To solve the problem, let's break it down step-by-step:
1. Initial Setup:
- Assume the initial area of the triangle is 1 (for simplicity).
- Let the initial height of the triangle be [tex]\( h \)[/tex] and the initial base be [tex]\( b \)[/tex].
2. Initial Area:
- The area of a triangle is given by [tex]\( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)[/tex].
- So, [tex]\( 1 = \frac{1}{2} \times b \times h \)[/tex].
3. Increase in Height:
- The height is increased by 40%.
- New height [tex]\( h_{\text{new}} = h \times (1 + \frac{40}{100}) = 1.4h \)[/tex].
4. New Area with Original Base:
- Keeping the base [tex]\( b \)[/tex] the same, the new area with the increased height is [tex]\( \text{Area}_{\text{new}} = \frac{1}{2} \times b \times 1.4h \)[/tex].
- Substituting the original area relationship [tex]\( 1 = \frac{1}{2} \times b \times h \)[/tex]:
[tex]\[ 1.4 \times \frac{1}{2} \times b \times h = 1.4 \times 1 = 1.4 \][/tex].
- So the new area with the increased height and same base is 1.4.
5. Maximum Allowed Area:
- The maximum allowed increase in area is 60%.
- Maximum area allowed [tex]\( \text{Area}_{\text{max}} = 1 \times (1 + \frac{60}{100}) = 1.6 \)[/tex].
6. Determine New Base for Maximum Allowed Area:
- To find the proportional increase in the base that keeps the area within the maximum allowed increase:
[tex]\[ 1.6 = \frac{1}{2} \times b_{\text{new}} \times 1.4h \][/tex].
- We know [tex]\( 1 = \frac{1}{2} \times b \times h \)[/tex], hence [tex]\( b_{\text{new}} \times 1.4 \times \frac{1}{2} \times h = 1.6 \times \frac{1}{2} \times b \times h \)[/tex].
- Simplifying the equation:
[tex]\[ b_{\text{new}} \times 1.4 = 1.6 \times b \][/tex].
- Solving for the new base [tex]\( b_{\text{new}} \)[/tex]:
[tex]\[ b_{\text{new}} = \frac{1.6 \times b}{1.4} = \frac{1.6}{1.4} \times b = \frac{8}{7} \times b \][/tex].
7. Calculate the Percentage Increase in the Base:
- The increase in the base is [tex]\( \frac{8}{7} \times b \)[/tex].
- The percentage increase in the base is:
[tex]\[ \left( \frac{\frac{8}{7} \times b - b}{b} \right) \times 100 \% = \left( \frac{8}{7} - 1 \right) \times 100 \% = \left( \frac{1}{7} \right) \times 100 \% = 14.28 \% \][/tex].
So, the maximum percentage increase in the length of the base to restrict the area increase to a maximum of 60% is:
c) 14.28%
