Question

if the height of a triangle is decreased by 40% and its base is increased by 40% the effect on its area is:
no change
8% decrease
16%increase
16% decrease

Answer 1

The area of a triangle is given by, $A=\frac{1}{2}bh$,
The change in area (if there is any) is given by, $(1+\frac{\delta A}{A})=(1+\frac{\delta b}{b})(1+\frac{\delta h}{h})$.
Here, $\frac{\delta b}{b}=+0.4$ and $\frac{\delta h}{h}=-0.4$.
Hence, $\frac{\delta A}{A}=-0.16$.
Therefore, Area decreases by 16%.

Answer 2

let initial base=b,
            height=h
area, A₁ = 0.5 bh

after height is decreased by 40% and base is increased by 40%,
base = 1.4b
height = 0.6h
Area, A₂ = 0.5×1.4b×0.6h = 0.42 bh

A₂ - A₁ = 0.42 bh - 0.5 bh = -0.08 bh
-ve sign indicates that area decreased.

% decrease = (0.08bh/0.5bh)×100 = 16%