Question 1011966
14 an extension ;

ladder rests {{{4m}}} up a wall
If the ladder is extended a further {{{0.8}}} without moving the foot of the ladder, then it will now rest {{{1m}}} further up the wall

 How long is the extended ladder?

let {{{L}}} = original length of the ladder
{{{a}}} = the distance of the foot of the ladder from the wall

{{{L^2 = a^2 + 4^2}}}
{{{L^2 = a^2 + 16}}}
{{{a^2 = L^2 - 16}}} <--- eq. 1

{{{(L+0.8)^2 = a^2 + (4+1)^2}}}

subs {{{a^2}}} from eq. 1

{{{(L+0.8)^2 = L^2 - 16 + 5^2}}}

{{{cross(L^2) + 1.6L + 0.64 = cross(L^2) - 16 + 25}}}

{{{1.6L + 0.64 = 9}}}

{{{1.6L = 8.36}}}

{{{highlight(L = 5.23m)}}}

{{{L+0.8 =5.23m+0.8m =6.03m}}} <---- length of extended ladder



15) 

an equilateral triangle has area {{{A=16sqrt(3)cm^2}}}. 
Find the length of its sides.

{{{A = (1/2)bh}}}

{{{16*sqrt(3) = (1/2)bh}}}

{{{bh = 32sqrt(3)}}}.........eq.1

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Find altitude  {{{h}}} in terms of side {{{b}}}:

Since the triangle is equilateral, all sides are "{{{b}}}":
Draw an altitude from the vertex to the base.
You have a right triangle with base = {{{b/2}}} and hypotenuse = {{{b}}}
The altitude is the 3rd side:

{{{b^2 = (b/2)^2 + h^2}}}
{{{b^2-(b/2)^2 = h^2}}}

{{{b^2-b^2/4 = h^2}}}

{{{(3/4)b^2 = h^2}}}

{{{sqrt((3/4)b^2) = h}}}

{{{(sqrt(3)/2)b = h}}}


substitute in eq.1

{{{bh = 32sqrt(3)}}}

{{{b*(sqrt(3)/2)b = 32sqrt(3)}}}

{{{b^2= 32sqrt(3)/(sqrt(3)/2) }}}

{{{b^2= (2*32cross(sqrt(3)))/cross(sqrt(3)) }}}

{{{b^2= 64 }}}

{{{b=sqrt(64)}}}

{{{highlight(b=8)}}}->the length of each side

and now we can find the length of altitude too:

{{{(sqrt(3)/2)8 = h}}}

{{{4*sqrt(3)= h}}}


check the area:

{{{A = (1/2)bh}}}

{{{A = (1/2)8*4*sqrt(3)}}}

{{{A = (1/cross(2)1)cross(8)4*4*sqrt(3)}}}

{{{A = 16sqrt(3)}}} which confirms our solution