
