Answer to Question #313826 in Analytic Geometry for Vick

Question

A trapezoid has an area of 36 sq. m and an altitude of 2m. Its two bases have ratio of 4:5. Find the perimeter of the trapezoid. Find the percentage increase in the volume if its edge is increased by 1%

Answer 1

$S=36m^2\\ BK=2m\\ BC:AD=4:5$

BK, CM - an altitude

Lets

$BC=4x, AD=5x\\ S=\frac{BC+AD}{2}\cdot BK\\ S=\frac{4x+5x}{2}\cdot2=36\\ 9x=36\\ x=4\\ BC=16m, AD=20m$

$AK=MD=\frac{AD-BC}{2}=\frac{20-16}{2}=2(m)\\$

$\triangle ABK, \angle K=90^0\\ AB^2=AK^2+BK^2\\ AB^2=2^2+2^2=4+4=8\\ AB=\sqrt{8}=2\sqrt{2}(m)\\ AB=CD=2\sqrt{2}m\\ P=AB+BC+CD+AD=\\ =2\sqrt{2}+16+2\sqrt{2}+20=36+4\sqrt{2}(m)$

If AB and CD is increased by 1%=0.01:

$AB'=1.01(2\sqrt{2})=2.02\sqrt{2}(m)\\ P'=16+20+2.02\sqrt{2}+2.02\sqrt{2}=36+4.04\sqrt{2}(m)\\ \frac{P'\cdot100\%}{P}=\frac{(36+4.04\sqrt{2})\cdot100\%}{36+4\sqrt{2}} \approx100.14\%$