Answer :
To determine the percentage change in the area of a triangle when its height is decreased by 40% and its base is increased by 40%, we can follow these steps:
1. Calculate the original area of the triangle:
- Let the original height of the triangle be [tex]\( h \)[/tex].
- Let the original base of the triangle be [tex]\( b \)[/tex].
- The area of a triangle is given by [tex]\(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)[/tex].
So, the original area, [tex]\( A_{\text{original}} \)[/tex]:
[tex]\[ A_{\text{original}} = \frac{1}{2} \times b \times h \][/tex]
2. Determine the new height and new base:
- The height is decreased by 40%. Hence, the new height is [tex]\( 60\% \)[/tex] of the original height [tex]\( h \)[/tex]:
[tex]\[ \text{New height} = 0.6h \][/tex]
- The base is increased by 40%. Hence, the new base is [tex]\( 140\% \)[/tex] of the original base [tex]\( b \)[/tex]:
[tex]\[ \text{New base} = 1.4b \][/tex]
3. Calculate the new area of the triangle:
- With the new base and new height, the area [tex]\( A_{\text{new}} \)[/tex] becomes:
[tex]\[ A_{\text{new}} = \frac{1}{2} \times \text{New base} \times \text{New height} \][/tex]
Substituting the new base and height:
[tex]\[ A_{\text{new}} = \frac{1}{2} \times 1.4b \times 0.6h \][/tex]
4. Simplify to find the relation between the new and original areas:
- Simplify the expression for the new area:
[tex]\[ A_{\text{new}} = \frac{1}{2} \times 1.4 \times 0.6 \times b \times h \][/tex]
[tex]\[ A_{\text{new}} = \frac{1}{2} \times 0.84 \times b \times h \][/tex]
[tex]\[ A_{\text{new}} = 0.84 \times \frac{1}{2} \times b \times h \][/tex]
[tex]\[ A_{\text{new}} = 0.84 \times A_{\text{original}} \][/tex]
5. Calculate the percentage change in area:
- The change in area is the difference between the new area and the original area:
[tex]\[ \text{Change in Area} = A_{\text{new}} - A_{\text{original}} \][/tex]
- The percentage change in the area can be calculated as follows:
[tex]\[ \text{Percentage Change} = \left(\frac{A_{\text{new}} - A_{\text{original}}}{A_{\text{original}}}\right) \times 100 \][/tex]
Substitute [tex]\( A_{\text{new}} = 0.84 \times A_{\text{original}} \)[/tex]:
[tex]\[ \text{Percentage Change} = \left(\frac{0.84 \times A_{\text{original}} - A_{\text{original}}}{A_{\text{original}}}\right) \times 100 \][/tex]
Simplify the expression:
[tex]\[ \text{Percentage Change} = \left(0.84 - 1\right) \times 100 \][/tex]
[tex]\[ \text{Percentage Change} = -0.16 \times 100 \][/tex]
[tex]\[ \text{Percentage Change} = -16\% \][/tex]
Therefore, the area of the triangle decreases by 16% when the height is decreased by 40% and the base is increased by 40%.
1. Calculate the original area of the triangle:
- Let the original height of the triangle be [tex]\( h \)[/tex].
- Let the original base of the triangle be [tex]\( b \)[/tex].
- The area of a triangle is given by [tex]\(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)[/tex].
So, the original area, [tex]\( A_{\text{original}} \)[/tex]:
[tex]\[ A_{\text{original}} = \frac{1}{2} \times b \times h \][/tex]
2. Determine the new height and new base:
- The height is decreased by 40%. Hence, the new height is [tex]\( 60\% \)[/tex] of the original height [tex]\( h \)[/tex]:
[tex]\[ \text{New height} = 0.6h \][/tex]
- The base is increased by 40%. Hence, the new base is [tex]\( 140\% \)[/tex] of the original base [tex]\( b \)[/tex]:
[tex]\[ \text{New base} = 1.4b \][/tex]
3. Calculate the new area of the triangle:
- With the new base and new height, the area [tex]\( A_{\text{new}} \)[/tex] becomes:
[tex]\[ A_{\text{new}} = \frac{1}{2} \times \text{New base} \times \text{New height} \][/tex]
Substituting the new base and height:
[tex]\[ A_{\text{new}} = \frac{1}{2} \times 1.4b \times 0.6h \][/tex]
4. Simplify to find the relation between the new and original areas:
- Simplify the expression for the new area:
[tex]\[ A_{\text{new}} = \frac{1}{2} \times 1.4 \times 0.6 \times b \times h \][/tex]
[tex]\[ A_{\text{new}} = \frac{1}{2} \times 0.84 \times b \times h \][/tex]
[tex]\[ A_{\text{new}} = 0.84 \times \frac{1}{2} \times b \times h \][/tex]
[tex]\[ A_{\text{new}} = 0.84 \times A_{\text{original}} \][/tex]
5. Calculate the percentage change in area:
- The change in area is the difference between the new area and the original area:
[tex]\[ \text{Change in Area} = A_{\text{new}} - A_{\text{original}} \][/tex]
- The percentage change in the area can be calculated as follows:
[tex]\[ \text{Percentage Change} = \left(\frac{A_{\text{new}} - A_{\text{original}}}{A_{\text{original}}}\right) \times 100 \][/tex]
Substitute [tex]\( A_{\text{new}} = 0.84 \times A_{\text{original}} \)[/tex]:
[tex]\[ \text{Percentage Change} = \left(\frac{0.84 \times A_{\text{original}} - A_{\text{original}}}{A_{\text{original}}}\right) \times 100 \][/tex]
Simplify the expression:
[tex]\[ \text{Percentage Change} = \left(0.84 - 1\right) \times 100 \][/tex]
[tex]\[ \text{Percentage Change} = -0.16 \times 100 \][/tex]
[tex]\[ \text{Percentage Change} = -16\% \][/tex]
Therefore, the area of the triangle decreases by 16% when the height is decreased by 40% and the base is increased by 40%.
