To explore how increasing both the base and height of a triangle by 5% affects its area, let's start by reviewing the formula for the area of a triangle:

$\text{Area of a triangle} = \frac{1}{2} \times \text{base} \times \text{height}$

Step 1: Define the original area

Let the original base of the triangle be $b$ and the original height be $h$. The original area $A_{\text{original}}$ is:

$A_{\text{original}} = \frac{1}{2} \times b \times h$

Step 2: Apply a 5% increase to both the base and height

If we increase both the base and the height by 5%, the new base $b_{\text{new}}$ and new height $h_{\text{new}}$ will be:

$b_{\text{new}} = b \times (1 + 0.05) = 1.05b$ $h_{\text{new}} = h \times (1 + 0.05) = 1.05h$

Step 3: Calculate the new area

The new area $A_{\text{new}}$ after the increase is:

$A_{\text{new}} = \frac{1}{2} \times b_{\text{new}} \times h_{\text{new}} = \frac{1}{2} \times (1.05b) \times (1.05h)$

Simplifying this expression:

$A_{\text{new}} = \frac{1}{2} \times 1.05^2 \times b \times h = \frac{1}{2} \times 1.1025 \times b \times h$

$A_{\text{new}} = 1.1025 \times A_{\text{original}}$

Step 4: Analyze the percentage increase in the area

The new area is 1.1025 times the original area, meaning the area increases by 10.25%, not by 5%. This shows that increasing both the base and height of a triangle by 5% results in a 10.25% increase in the area, which is more than 5%.

Conclusion:

The increase in the area is not 5%, but rather 10.25%. This happens because both the base and the height are being increased, and the area depends on the product of these two dimensions. When both dimensions are scaled by 5%, their combined effect leads to a compounded increase in the area.

Key Takeaway:

Increasing both the base and height by a percentage leads to a larger increase in the area than the original percentage due to the multiplicative relationship between the base and height in the area formula.

Would you like more details on any of these steps or further explanations?

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Here are 5 related questions to expand on this topic:

1. How would the area change if only the base or only the height were increased by 5%?
2. What if the base and height were increased by different percentages (e.g., 5% for the base and 10% for the height)?
3. How would a 5% decrease in both base and height affect the area?
4. Can you generalize the percentage increase in the area if the base and height are both increased by any arbitrary percentage $p \%$?
5. How does this concept apply to other shapes, like rectangles or parallelograms?

Tip: When solving geometric problems involving area, pay attention to the fact that changing two related dimensions can lead to compounded effects, not just linear increases.